W ELL -P OSEDNESS AND R EALIZ A TION T HEOR Y FOR D EL A Y
D IF FERENTIAL -A LG E B R A I C E QUA T I O N S
vorgelegt v on
M. Sc.
Benjamin U nger
OR CID: 0000-0003-4272-1079
an der F akultät II - Mathematik und N atur wissenschaften
der T echnischen U niversität Berlin
zur Erlangung des akademischen Gr ades
Doktor der N atur wissenschaften
- Dr . r er . nat. -
genehmigte Dissertation
Pr omotionsausschuss:
V orsitzender: Pr of. Dr . Martin H enk
Gutachter: Pr of. Dr . V u H oang Linh
Gutachter: Pr of. Dr . V o lker M ehrmann
Gutachter: Pr of. Dr . Wim Michiels
T ag der wissenschaftlichen A ussprache: 22.10.2020
Berlin 2020

ii

Abstract
This thesis is dedicated to delay differ ential-algebraic equations (DDAEs), i.e., constr aint dynamical
systems wher e the rate of change depends on the curr ent state and its past. T ypical applications
include
(i)
feedback control, wher e the delay is a direct consequence of the time r equ ir ed to measure the
curr ent state, compute the feedback, and implement the contr ol action,
(ii)
hybrid numerical-exper imental testing environments , used for instance in earthquake engi-
neering,
(iii)
transmission and pr opagation delays, encountered for example in ch emical r eactions con-
nected in series and wide-area po wer -networks, an d
(iv) as a mathematical tool to analyze hyperbolic equations and time-integr ation schemes.
The fact that algebraic equations can be included in the implicit system description fosters a r apid
model development since complex models can be assembled fr om a library of existing models with
well-defined interaction v ar iables .
F rom a mathematical point of view , DDAEs do not only feature difficulties alr eady kno wn from the
theor y of differ ential-algebraic equations (DAEs) and delay differ ential equations (DDEs) but pose
additional challenges . F or instance, initial tr ajector y problems for DDAEs may not be causal. Thus,
even in a distributional solution space , they may not have a solution for all initial tr ajector ies . This
fact, combined with the infinite-dimensional char acter of delay equations and the high sensitivity
to perturbation kno wn from the theor y of DAEs, r enders DDAEs a challenging mathematical object.
Consequently , the analysis of DDAEs is far from complete . The aim of this thesis is to addr ess some
of the many open problems .
I n the first par t of the thesis, initial tr ajector y problems for linear time-invariant (L TI) and nonlinear
DDAEs ar e discussed. W e start our analysis with a distr ibutional solution concept and establish
the existence and uniqueness of solutions, whenever the DDAE is delay -regular . J umps and Dirac-
iii

iv
impulses in the solution can be avoided if the coefficient matrices of the L TI DDAE satisfy some
algebraic conditions , which are obtained b y tracking so-called primary discontinuities. W e extend
some of the r esults to the nonlinear setting resulting in existence and uniqueness r esults for a large
class of nonlinear DDAEs.
The second part of the thesis is dedicated to constructing a DDAE solely from a prescribed set of data
points . Having a time-delay in the r ealization allo ws us to build an infinite-dimensional system from
finitely many points capable of r eproducing the transcendental char acter of the transfer function of
a distributed parameter subsystem that models convection or tr anspor t. W e construct a realization
so that it interpolates the data set in the fr equency domain and demonstrate its applicability with
several numerical examples .

Zusammenfassung
Diese Arbeit befasst sich mit zeitv e rzögerten Differential-A lgebraischen G lei chungen (DDAEs), das
heißt mit Differ entialgleichungen mit Z wangsbedingungen, bei denen die Änderungsrate so wohl
vom aktuellen Zustand als auch v on der V ergangenheit abhängt. T ypische Anwendungen umfassen
(i)
Rückkopplungssteuerung, wobei sich die V er zögerung aus der benötigten Zeit zur M essung
des aktuellen Zustands , Ber echnung der Rückkopplung und Implementierung selbiger ergibt,
(ii)
hybride numerisch-exper imentelle T estverfahr en, wie sie beispielsweise in der Er dbebenfor -
schung eingesetzt wer den,
(iii)
Übertragungs- und A usbreitungsverzögerungen, die beispielsweise bei in R eihe geschalteten
chemischer R eaktionen so wie bei großflächigen Str omnetzen auftreten und
(iv)
als mathematisches W erkzeug zur Analyse hyperbolischer Gleichungen und Z eitintegrations-
verfahr en.
Die T atsache , dass algebraische Gleichungen in das implizite S ystem aufgenommen wer den können,
begünstigt eine schnelle M odellentwicklung, da komplexe M odelle nach dem Baukastenprinzip
aus einer Bibliothek von M odellen mit genau definierten Inter aktionsvariablen zusammengesetzt
wer den können
A us mathematischer Sicht w eisen DD AEs nicht nur Schwierigkeiten auf, die bereits aus der Theo-
rie der Differential-A lgebraischen G leichungen (DAEs) und zeitverzögerten Differentialgleichungen
(DDEs) bekannt sind, sondern stellen zusätzliche H erausfor der ungen ber eit. Beispielsweise kön-
nen Anfangstrajektorienprobleme für DDAEs akausales V erhalten aufweisen. Dies führt dazu, dass
selbst bei linear en DDAEs mit einem distr ibutionellen Lösungskonzept nicht notwendigerwei-
se für alle Anfangstrajektorien eine Lösung existiert. Dieses Phänomen in K ombination mit dem
unendlich-dimensionalen Charakter von z eitver zögerten Differ entialgleichungen so wie der aus der
DAE-Theorie bekannten hohen Sensitivität gegenüber S törungen machen DDAEs zu einem heraus-
for der nden mathematischen Objekt. F olglich gibt es zahlreiche nicht gelöste F orschungsfragen im
v

vi
Zusammenhang mit DDAEs . Das Ziel dieser Arbeit ist es , einige dieser Fragen zu beantworten.
I m ersten T eil der Arbeit wer den Anfangstrajektorienprobleme für linear e zeitinvariante (L TI) und
nichtlinear e DDAEs diskutier t. Wir beginnen unser e Analyse mit einem distributionellen Lösungs-
konzept und beweisen E xistenz- und Eindeutigkeitsresultat für sogenannte delay-r egul är e DDAEs.
U m Sprünge , Dirac-I mpulse und Ableitungen von D irac-Impulsen in der Lösung ausschließen zu
können, müssen die K oeffizientenmatrizen der L TI DDAE bestimmte algebraische Bedingungen
erfüllen. Diese Bedingungen können dur ch eine N achverfolgung sogenannter pr imär er U nstetig-
keitsstellen hergeleitet wer den. T eilweise können die erhaltenen Ergebnisse auf nichtlinear e DD AEs
verallgemeinert wer den, was zu neuen Existenz- und Eindeutigkeitsr esultaten führt.
Der zweite T eil der Arbeit befasst sich mit der K onstr uktion einer DDAE aus einem vorgegebenen D a-
tensatz, einer sogenannten R ealisier ung. Dabei liefert das Zeitverzögerungsglied in der Realisierung
den V orteil, dass aus endlich vielen Datenpunkten ein unendlich-dimensionales S ystem konstr uiert
wer den kann. Dieses S ystem ist dann in der Lage, den tr anszendenten Charakter , der beispielsweise
bei T ransportgleichungen vorkommt, einer Ü ber tragungsfunktion abzubilden. U nser e K onstr uktion
basiert darauf, dass die gegebenen Daten im F requenzber eich interpoliert wer den. Die Effektivität
des V erfahrens wir d anhand zahlr eicher numer ischer Beispiele demonstriert.

A ckno wledgments
I would like to thank Pr of. V olker M ehrmann for his guidance, supervision, and the pro vided
fr eedom in the choice of my resear ch directions . Further , I am thankful to Prof. W im Michiels and
Pr of. V u H oang Linh for ser ving on my thesis committee. I also want to thank my co-authors I nes
Ahr es, Robert Altmann, Christopher Beattie, F elix B lack, E lliot F osong, Serkan G ugercin, Roland
Maier , Philipp Schulze , and S tephan T renn for many successful cooper ations. Besides, I thank
my colleagues Christian, Christoph, Daniel, F redi, M arine, M atthias, Mathieu, Lena, P aul, Phi,
Philipp , Riccardo , U te, and Viola for valuable hints , lively discussions, and friendship . I ackno wledge
funding from the Deutsche F orschungsgemeinschaft through the C ol laborativ e Resear ch Center
910 Control of self-organizing nonlinear systems: Theor etical methods and concepts of application
(project number 163436311) and the Berlin M athematical School (BMS). Finally , I want to thank my
wife and my par ents for their constant suppor t and unconditional lo ve.
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viii

Contents
List of Abbr eviations xi
List of F igures xiii
List of T ables xv
1 I ntroduction 1
1.1 M otivating examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 R eal-time dynamic substructuring . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.2 T ime-delayed feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 T ransmission and propagation delays . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 R efor mulation of hyperbolic problems as delay equations . . . . . . . . . . . . 7
1.1.5 Delay differ ential-algebraic equations as mathematical tool . . . . . . . . . . . 9
1.2 Scope and synopsis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 Challenges and state of the art . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Pr eviously published results and joint work . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 N otation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
I Classification and well-posedness 19
2 Differ ential-algebraic equations and pr eliminar y results 21
2.1 Classical solutions for linear time-invariant DAEs . . . . . . . . . . . . . . . . . . . . . 22
2.2 Distributional solutions for inconsistent initial values . . . . . . . . . . . . . . . . . . 28
2.3 S trangeness-index for nonlinear DAEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Distributional solutions for linear time-inv ar iant DDAEs 41
3.1 Distributional shift operator and delay -regularity . . . . . . . . . . . . . . . . . . . . . 41
3.2 I nterlude: F eedback r egularization of DAEs with delay . . . . . . . . . . . . . . . . . . 53
ix

x
3.3 Delay - equivalence and the compr ess- and-shift algorithm . . . . . . . . . . . . . . . . 55
4 Classical solutions and discontinuity propagation 67
4.1 Continuous solutions and classification . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 I mpact of splicing conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.3 Comparison to the existing classification . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5 N onlinear DDAEs 85
5.1 H ybr id numerical-exper imental system . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
5.2 The method of steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.3 Solv ability of the hybr id model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
II S tru ctur ed realization t heor y 101
6 Pr oblem setting and background 103
6.1 Pr oblem setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2 The Loewner framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7 S tr uctur ed interpolator y realizations 113
7.1 I nterpolation conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7.2 S tr uctur ed Loewner realizations for the case K = 2 . . . . . . . . . . . . . . . . . . . . 116
7.3 S tr uctur ed realization for the case K ≥ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.3.1 I nterpolation at additional points . . . . . . . . . . . . . . . . . . . . . . . . . . 119
7.3.2 Matching derivative data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
7.4 T runcation of r edundant data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7.5 An algorithm for structur ed realization . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
7.6 Connection to structure-pr eser ving interpolatory projections . . . . . . . . . . . . . . 131
7.6.1 The case K = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
7.6.2 The case K ≥ 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.7 N umerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8 F rom time-domain data to str uctur ed realizations 143
8.1 T ime-domain data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
8.2 I mplementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8.3 Estimation of parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
8.4 A case study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9 S ummar y and outlook 157
Bibliography 159

List of Abb reviations
BIBO bounded-input/bounded-output
CSTR continuous stirr ed-tank reactor
DAE differ ential-algebraic equation
DDAE delay differ ential-algebraic equation
DDE delay differ ential equation
ETFE empirical transfer function estimate
FFT fast F our ier transform
ITP initial trajectory problem
IVP initial v alue problem
lsTFE least-squar es transfer function estimate
L TI linear time-invariant
MBS multibody system
MIMO multiple-input/multiple-output
MOR model or der reduction
NDDE neutral delay differ ential equation
ODE or dinar y differ ential equation
PDE partial differential equation
RL C resistor -inductor-capacitor
R OM r educed order model
SISO single-input/single-output
SVD singular value decomposition
xi

xii List of Abbr eviations

List of Figures
1.1 Composition of a physical system by sever al subsystems . . . . . . . . . . . . . . . . . 1
1.2 R eal-time dynamic substructuring for a coupled system . . . . . . . . . . . . . . . . . 4
1.3 A simplified model for a two-dimensional gantr y crane . . . . . . . . . . . . . . . . . 6
1.4 A two-stage continuous stirred-tank r eactor sy stem . . . . . . . . . . . . . . . . . . . . 7
1.5 Acoustic tr ansmission in a fluid-filled duct . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Discontinuity propagation in DDAE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
5.1 Decomposition of a physical system into substructur es . . . . . . . . . . . . . . . . . 86
6.1 I nput-output mapping of a black-box system . . . . . . . . . . . . . . . . . . . . . . . 103
7.1 Example 7.26 – T ransfer functions of the differ ent r ealizations with n = 4 . . . . . . . 135
7.2 Example 7.28 – Bode and r elative error plot for n = 16 . . . . . . . . . . . . . . . . . . 137
7.3 Example 7.29 – Decay of the (nor malized) singular values . . . . . . . . . . . . . . . . 138
7.4 Example 7.29 – Relative err or for the different r ealizations . . . . . . . . . . . . . . . . 138
7.5 Example 7.30 – T ransfer functions of the differ ent r ealizations with n = 4 . . . . . . . 139
7.6 Example 7.31 – Entry -wise Bode plot of H , ˜︁
H L and ˜︁
H A with n = 32 . . . . . . . . . . . 140
7.7 Example 7.31 – T ransfer functions of the r ealizations with n = 32. . . . . . . . . . . . . 141
8.1 Estimation of the transfer function with least-squares . . . . . . . . . . . . . . . . . . 150
8.2 T ransfer function for str uctur ed time-domain r ealization . . . . . . . . . . . . . . . . 151
8.3 Output comparison for the original model and the structured r ealizations . . . . . . 152
8.4 S ampling of the least-squares error (8.14) o ver the delay time τ . . . . . . . . . . . . . 153
8.5 Eigenvalues of the r ealization with largest r eal part . . . . . . . . . . . . . . . . . . . . 155
xiii

xiv LIST OF FIGURES

List of T ables
6.1 Examples for system structur es with output mapping y ( t ) = C x ( t ) . . . . . . . . . . . 104
7.1 Example 7.26 – H ∞ errors of the differ ent realizations . . . . . . . . . . . . . . . . . . 134
7.2 Example 7.30 – C ondition numbers of the linear system and the decoupled systems 136
7.3 Example 7.30 – H ∞ errors for the differ ent realizations . . . . . . . . . . . . . . . . . . 140
7.4 Example 7.31 – N ear pole-zero cancellation for the lar gest unstable poles . . . . . . . 142
8.1 S imulation param eters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
8.2 I nterpolation data: least-squares estimates of the tr ansfer function . . . . . . . . . . 150
8.3 T est data: least-squares estimates of the transfer function . . . . . . . . . . . . . . . . 150
8.4 Error measur ements for the validation inputs with kno wn delay . . . . . . . . . . . . 152
8.5 Error measur ements for the validation inputs based on the estimated delay . . . . . 153
xv

xvi LIST OF T ABLES

1
Intro duction
Complex physical or chemical systems often comprise sever al subsystems that interact with each
other and the environment. F or instance, an electrical gr id is a network of po wer generators,
transmission lines , and consumers — each of which may again be a composition of subsystems.
Let us emphasize that also the inter action with the environment, for instance , via external forces
or dissipation of energy , might be r epresented as another (sub-)system that is inter connected with
the physical system; see Figure 1.1 for a schematic r e pr esentation where the inter action of the
subsystems is r epresented with arro ws. I n a simulation-driven environment, it is standar d to model
the physical system under investigation in terms of (partial) differ ential equations that describe
the evolution of the system. I nstead of deriving the equations of motions for the complete system
at once , a bottom-up approach models each subsystem separately and then connect the models
for the subsystems via suitable inter connections. An easy way to model such an inter connection is
given b y an algebraic equation, thus making the complete model a (partial) differ ential-algebraic
equation (DAE). Although, in principle, it may be possible to r esolve the algebraic equations and
hence r ewr ite the r esulting system as a (par tial) differ ential equation, it is a pr iori not clear , whether
this would be r easonable from a computational perspective or a modeling perspective . An example
S ubsystem 1
S ubsystem 2
S ubsystem 3
P HYSICAL S YSTEM

F igure 1.1 – Composition of a physical system b y several subsystems
1

2 CHAPTER 1. INTR ODUCTION
of the latter aspect is a chain of
n ≥
2 mathematical pendulums; see , for instance, [12, Ex ample 2.2].
As a consequence , we keep all algebraic constraints and wor k directly with the DAE.
If an extensive libr ar y of models with well-defined interaction ports of small components is available ,
then the modeling process of a networ k of some of these components can be automated, facilitating
a quick modeling process . Examples of tools that use such an idea are S I M U L I N K and M O D E L I C A . In
some applications, for instance , in earthquake engineering, the simulation and model capacities
ar e limited, such that there is still a need for actual experiments [218]. T o remedy the high cost
that comes with such experiments, it is common in the dynamical testing community to emplo y
a hybrid numerical-e xperimental setup , see [45] and the refer ences therein. I n more detail, only
a small physical subsystem that featur es the key region of inter est is experimentally tested, while
the r emainder of the system is simulated numer ically . The interconnection of th e physical and
numerical domain requir es a transfer system, which is typically a set of actuators [44]. The transfer
of information from the numerical system to the actuators is intrinsically non-instantaneous [129],
which introduces a time delay in the o verall system. A detailed example is pr esented in section 1.1.1.
The complete system dynamics may thus be modeled as an implicit system of equations of the form
0 = F ( t , x ( t ), ˙
x ( t ), x ( t − τ )), (1.1)
wher e
x
(
t
) denotes the unkno wn state and
τ >
0 the time delay . W e emphasize that in gener al, the
partial derivative of
F
with r espect to
˙
x
is allo wed to be singular , and hence the implicit function
theor em may not be used directly to r efor mulate
(1.1)
as a delay differ ential equation (DDE). One
possible sour ce for the singular ity of the partial deri vative of
F
with r espect to
˙
x
is due to the
inter connections, which ar e usual ly described via algebraic equations . H ence, w e refer to
(1.1)
as
a delay differ ential-algebraic equation (DDAE). Clearly , it is important to understand the effect of
the delay that is introduced due to the hybrid numerical-experimental setting and thus a thorough
analysis of the DDAE
(1.1)
, which ser ves as the object under investigation in this thesis, is of
paramount importance . W e stress that further delays may arise in the modeling process of the
system depicted in Figur e 1.1:
(i)
If the physical system itself is a controlled plant , then one may think of one of the subsystems
as a controller . The controller interacts with the plant b y measur ing some quantity of inter est,
compute (in some sense) a control action, and implement this action for the system. I f any of
these steps r equires some time , the controller induces an intrinsically necessar y time delay .
F or instance, in a chemical process , one may take a sample, analyze it, and, based on the r esult,
decide to modify the process . Another example from mechanical engineering is presented
in section 1.1.2. W e note that sometimes it may even be advantageous to implement a small
time-delay to impro ve the control action [105, 149] or to unco ver unstable periodic orbits in
nonlinear dynamical systems [176].
(ii)
If the subsystems ar e physically separated, then the interaction betw een subsy stems in the
form of exchange of energy , infor mation, or data may r equire a non-neglectable amount of
time , which introduces another source for a time-delay in the s ystem. S uch a communication
or propagation delay appears for instance in modern electric po wer grids [2], satellite com-
munication [195], or in a chemical process [64, 161]. The latter is illustr ated in more detail in

1.1. MO TIV A TING EXAMPLES 3
section 1.1.3.
(iii)
One of the subsystems itself might be modeled with a delay equation. F or instance, it is
well-kno wn that the linear advection equation can be r ewritten as a delay equation. The
r efor mulation of a hyperbolic equation as a delay equation not only offers a differ ent approach
to existence and uniqueness r esults [38] , and the development of different mathemat ical
models [37], but also is cheaper to solve numerically . This fact also serves as a motivation to
r ealize a transport process with a DDAE [77, 189, 191]. The process of r ewriting a hyperbolic
equation as a delay equation is detailed in section 1.1.4.
Besides the r elevance of DDAEs in modern modeling frameworks , we emphasiz e that DDAEs are also
a po werful mathematical tool that can be used in the analysis and design of numerical algorithms,
cf. section 1.1.5, or in assessing and classification of data, which is, for inst ance , used to relate finger
tapping with the severity of P arkinson ’ s disease [131, 132].
Besides the alr eady mentioned examples, futher applications include slo w-fast systems, such as
electro-optic oscillators [169, 217] and optical networks [84], electric circuits [183], applications
in biology [37, 65], chemical kinetics [74], human balance control [160, 198], and machine tool
vibrations [119, 163, 164]. F or additional phenomena that feature time delay we r efer to the mono-
graphs [75, 122] and the refer ences ther ein.
1.1 M otivating examples
W e present some of the examples mentioned abo ve in mor e detail in this section to stress the
r elevance of DDAEs in applications. The notation is simplified whenever it is deemed r easonable b y
omitting the explicit dependency on the time variable .
1.1.1 R eal-time dynamic substr ucturing
I n some applications, the description of a physical system with a mathematical model is difficult
due to its complex natur e or uncer tainty [218]. Since testing of a complete pr ototype may be
prohibitively expensiv e , it is desirable to incorporate the benefits of actual testing with the b enefits
of numerical simulation. This is accomplished b y testing only a substr uctur e (or subsystem in the
sense of Figur e 1.1) and connect the experiment via a transfer system with the r emaining system,
which is simulated numerically . S uch a hybrid experimental-numer ical approach is called r eal-
time dynamic substructuring or har dware-in-the-loop testing [45]. The transfer system is typically
r ealized with a set of hydraulic actuators . S ince the dynamic behavior of any actuator includes
a r esponse delay [107, 216], the resulting system is a DDAE. Let us emphasize that further delays
might be pr esent, which arise, for instance , from data acquisition, computation, or digital signal
processing. I n many applications, these delays ar e small compared t o the actuator delay and may
thus be neglected in the modeling process; for more details , we refer to [129] and the r eferences
ther ein.
W e illustrate such a hybrid experimental-numer ical setup with a coupled pendulum-mass-spring-
damper system, as described in [129, 212]. F or ou r example , we consider the mass-spring-damper

4 CHAPTER 1. INTR ODUCTION
M
( x 1 , y 1 )
K C
x
y
m
( x 2 , y 2 )
ℓ

(a) Fully coupled system
F ext
M
K C
N U M E R I C A L
M O D E L
F pendulum
m
ℓ
E X P E R I M E N T
+ A C T U AT O R
adjust
position
send
F pendulum

(b) H ybr id numerical-experimental setup
F igure 1.2 – Real-time dynamic substructuring for a coupled pendulum-mass-spring-damper system
system as the numerical simulation and the pendulum as the experiment; see Figur e 1.2 for an
illustration. F or our numerical model, we assume that the mass
M
is mounted on a linear spring
and a linear viscous damper . The resulting equation of motion for the mass-spring-damper system
is given b y
M ¨
y 1 + C ˙
y 1 + K y 1 = F ext , (1.2)
wher e
C
and
K
denote the damping and the stiffness coefficient, r espectively , and
y 1
denotes the
vertical displacement of the center of mass with r espect to the equilibr ium position. W e assume
that ther e is no hor izontal displacement, i.e ., the mass
M
can only mo ve up wards and do wnwards .
W e thus set
x 1 : =
0. The external force , which in this scenar io will be pro vided b y the pendulum,
is given b y
F ext
. W e assume that the pendulum is given b y a point mass
m
that is attached to the
spring-mass-damper system via a massless rod of length
ℓ
. Assuming no friction, the model for the
pendulum is given b y
m ¨
x 2 = − 2 λ x 2 ,
m ¨
y 2 = − 2 λ ( y 2 − y 1 ) − m g ,
0 = x 2
2 + ( y 2 − y 1 ) 2 − ℓ 2 ,
(1.3)
with gravitational constant g and Langr ange multiplier
λ
. By N ewton ’ s second law , the force that the
pendulum generates in
y
-dir ection is given by
F pendulum = −
2
λ
(
y 2 − y 1
)
− m
g . Consequently , the
equations of motion for the fully coupled system (as depicted in Figur e 1.2a) are giv en by
M ¨
y 1 + C ˙
y 1 + K y 1 = − 2 λ ( y 2 − y 1 ) − m g ,
m ¨
x 2 = − 2 λ x 2 ,
m ¨
y 2 = − 2 λ ( y 2 − y 1 ) − m g ,
0 = x 2
2 + ( y 2 − y 1 ) 2 − ℓ 2 ,
(1.4)
with unkno wn functions
y 1 , x 2 , y 2
, and
λ
. I n the hybr id numerical-exper imental setup (cf. Fig-
ur e 1 .2b), the actuator introduces a time-delay
τ >
0 into the system, which is assumed constant.
The delay can be understood as an offset in time between the mass-spring-damper dynamics
(1.2)
and the pendulum dynamics
(1.3)
. I n par ticular , we hav e to replace
t
b y
t − τ
in the pendulum

1.1. MO TIV A TING EXAMPLES 5
dynamics
(1.3)
and in the for ce
F pendulum
. Thus, the complete mathematical descr iption for the
hybrid numerical-exper imental setup is given b y
M ¨
y 1 + C ˙
y 1 + K y 1 = − 2 λ ( · − τ )( y 2 ( · − τ ) − y 1 ( · − τ )) − m g , (1.5a)
m ¨
x 2 ( · − τ ) = − 2 λ ( · − τ ) x 2 ( · − τ ), (1.5b)
m ¨
y 2 ( · − τ ) = − 2 λ ( · − τ )( y 2 ( · − τ ) − y 1 ( · − τ )) − m g , (1.5c)
0 = x 2 ( · − τ ) 2 + ( y 2 ( · − τ ) − y 1 ( · − τ )) 2 − ℓ 2 . (1.5d)
If we intr oduce new variables for
˙
y 1
,
˙
x 2
, and
˙
y 2
, then we can rewrite
(1.5)
in the form
(1.1)
. Let
us emphasize that in or der to solve
(1.5)
, for instance with a numerical method, we have to shift
equations (1.5b) to (1.5d) in time .
1.1.2 T ime-delayed feedback control
I n several (engineering) applications, it is a central goal to enfor ce a specific behavior wi thin the
system, for instance , to stabilize an unstable equilibrium, su ch as the up ward position in an inverted
pendulum. The goal is typically achieved b y implementing a controller into the system. Given
Figur e 1.1, the controller can be interpr eted as a subsystem. If the dynamics of the physical system
ar e complex and thus the resulting model is subject to uncertainties or further exter nal for ces may
act on the system, it is common to design the contr oller based on the current state of the system.
Ther e are two differ ent sources for time-delays in such a feedback loop: (a) the controller r equires
some time to measur e the quantity of interest, compute the feedback law , and implement the
control action, or (b) a time-delay is implemented on purpose to facilitate some desir ed behavior . A
popular control str ategy that falls into the second categor y is called P yr agas control [175]. P rominent
examples include stabilization of unstable periodic orbits in chaotic electrical networks [176],
control of a T aylor -Couette Flo w [140], microcantilev er s [223], or semiconductor lasers [185], and
sway r eduction for container cranes [105, 1 49]. W e pr esent the latter application in more detail in
the follo wing.
A simplified model of a two-dimensional container crane (also called gantry crane) is given b y
a mathematical pendulum that is attached to a mo ving cart, the so-called tr olley ; see F igure 1.3.
H ereb y , both the payload and the cart are modeled as point masses . W e may control the crane b y
applying a horizontal for ce to the car t and b y changing the length of the rope . Assuming a frictionless
mo vement of the cart, the simplified model in Figur e 1.3 can be consider ed as a controlled multibody
system (MBS). The equation of motions can be derived from a variational principle and H amilton ’ s
principle of least action, resulting in the DAE
m 1 ¨
x 1 = 2 λ ( x 2 − x 1 ) + f ,
m 2 ¨
x 2 = − 2 λ ( x 2 − x 1 ),
m 2 ¨
y 2 = − 2 λ ( y 2 − y 1 ) − m 2 g ,
0 = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 − ℓ 2 ,
0 = y 1 ,
(1.6)
see [12] and the r eferences ther ein. A typical control task is to mo ve the payload from an initial
position (
x 2
(0)
, y 2
(0)) to a given position (
˜
x 2 , ˜
y 2
) as fast a possible . H o wever , a rapid mo vement of the

6 CHAPTER 1. INTR ODUCTION
x
y
trolley
( x 1 , y 1 )
m 1
f
g
( x 2 , y 2 )
m 2
payload
ℓ

m 1 mass of the trolley
m 2 mass of the payload
ℓ length of the rope
x 1 horizontal position of the trolley
y 1 vertical position of the trolley
x 2 horizontal position of the payload
y 2 vertical position of the payload
g gravity constant
f horizontal applied for ce
F igure 1.3 – A simplified model for a two-dimensional gantr y crane
crane may r esult in a swaying payload, maneuver ing the gantry crane in a potentially dangerous state .
F oll o wi ng [105, 148] the sway can be reduced b y applying a delayed position feedback controller , i.e.,
the external force f is generated b y a controller κ of the form
f ( t ) = κ ( t , x 1 ( t − τ ), y 2 ( t − τ ), x 2 ( t − τ )).
F or details we r efer to [75, Section 1.8].
1.1.3 T ransmission and propagation delays
The exchange of ener gy , information, or data in the interaction between two or mor e subsystems in
Figur e 1.1 is often modeled as an instantaneous process to happen instantaneously . If, ho wever ,
some of the subsystems ar e physically separated, then this appro ximation cannot r easonably be
made , and the modeling process has to account for the r esulting transmission, propagation, or
communication delays . Examples are wide-ar ea po wer networks [2], synchronization of distant
brain r egions [170], chemical processes [64, 161] or controlling a satellite in outer space [195]. I n this
subsection we illustrate a p ropagation delay via an irr eversible reac tion
A → B
that is coupled with
the r eversible reaction
B ⇌ C
in a continuous stirr ed-tank reactor (CSTR) with reaction rates
r A → B
and
r B ⇌ C
, which depend on the reactant concentr ations
c A , c B , c C
, and the temperatur e
T
in
the tank. The reversible r eaction
B ⇌ C
is assumed to happen much faster than the irr eversible
r eaction
A → B
, which implies that the fast r eaction is essentially at equilibrium with equilibrium
constant
K B ⇌ C
. N otice that Le Chatelier ’ s pr inciple implies that the equilibrium constant depends
on the temperatur e T in the tank, i.e ., K B ⇌ C = K B ⇌ C ( T ).
T o facilitate the transformation from
A
to
C
, we process the r eaction in two CSTRs. In the first
r eactor , we use a high temperatur e
T 1
to promote the conversion fr om
A
to
B
. Since the thermo-
dynamic equilibrium limits the production of
C
(see the discussion abo ve), we pr escr ibe a lo wer
temperatur e
T 2
in the second tank to promote the pr oduction of
C
. The two tanks ar e linked via a

1.1. MO TIV A TING EXAMPLES 7
A → B ⇌ C
c A ,1 c B ,1 c C ,1
temperatur e T 1
R E A C T O R 1
c A ,2 c B ,2 c C ,2
A → B ⇌ C
temperatur e T 2
R E A C T O R 2
u = c A ,0
transmission delay τ
c A ,1 , c B ,1 , c C ,1

F igure 1.4 – A two-stage continuous stirred-tank r eactor system
lossless transmission line (cf. F igure 1.4). As a consequence , the inflo w of the second tank at time
t
equals the outflo w of the first tank at time
t − τ
. H ereb y ,
τ >
0 describes the time that is requir ed to
travel thr ough the transmission line.
Assuming a r eaction equilibr ium for the fast r eaction and constant volumes in both tanks , i.e., the
inflo w and outflo w rates ar e identical, the set of DDAEs
˙
c A ,1 = κ 1 ( u − c A ,1 ) − r A → B ,1 , ˙
c A ,2 = κ 2 ( c A ,1 ( · − τ ) − c A ,2 ) − r A → B ,2 ,
˙
c B ,1 = − κ 1 c B ,1 + r A → B ,1 − r B ⇌ C ,1 , ˙
c B ,2 = κ 2 ( c B ,1 ( · − τ ) − c B ,2 ) + r A → B ,2 − r B ⇌ C ,2 ,
˙
c C ,1 = − κ 1 c C ,1 + r B ⇌ C ,1 , ˙
c C ,2 = κ 2 ( c C ,1 ( · − τ ) − c C ,2 ) + r B ⇌ C ,2 ,
0 = K B ⇌ C ( T 1 ) c B ,1 − c C ,1 , 0 = K B ⇌ C ( T 2 ) c B ,2 − c C ,2 ,
0 = r A → B ,1 − k A → B ( T 1 ) c A ,1 , 0 = r A → B ,2 − k A → B ( T 2 ) c A ,2 ,
(1.7)
describes the complete model with unkno wns
c A , i , c B , i , c C , i , r A → B , i , r B ⇌ C , i
for
i =
1
,
2. The con-
stant
κ i
is given b y the flo w rate divided b y the volume . The reactant
A
is fed to the first CSTR with
concentration
u = c A ,0
. N ote that the reaction
r A → B , i
is explicitly described b y the algebraic equation
and the pr escr ibed function
k A → B
, which is for example given b y the Arrhenius equation [130 , p . 153].
I n contrast, the r eaction
r B ⇌ C , i
is only implicitly given b y the equilibr ium equation with pr escr ibed
equilibrium ratio K B ⇌ C ( T ).
R emark 1.1.
The DDAE
(1.7)
can formally be obtained b y accounting for the different time scales
via a singular perturbation and a for mal limit [123]. S uch a process is typical for dynamics with fast
and slo w time scales, and it is essential to understand the limiting situation for the construction of
numerical methods [127]. As a consequence, DDAEs also arise as the limiting situation of singularly
perturbed DDEs. ♣
1.1.4 R efor mulation of hyperbolic problems as delay equations
N ot only the interaction between subsystems in F igure 1.1 may induce delays into the o verall
system dynamics, but also subsystems themselves might f eatur e a descr iption with delayed vari-
ables . F or instance, it is well-kno wn that many first-order hyperbolic partial differ ential equations

8 CHAPTER 1. INTR ODUCTION
0 L
ξ 0
L length of the duct
ρ 0 r eference density
c speed of sound
v velocity
p pr essure

F igure 1.5 – Acoustic transmission in a fluid-filled duct
(PDEs) can be r ewr itten as delay differ ence equations by exploiting in some sense the method
of characteristics [61, 120]. This technique is applied for instance to circuits that inv olve lossless
transmission lines [39, 139], str uctur ed population models [ 36], mining ventilation [219], and blood
flo w systems [38]. W e exemplify the transformation from a hyperbolic problem to a delay equation
b y consider ing acoustic transmission in a fluid-filled duct of length
L
that has an acoustic driver
positioned at one end (cf. Figur e 1.5). F ollo wing [63], the pressur e
p
(
t , ξ
) and the fluid velocity
v
(
t , ξ
)
at a point ( t , ξ ) ∈ (0, T ) × (0, L ) satisfy the coupled PDE
1
c 2
p ( t , ξ )
∂ t = − ρ 0
∂ v ( t , ξ )
∂ξ , ρ 0
∂ v ( t , ξ )
∂ t = − ∂ p ( t , ξ )
∂ξ , 0 < t < T , 0 < ξ < L , (1.8a)
v ( t , 0) = u ( t ), p ( t , L ) = 0, 0 < t < T , (1.8b)
wher e
c >
0 denotes the speed of sound (which is assumed to be constant within the duct) and
ρ 0 >
0 the r eference density . The boundary condition imposed by the acoustic driver at
ξ =
0 is
given b y the control input
u
. S ince
(1.8a)
r esembles a wave equation, the general solution of the
PDE (1.8a) is given b y
v ( t , ξ ) = φ ( ξ − c t ) + ψ ( ξ + c t ), p ( t , ξ ) = ρ 0 c (︁ φ ( ξ − c t ) − ψ ( ξ + c t ) )︁ , (1 .9)
wher e
φ
is a wave trav eling to the r ight and
ψ
a wave trav eling to the left. Similar to [39, 102], we
r ewr ite (1.9) to obtain
2 φ ( ξ − c t ) = v ( t , ξ ) + 1
ρ 0 c p ( t , ξ ), 2 ψ ( ξ + c t ) = v ( t , ξ ) − 1
ρ 0 c p ( t , ξ ).
U sing
2 φ ( − c t ) = 2 φ (︁ L − c (︁ t + L
c )︁)︁ = v (︁ t + L
c , L )︁ + 1
ρ 0 c p (︁ t + L
c , L )︁ ,
2 ψ ( c t ) = 2 ψ (︁ L + c (︁ t − L
c )︁)︁ = v (︁ t − L
c , L )︁ − 1
ρ 0 c p (︁ t − L
c , L )︁
and the boundar y conditions (1.8b) at t − L / C we obtain
u (︁ t − L
c )︁ = v ( t − L
c , 0) = 1
2 (︂ v ( t , L ) + v ( t − 2 L
c , L ) + 1
ρ 0 c (︁ p ( t , L ) − p ( t − 2 L
c , L ) )︁ )︂
= 1
2 (︁ v ( t , L ) + v ( t − 2 L
c , L )︁ .
Setting x ( t ) : = v ( t , L ), ˜
u ( t ) : = u ( t − L / c ), and τ : = 2 L / c we get the linear differ ence equation
x ( t ) = − x ( t − τ ) + 2 ˜
u ( t ),
which is a special case of a linear DDAE.

1.1. MO TIV A TING EXAMPLES 9
1.1.5 Delay differ ential-algebraic equations as mathematical tool
Besides the various application ar eas outlined abo ve, DDAEs ar e also a po werful mathematical tool,
which we highlight with sever al examples. F or instance, DDAEs ar e used to design numerical time-
integration methods for neutr al delay differ enti al equations (NDDEs), i.e., differ ential equations
wher e the rate of change at time
t
depends on the rate of change at time
t − τ
. If the NDDE
featur es a par ticular structure [24], the DDAE formulation rev eals that the problem consists of a
differ ential equation coupled with an algebraic recursion for the delay . This r efor mulation benefits
the theor etical and numer ical investigation of the NDDE, see , for instance, [24, 25]. DDAEs are also
used to establish existence and uniqueness of solutions . F or example, in [38] the authors study a
hyperbolic equation that is coupled via its boundary conditions to a switched DAE. By r ewriting
the hyperbolic equation as a delay equation (cf. section 1.1.4), the existence of solutions is a mer e
consequence of the existence of solutions of the corr esponding switched DDAE. Another application
of DDAEs is pr esented in [4, 5] in the analysis of a semi-explicit scheme for a linear poroelasticity
problem that arises in the modeling of deformation resulting fr om tumor gro wth in the brain [182].
W e detail this example to explain the line of r easoning. In mor e detail, the mathematical model of
this problem is an elliptic equation for the displacement that is coupled to a par abolic equation
for the pr essure . Semi-discr etization in space yields (under reasonable assumptions on the sp atial
discr etization) to a DAE of the for m
K u u ( t ) − D T p ( t ) = f ( t ), (1.10a)
D ˙
u ( t ) + M p ˙
p ( t ) + K p p ( t ) = g ( t ). (1.10b)
H er eb y ,
u
denotes the displacement,
p
the pr essure ,
K u
and
K p
the stiffness matrices corresp onding
to
u
and
p
,
M p
the mass matrix for the pr essu r e, and
D
the coupling matrix. F or the numerical
integration in time , we consider a step size
τ >
0 and the time grid
t k : = k τ
. The equations
(1.10a)
and (1.10b) can be decoupled b y emplo ying a semi-explicit Euler discr etization in time , given b y
K u u k + 1 − D T p k = f k + 1 , (1.11a)
D ( u k + 1 − u k ) + M p ( p k + 1 − p k ) + τ K p p k + 1 = g k + 1 (1.11b)
with appro ximations
u k ≈ u
(
k τ
),
p k ≈ p
(
k τ
), and
f k : = f
(
k τ
),
g k : = g
(
k τ
). N ote that
p k
appears in
(1.11a)
(instead of
p k + 1
), which r enders the scheme semi-explicit. Clearly , the decoupling has the
advantage that two smaller subsystems with a nice structur e have to be solved in each time-step . A
key tool for the analysis of the semi-explicit scheme is the obser vation that
(1.11)
corr esponds to an
implicit Euler discr etization of the DDAE
K u u ( t ) − D T p ( t − τ ) = f ( t ), (1.12a)
D ˙
u ( t ) + M p ˙
p ( t ) + K p p ( t ) = g ( t ). (1.12b)
I n par ticular , the semi-explicit scheme only converges if the DDAE
(1.12)
is asymptotically stable ,
which imposes a weak coupling condition, i.e .,
D
is in some sense small compar ed to
K u
and
M p
.
W e refer to [4] for further details .

10 CHAPTER 1. INTR ODUCTION
1.2 Scope and synopsis
S ummar izing the motivating examples fr om the previous section, the main object under investiga-
tion in this thesis is the nonlinear DDAE
0 = F ( t , x ( t ), ˙
x ( t ), x ( t − τ ), u ( t )), (1.13)
wher e
x
(
t
)
∈ F n x
and
u
(
t
)
∈ F n u
denote , respectively , the stat e and control of the system. H ereb y ,
F
denotes either the field of r eal or complex numbers, i.e.,
F ∈ { R , C }
. The function
F
is defined on the
time inter val I : = [ t 0 , t f ] and open sets D x , D ˙
x , D σ τ x ⊆ F n x and D u ⊆ F n u via
F : I × D x × D ˙
x × D σ τ x × D u → F m
and is assumed to be sufficiently smooth. A special case of
(1.13)
is the so-called linear time-varying
DDAE
E ( t ) ˙
x ( t ) = A 1 ( t ) x ( t ) + A 2 ( t ) x ( t − τ ) + B ( t ) u ( t ) + f ( t ), (1.14)
with
E , A 1 , A 2 : I → F m × n x
,
B : I → F m × n u
, and inhomogeneity
f : I → F m
. If the matrix functions
E , A , D
, and
B
ar e constant on
I
, then
(1.14)
is called linear time-invariant (L TI), and b y abuse of
notation written as
E ˙
x ( t ) = A 1 x ( t ) + A 2 x ( t − τ ) + B u ( t ) + f ( t ), (1.15)
with matrices E , A 1 , A 2 ∈ F m × n x and B ∈ F m × n u .
An important feature that distinguishes the DDAE
(1.13)
from a r etarded DDE is that w e allo w
∂
∂ ˙
x F
to
be pointwise singular . While the potential singularity of
∂
∂ ˙
x F
allo ws to include algebraic constr aints
and thus for a very flexible modeling approach, it comes with additional difficulties in the theor etical
and numerical analysis. This is well-kno wn for DAEs, see for instance [171], and accor dingly , these
difficulties ar e transferred dir ectly to DDAEs [13, 53, 98].
R emark 1.2.
The formulation of the DDAE
(1.13)
is not r estr icted to one single delay , since multiple
commensurate delays [85] may be r ewritten as a single delay by intr oducing new vari ables [94].
M ore pr ecisel y , a DDAE with multiple commensurate delays may be written as
0 = F ( t , x ( t ), ˙
x ( t ), x ( t − τ ), x ( t − 2 τ ), . . . , x ( t − k τ ), u ( t )). (1.16)
I ntroducing the new variables
z i
(
t
)
= z i − 1
(
t − τ
) for
i =
1
, . . . , k
with
z 0
(
t
)
= x
(
t
) allo ws to write
(1.16)
as ⎡
⎢
⎢
⎢
⎢
⎣
0
0
.
.
.
0
⎤
⎥
⎥
⎥
⎥
⎦ = ⎡
⎢
⎢
⎢
⎢
⎣
F ( t , z 0 ( t ), ˙
z 0 ( t ), z 1 ( t ), . . . , z k ( t ), u ( t ))
z 1 ( t ) − z 0 ( t − τ )
.
.
.
z k ( t ) − z k − 1 ( t − τ )
⎤
⎥
⎥
⎥
⎥
⎦ ,
which is again of the form
(1.13)
. N ote that if the DDAE depends also on derivatives of the past
argument, i. e . on
x ( ℓ )
(
t − τ
) for some
ℓ ∈ N
, one can use a similar procedur e to recast this pr oblem
in the form (1.13). ♣

1.2. SCOPE AND SYNOPSIS 11
A standar d question to ask is whether for a given control function
u
, the DDAE
(1.13)
(r esp .
(1.14)
)
possesses a unique solution. As it is alr eady kno wn from the theor y of or dinar y differential equations
(ODEs), this is in general not the case . F or ODEs, this can be fixed b y imposing a constraint on the
state in form of an initial or final condition (and some further technical assumptions). In t his thesis
we focus on pr escribed initial conditions, which for delay equations take the form
x ( t ) = φ ( t ) for t ∈ [ − τ , 0 ]. (1.17)
The equations
(1.13)
and
(1.17)
together ar e referr ed to as initial value problem (IVP). After estab-
lishing conditions on
F
and
φ
for a solution to exist, the next question to ask is whether the solution
is unique and whether its dependency on the data is continuous . M ore pr ecisely , we are inter ested
in the question, whether for a given control u the oper ator equation
K ( x ) : = [︄ F ( · , x , ˙
x , σ τ x , u )
x [ − τ ,0] ]︄ = [︄ p
φ ]︄ , (1.18)
with shift operator ( σ τ x )( t ) = x ( t − τ ) is well-posed in the sense of Hadamar d [88, 100, 147], i.e ., if
(i) for each ( p , φ ) ∈ P × Φ , ther e is a solution x ∈ X of (1.1 8),
(ii) this solution is unique in X and
(iii) the dependency of x upon ( p , φ ) is continuous .
H ereb y , the operator
K
maps the topological space
X
into the topological space
Z : = P × Φ
. T o
answer the question whether
(1.18)
is well-posed, a number of other questions need to be answer ed
first, ranging fr om the solu tion concept used for
(1.13)
(thus fixing the space
X
), to the smoothness
r equirements for
F
. These questions ar e addressed in detail in the first part of the thesis . M or e
pr ecisely , we have the follo wing results:
(i)
The forthcoming Examples 1.4 and 1.5 r eveal that in general we cannot expect a classical
solution to exist. W e thus start our analysis b y seeking solutions in the space of piecewise-
smooth distributions (see Definition 3.3). Our first main r esult — Theorem 3.5 — details that
a r egular matr ix pair (
E , A 1
) in
(1.15)
is a sufficient condition for existence and uniqueness of
solutions for any initial trajectory , input, and exter nal for cing.
(ii)
I n order to obtain a necessary condition, we define the notion of delay-r egularity (Defini-
tion 3.8) and establish in Theor em 3.20 e xistence and uniqueness of solutions for all external
for cing signals if and only if
det
(
s E − A 1 − ω A 2
)
≡
0. As a mere consequence , we sho w that a
linear DAE can be r egularized via feedback if and only if it can be regularized b y a delayed
feedback. The details are pr esented in section 3.2.
(iii)
I n Definition 3.28 we introduce a no vel equivalence r elation, called delay-equivalence , which
allo ws us to establish that whenever the DDAE
(1.15)
is delay -regular , then it can be trans-
formed to a delay -equivalent DDAE with regular matrix pencil, see Theorem 3.37 and R e-
mark 3.39. The delay -equivalent DDAE can be obtained b y a simple compress-and-shift
algorithm, which was previously suggested in the liter ature [53]. Our analysis rev eals that with

12 CHAPTER 1. INTR ODUCTION
a simple modification of the algorithm it can detect along the way if the DDAE is delay -r egular ,
and terminate other wise.
(iv)
U sing the previous r esults, we introduce a new classification of L TI DDAEs in section 4.1 that
is based on the propagation of so-called primary discontinuities [26]. The main motivation
for such a classification is to establish conditions that ensur e existence of solutions that are
continuously differ entiable almost ever ywhere . W e present a complete algebr aic characteriza-
tion of the differ ent classes in Theorem 4.16, which in turn allo ws us to for mulate a general
existence r esult for DDAEs (cf. Theorem 4.18).
(v)
If we impose some additional r egularity assumptions on the histor y function
φ
, that is, w e
impose that the history function is linked smoothly to the solution of the DDAE, then we can
further impro ve the r esult of Theorem 4.18. In mor e detail, the impact of so-called splicing
conditions is analyzed in section 4.2, wher e we sho w the existence of a continuous solutions
for a higher index DDAE in Theor em 4.27.
(vi)
I n chapter 5 we sho w that the results fr om the L TI case can in parts be translated to nonlinear
DDAEs. The main obser vation is that the results in Chapter 4 can be formulated in terms of
the underlying DDE, which can also be defined for nonlinear DDAEs. With this we establish
existence and uniqueness r esults for a class of nonlinear DDAEs in Theorem 5.24.
(vii)
W e conclude our analysis with a detailed investigation of the hybrid numerical-exper imental
system pr esented in section 1.1.1. I n par ticular , we sho w that the compress-and-shift algo-
rithm (Algorithm 1) from chapter 3 can be applied to the hybrid system even in the nonlinear
case . The algori thm terminates after a single shift with a regular DDAE, whenev e r the two
subsystems ar e regular DAEs. The details are pr esented in Lemma 5.10, Theorem 5.15 and
Theor em 5.17. U sing the solution theor y developed in chapter 5 we pr ov e that the hybr id
system is solvable whenever the experimental and numerical part are both str angeness-free ,
see Cor oll ary 5.25.
I n the second par t of the thesis we invert the problem b y asking whether we can determine an
operator ˜︂
K : X → Z that minimizes
∥ ˜︂
K ( x ) − p ∥ (1.19)
for some given
x ∈ X
,
p ∈ Z
and some suitable norm
∥ · ∥
. In this case ,
˜︂
K
is called a r ealization
for the data pair (
x , p
). The simplest r ealiz ation is of course to just map
x
onto
p
. H o wever , this
r ealization is o nly valid for the specific data pair (
x , p
) and any variation in the data pair r esults in
a differ ent realization. Instead, we may want to ask for a r ealization
˜︂
K
that minimizes
(1.19)
for
all data pairs (
x , p
). S ince the class of all possible r ealizations is hard to para meterize , we r estr ict
ourselves to the case that
˜︂
K
is linear , i.e., we consider only linear time-invariant DDAEs of the
form
(1.15)
. Often in practical application, the state
x
itself may not be available (or of inter est) and
instead, only an obser vation of the state in form of an output equation
y ( t ) = C x ( t ) (1.20)
wher e the matr ix
C ∈ F n y × n x
is available . Consequently , we assume that ther e exists a dynamical sys-
tem, exemplified b y an operator
S
that maps inputs
u
to outputs
y
. H ereb y the standing assumption

1.2. SCOPE AND SYNOPSIS 13
is that we can evaluate
S
for given inputs but do not have access to a state-space r ealization of
S
,
i.e .,
S
acts as a black-bo x. The DDAE realization problem can thus be stated as f ollo ws: F or given
trajectories ˆ︁
u and ˆ︁
y , construct a DDAE
˜︁
S : {︄ ˜︁
E ˙
x ( t ) = ˜︁
A 1 x ( t ) + ˜︁
A 2 x ( t − τ ) + ˜︁
B u ( t ),
y ( t ) = ˜︁
C x ( t ) (1.21)
such that
∥ S
(
u
)
− ˜︁
S
(
u
)
∥
is minimized for all admissible inputs
u
. If we choose the norm to be the
L ∞
norm then it is well-kno wn (cf. [20]) that the error is bounded b y the
H 2
norm of the error system
multiplied with the norm of the input, i.e.,
  y − ˜︁
y   L ∞ ≤ ∥ S − ˜︁
S ∥ H 2 ∥ u ∥ L 2 ,
pro vided that all quantities are well-defined. I f in addition we are seeking for a standar d state-
space system, i.e.,
E = I n x
and
A 2 =
0, then the
H 2
error is minimized if the tran sfer function
associated with
˜︁
S
interpolates the transfer function associated with
S
at certain frequencies – see for
instance [20]. Thus, we propose to tackle the DDAE r ealization problem b y constructing a realization
of the form (1.21), such that its transfer function
˜︁
H ( s ) = ˜︁
C (︁ s ˜︁
E − ˜︁
A 1 − exp( − τ s ) ˜︁
A 2 )︁ − 1 ˜︁
B (1.22)
interpolates the transfer function associated with
S
at given fr equencies. W e refer to Problem 6.6
and Problem 6.9 for a pr ecise problem description. W e obtain the follo wing results .
(i)
W e present necessary and sufficient conditions for interpolation in Theor em 7.1 by analyzing
the mor e general class of r ealizations of the form
˜︁
H
(
s
)
= ˜︁
C (︂ ∑︁ K
k = 1 h k ( s ) ˜︁
A k )︂ − 1 ˜︁
B
with linear
independent family
{ h 1 , . . . , h K }
of meromorphic functions mapping the complex plane into
itself.
(ii)
The interpolation conditions from Theor em 7.1 rev eal that the situation is different for
K =
2
and
K ≥
3. F or
K =
2 we pro vide a direct solution of the pr oblem (see Theorem 7.5) and sho w
its close connection to the Loewner framewor k [15 0] in Cor ollar y 7.7.
(iii)
F or the case
K ≥
3, which is the case for the DDAE realization problem, we present two
strategies to handle the r emaining degrees of fr eedom: First b y interpolation of additional
data (cf. section 7.3.1), and second b y interpolation of derivative information of the transfer
function (cf. section 7.3.2). I n both cases we do not increase the dimension of the inv olved
matrices. The main r esults ar e presented in Theor em 7.12 and Theorem 7.16.
(iv)
T o obtain data in the frequency domain w e consider in chapter 8 the estimation of frequency
data from time series, i.e ., from a sampling of
u
and
y
in the time-domain. W e use the least-
squar es transfer function estimate (lsTFE) framework introduced in [168] and generaliz e the
r equired r esults to our setting, which includes continuous time systems and general system
structures . The r esul ting method is summarized in Algorithm 5.
(v)
Based on additional data points , we pr esent a least-squares approach (see section 8.3) to
estimate possibly unkno wn parameters in the r ealization as, for instance , the delay parameter .
W e demonstrate the r esults with a complete case study with a delay example in section 8.4.

14 CHAPTER 1. INTR ODUCTION
1.3 Challenges and state of the ar t
S ince DAEs and DDEs are special cases of the DDAE
(1.13)
, it is clear that the problems specific
to these sub classes ar e also inherent to DDAEs. F or instance , DAEs r equire a so-called consistent
initialization [127, 167], since the class of possible initial conditions is r estr icted (see the forthcoming
Chapter 2). F or DDE s, one needs to put special emphasis on the tracking of so-called br eaking
points [90, 91], which r esult from the fact that the histor y function
φ
may not be linked smoothly to
the solution x at t = 0, i.e . we have
lim
t ↗ 0
˙
φ ( t ) = lim
t ↘ 0 ˙
x ( t ) (1.23)
in general. The follo wing examples suggest, that in some cases this so-called primar y discontinu-
ity [26] may be smoothed out o ver time , w hile in other cases, the discontinuity is pr opagated o ver
time or even amplified.
Example 1.3.
The IVP
˙
x
(
t
)
= − x
(
t −
1) with histor y function
φ ≡
1 can be solved b y integration on
successive time intervals (see chapter 2 for more details), which yields for 0 ≤ t ≤ 4 the solution
x ( t ) = ⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
1 − t , if 0 ≤ t ≤ 1,
1
2 t 2 − 2 t + 3
2 , if 1 ≤ t ≤ 2,
− 1
6 t 3 + 3
2 t 2 − 4 t + 17
6 , if 2 ≤ t ≤ 3,
1
24 t 4 − 2
3 t 3 + 15
4 t 2 − 17
2 t + 149
24 , if 3 ≤ t ≤ 4,
which is depicted as the blue line in Figur e 1.6. N ote that
lim t ↗ 0 ˙
φ
(
t
)
=
0
= −
1
= lim t ↘ 0 ˙
x
(0), i.e . the
solution is not continuously differ entiable at t = 0. Str aight for ward computations sho w
lim
t ↗ 1
d
d t ( 1 − t ) = − 1 = lim
t ↘ 1
d
d t (︂ 1
2 t 2 − 2 t + 3
2 )︂ ,
lim
t ↗ 2 (︃ d
d t )︃ 2 (︃ 1
2 t 2 − 2 t + 3
2 )︃ = 1 = lim
t ↘ 2 (︃ d
d t )︃ 2 (︃ − 1
6 t 3 + 3
2 t 2 − 4 t + 17
6 )︃ ,
lim
t ↗ 3 (︃ d
d t )︃ 3 (︃ − 1
6 t 3 + 3
2 t 2 − 4 t + 17
6 )︃ = − 1 = lim
t ↘ 3 (︃ d
d t )︃ 3 (︃ 1
24 t 4 − 2
3 t 3 + 15
4 t 2 − 17
2 t + 149
24 )︃ ,
and thus the solution becomes smoother o ver time . ♠
Example 1.4. The IVP 0 = x ( t ) + x ( t − 1) + 1 with histor y function φ ( t ) = t has the solution
x ( t ) = ⎧
⎨
⎩
k − 1 − t , if k − 1 ≤ t ≤ k and k ∈ N odd,
t + k , if k − 1 ≤ t ≤ k and k ∈ N even.
I n par ticular , the solu tion
x
(plotted as dashed r ed line in Figure 1.6) is continuous but
˙
x
is discon-
tinuous at every t = k and thus no smoothing occurs . ♠
Example 1.5. C onsider the DDAE
[︄ 1 0
0 0 ]︄ [︄ ˙
x 1 ( t )
˙
x 2 ( t ) ]︄ = [︄ 0 1
1 0 ]︄ [︄ x 1 ( t )
x 2 ( t ) ]︄ + [︄ 0 0
0 − 1 ]︄ [︄ x 1 ( t − 1)
x 2 ( t − 1) ]︄ (1.24)

1.3. CHALLENGES AND ST A TE OF THE AR T 15
− 1 01234
− 1
0
1
2
t

F igure 1.6
– Discontinuity propagation in DDAEs. The solid blue line repr esents the solution of the IVP
in Example 1.3, the dashed r ed line the solution der ived in Example 1.4, and the first component of the
solution of Example 1.5 is pr esented with the dotted yello w line.
with initial condition
x ( t ) = φ ( t ) = [︄ 1
3 t 3 + t 2 − 1,
1
3 ( t − 1) 3 + ( t − 1) 2 − 1 ]︄ for − 1 ≤ t ≤ 0.
N ote that
x 1
(
t
)
= x 2
(
t −
1) and thus it is sufficient to compute the solution
x 1
(the dotted yello w line
in Figur e 1.6), which is given b y
x 1 ( t ) = ⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
t 2 − 1, t ∈ [0, 1],
2 t − 2, t ∈ [1, 2],
2, t ∈ [2, 3),
0, t ≥ 3.
I n par ticular , the solution becomes less smooth at multiples of the time delay and even discontinu-
ous at t = 3. ♠
The study of primar y discontinuities of the scalar DDE
a 0 ˙
x ( t ) + a 1 ˙
x ( t − τ ) + b 0 x ( t ) + b 1 x ( t − τ ) = f ( t ) (1.25)
is based on the the classification proposed in [27]: The DDE
(1.25)
is said to be of r etarded type
if
a 0 =
0 and
a 1 =
0, of neutral type if
a 0 =
0 and
a 1 =
0, and of advanced type if
a 0 =
0 and
a 1 =
0.
F ollo wing this classification, we observe that the DDAE in Example 1.3 is of retar ded type , the
DDAE Example 1.4 is of neutral type (if we differentiate the equation), while the first compon ent in
Example 1.5 satisfies a DDE of advanced t ype . In particular , the DDAE
(1.13)
may contain scalar
delay differ ential equations of any of the three types . As a consequence, a history function
φ
for the
IVP
(1.13)
,
(1.17)
may be r equired to satisfy so-called splicing conditions [26]. W e r efer to section 4.2
for further details.
The classification of
(1.25)
was extended in the series of papers [97
–
99] to linear time-varyi ng DDAEs
of the form
(1.14)
, using the so-called underlying DDE (see sections 4.1 and 4.3 for mor e details).

16 CHAPTER 1. INTR ODUCTION
Loosely speaking, the underlying DDE is obtained b y differentiating (and possibly shifting) p arts
of the DDAE
(1.13)
until one is able to solve for
˙
x
. H er eby , the number of differentiations that is
r equired during this process is used as a classification for the difficulties associated with solving the
DDAE
(1.13)
analytically or numerically . Differ ent technical aspects and different ways of counting
have led to sever al so-called index concepts for DAEs, for instance the differ entiation index [54],
the perturbation index [101], the tr actability index [144
–
146], the geometric index [179, 181] , the
structur al index [167, 174] and the strangeness inde x [124, 127]. F or a comparison we refer to [153].
Besides the fact that neutr al or even advanced equations may be hidden in the DDAE
(1.13)
, the
solution of the DDAE
(1.13)
may also depend on futur e evaluations of the DDAE. M ore pr ecisely , the
solution x ( t ) at the time point t may depend on
0 = F ( t + k τ , x ( t + k τ ), ˙
x ( t + k τ ), x ( t + ( k − 1) τ ), u ( t + k τ ))
with
k ∈ N
. A simple example for such a situation is given if we choose
E = A 1 =
0, and
A 2 = I n x
in
(1.15)
. Of course, in r eality , a dependence on the future is not possible , and theref or e, one may
question the utility of a DDAE whose solution depends on futur e values. H o wever , besides its
mathematical r elevance, the futur e evaluation of
F
may be interpr eted as a prediction of that futur e
value . In any case , the potential acausality causes some fur ther difficulties in the analytical and
numerical treatment of (1.13):
•
The method of steps (see [26] and the forthcoming chapter 2), which is commonly used to
solve the IVP
(1.13)
,
(1.17)
, cannot be used without pre-pr ocessing of the DDAE
(1.13)
. The
pr e-processing requir es to shift some of the equations of
(1.13)
to futur e time points [1, 53, 99].
The minimal number of shifts that is r equired to construct the solution is called the shift index .
W e refer to [98] for a pr ecise definition.
•
Due to the combination of differ entiation and shifting the DDAE
(1.13)
may include higher -
or der differential equations [53, 97, 206].
•
The shifting imposes r estr ictions on the set of history functions for which the IVP has a
solution. I n contrast to the DAE theor y , wher e we expect a restriction only at the time points
t = 0 and t = − τ , the restriction applies to all t ∈ [ − τ , 0].
So far , a general analysis of
(1.13)
is not available and most of the liter atur e addresses only special
cases . F or instance, a classical solution theory for DDAEs that need not be shifted is developed
in [13, 62] for nonlinear DDAEs with a special structure and for linear time-invariant DDAEs in [50].
Shifting and its consequences ar e studied in [1, 53, 94, 96
–
99, 17 3, 206]. N umer ical time integration
methods ar e developed and analyzed in [16, 33, 59, 90, 91, 98, 103, 108, 109, 136, 194, 201]. M ost of
the r eferences for the numerical methods r equire that ther e is no need to shift equations and that
the DAE that is obtained b y substituting a smooth function parameter
λ
for the delayed variable
has differ entiation index one. N otable exceptions ar e pro vided for instance in [13, 16, 98, 103]. The
stability and asymptotic stability of certain classes of DDAEs is studied in [ 41, 56, 69, 79, 95 , 135,
138, 141, 151, 156, 222, 224, 225 ]. S urpr isingly , it is not sufficient for asymptotic stability that all
eigenvalues of the L TI system
(1.15)
have negative r eal part [68]. The main r eason for this is that
the solution fails to exists after some time . This is due to the fact that an advanced equation may

1.4. PREVIOUSL Y PUBLISHED RESUL TS AND JOINT WORK 17
be hidden in the DDAE
(1.15)
, see the discussion abo ve . Closely related to stability is the question
wether a system can be stabilized via a suitable contr oller . This and further control theor etical topics
ar e discussed i n [3, 80, 81, 92, 157, 193, 194].
1.4 Pr eviously published r esults and j oint wor k
Some of the contents of this t hesis ar e already published.
•
The connection between the semi-explicit E u ler discr etization applied to linear poroelasticity
and a suitable DDAE, as pr esented in section 1.1.5, is joint work with R. Altmann and R. M aier
and published in [4]. An extension to nonlinear poroelasticity is consider ed in [5].
•
The analysis of linear DDAEs within the space of piecewise-smooth distributions is a result
of a collaboration with S. T renn, which r esulted in the conference pr oceedings [206] and the
pr epr ints [207, 208]. The results ar e presented in chapter 3.
•
M ost of the r esults from chapter 4 ar e published in [211], which additionally contains parts of
section 2.1. The extension of these r esults to nonlinear DDAEs is co ver ed in chapter 5 and in
parts published in the preprint [212].
•
The r ealization theor y for DDAEs, discussed in chapter 7, is published in the journal ar ticles
[189] (together with P . Schulze) and [191] (with P . Schulze , C. Beattie , and S. Guger cin). The
extension to time-domain data, co ver ed in chapter 8, is based on the results of a collabor ation
with E. F osong and P . Schulze , which are published in th e pr epr int [77].
R esults on str uctural analysis for DDAEs [1] (together with I. Ahr ens), Kolmogor ov
n
-widths for
linear dynamical systems [213] (with S. Guger cin), model reduction for tr ansport problems [35]
(joint work with F . Black and P . Schulze), and model reduction for switched systems [190] (with
P . Schulze) ar e only briefly mentioned in this thesis.
1.5 N otation
The symbols
Z
,
Q
,
R
, and
C
denote the integers, the r ational numbers, the r eal numbers, and the
complex numbers, r espectively . The natural numbers ar e the positive integers and ar e denoted
with
N : = { n ∈ Z | n >
0
}
. F or a field
F
and natural numbers
n , m ∈ N
we denote the set of all
n × m
matrices o ver
F
b y
F n × m
. The r ing of polynomials with coefficients from a field
F
is denoted b y
F
[
s
] with
s
being the indeterminate. The polynomial ring
F
[
s
] is naturally embedded in the field of
rational functions , denoted by
F
(
s
). Ther efor e, we can also consider matrices with entries in the ring
F [ s ]. The set of n -dimensional nonsingular matrices o ver the field F is denoted with
GL n ( F ) : = { A ∈ F n × n | A nonsingular },
which together with the standar d multiplication for matrices for ms a group , the so-called general
linear group . The neutral element of
GL n
(
F
) is denoted with
I n
and the inverse of
A ∈ GL n
(
F
) is
denoted with
A − 1
. I n par ticular , we have
A A − 1 = A − 1 A = I n
. The
i
th column of
I n
, i.e . , the
i
th

18 CHAPTER 1. INTR ODUCTION
unit vector , is denoted b y
e i ∈ F n
. If
A
(
s
)
∈ GL n
(
F
[
s
]) and
A
(
s
)
− 1 ∈ GL n
(
F
[
s
]), then
A
(
s
) is called
unimodular and it is easy to see that
A
(
s
)
∈ F
[
s
]
n × n
is unimodular if and only if
det
(
A
(
s
)) is a nonzero
constant, i.e .,
det
(
A
(
s
))
∈ F \ {
0
}
. The rank of a matrix
A ∈ GL n
(
F
) is denoted with
rank F
(
A
) or simply
with
rank
(
A
) if the field
F
is clear from the context. F or polynomial matr ices
A
(
s
)
∈ F
[
s
]
n × m
we
adopt the notation from the liter ature and write
rank F [ s ]
(
A
(
s
))
: = rank F ( s )
(
A
(
s
)). The transpose and
conjugate transpose of a matrix A are denoted with A T and A H .

P art I
Classification and w ell-posedness
19

2
Differential-algeb raic equations and p relimina ry results
A standar d approach to solve a differ ential equation with delayed argument, such as the delay
differ ential-algebraic equation (DDAE)
0 = F ( t , x ( t ), ˙
x ( t ), x ( t − τ ), u ( t )) for t ∈ [ t 0 , t f ) (2.1a)
introduced in section 1.2 (cf. (1.13)) with initial condition
x ( t ) = φ ( t ) for t ∈ [ − τ , 0 ] (2.1b)
is via successive integration of
(2.1)
on the time inter vals [(
i −
1)
τ , i τ
), which is sometimes r eferred
to as the method of steps [98], see also [26, 50]. First we assume that
M
is the smallest integer such
that t f < M τ and introduce for i ∈ I : = {1, . . . , M } the functions
x [ i ] : [0, τ ] → F n x , t ↦→ x ( t + ( i − 1) τ ),
u [ i ] : [0, τ ] → F n u , t ↦→ u ( t + ( i − 1) τ ),
F [ i ] : [0, τ ] × D x × D ˙
x × D σ τ x × D u → F m , ( t , x , y , z , u ) ↦→ F ( t + ( i − 1) τ , x , y , z , u ),
x [0] : [0, τ ] → F n x , t ↦→ φ ( t − τ ).
(2.2)
Then for each i ∈ {1, . . . , M } we have to solve the differ ential-algebraic equation (DAE)
0 = F [ i ] ( t , x [ i ] ( t ), ˙
x [ i ] ( t ), x [ i − 1] ( t ), u [ i ] ( t )), t ∈ [0, τ ), (2.3a)
x [ i ] (0) = x [ i − 1] ( τ − ), (2.3b)
with right continuation
x [ i − 1] ( τ − ) : = lim
t ↗ τ x [ i − 1] ( t ). (2.4)
The analysis of DDAEs r equir e s an in-depth understanding of the theory of DAE, which we thus
r ecall in this chapter . F or the analysis of
(2.3)
we emplo y the follo w ing solution concept from [127].
Definition 2.1.
A function
x [ i ] ∈ C 1
([0
, τ
];
F n x
) is called a (classical) solution of
(2.3a)
, if it satisfies
(2.3a)
pointwise . The function
x [ i ] ∈ C 1
([0
, τ
];
F n x
) is called a (classical) solution of the initial value
problem
(2.3)
if it is a solution of
(2.3a)
and satisfies
(2.3b)
. An initial condition
x [ i − 1]
(
τ
) is called
consistent , if the initial value problem (2.3) has at least one solution.
21

22 CHAPTER 2. DIFFERENTIAL-ALGEBRAIC EQ U A TIONS AND PRELIMINAR Y RESUL TS
N ote that the partial derivative of
F [ i ]
with r espect to
x [ i ]
in
(2.3a)
is allo wed to be singular , resulting
in significant differ ences to the theor y for or di nary diff er ential equations (ODEs), see al so [171]. The
main differ ences are illustrated in the next example , taken from [205].
Example 2.2. C onsider the linear time-invariant (L TI) DAE
⎡
⎢
⎣
0 1 0
0 0 0
0 0 0 ⎤
⎥
⎦ ⎡
⎢
⎣
˙
x 1 ( t )
˙
x 2 ( t )
˙
x 3 ( t ) ⎤
⎥
⎦ = ⎡
⎢
⎣
1 0 0
0 1 0
0 0 0 ⎤
⎥
⎦ ⎡
⎢
⎣
x 1 ( t )
x 2 ( t )
x 3 ( t ) ⎤
⎥
⎦ + ⎡
⎢
⎣
f 1 ( t )
f 2 ( t )
f 3 ( t ) ⎤
⎥
⎦ .
The thir d equ ation implies
f 3 ≡
0, which means that even for arbitra rily smooth ri ght-hand sides,
we may not have existence of solutions . The second equation
x 2
(
t
)
= − f 2
(
t
) sho ws that the set of
initial conditions is r estr icted. U sing the second equation we obtain the solution
x 1 ( t ) = ˙
x 2 ( t ) − f 1 ( t ) = − ˙
f 2 ( t ) − f 1 ( t )
for the first variable , which is thus less smooth than the inhomogeneity
f
. Note that
x 3
is not
specified at all and thus we do not have uniqueness of the solution. ♠
S ince a solution of the DAE
(2.3a)
(and hence also a solution of the initial value problem (IVP)
(2.1)
)
may depend on derivatives of
F [ i ]
and consequently on derivatives of the history function
φ
, we
make the follo wing assumption throughout the text.
Assumption 2.3.
The function
F [ i ]
in
(2.3a)
and the histor y function
φ
(and thus also the function
F in (2.1a) ) ar e sufficiently smooth.
R emark 2.4.
The theor y of DAEs is alr eady quite mature with a large body of liter ature , see for
instance the monographs [42, 101, 127, 134, 197]. F ur fur ther details we r efer to the collection of
sur vey articles [49, 111–114, 188] and the r eferences ther ein. ♣
2.1 Classical solutions for linear time-invariant DAE s
As pointed out in Example 2.2, many aspects of the theory for DAEs are alr eady pr esent for L TI DAEs
and hence we start our brief sur vey of DAE theor y with linear systems of the form
E ˙
x = A 1 x + ˜
f , (2.5)
with
E , A 1 ∈ F m × n x
and
˜
f
: [0
, t f
)
→ F m
, wher e we omit the time dependency of
x
and
f
for the ease
of pr esentation. As before , we frequently make the assumption th at
˜
f
is smooth enough, i.e . that
˜
f
satisfies the follo wing assumption.
Assumption 2.5. The inhomogeneity ˜
f is infinitely many times continuously differ entiable.
R emark 2.6.
A pplying the method of steps to the DDAE
(1.15)
r esults in the DAE
(2.5)
with
x = x [ i ]
and
˜
f = A 2 x [ i − 1] + f
. In particular , the inhomogeneity
˜
f
depends on the solution on the pr evious
inter val and hence Assumption 2.3 does not imply Assumption 2.5. ♣

2.1. CL ASSICAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DAES 23
The solvability of
(2.5)
is closely r elated to the properties of the matrix pair (
E , A 1
), see for in-
stance [127] for further details. If
m = n x
and
det
(
λ E − A 1
)
∈ F
[
λ
]
\ {
0
}
, then the matrix pair (
E , A 1
) is
called r egular . If a matrix pair is not r egular , it is referr ed to as singular . The follo wing result [127, The-
or em 2.14] sho ws that the r egular ity of the matrix pair is a necessar y condition to ensur e existence
and uniqueness of solutions for the DAE (2.5), see also [32].
Theor em 2.7. Let E , A ∈ C m × n x and su ppose that ( E , A 1 ) is a singular matrix pair .
(i) If r ank( λ E − A 1 ) < n x for all λ ∈ C , then the homogeneous initial value problem
E ˙
x = A 1 x , x (0) = 0
has a nontrivial solution.
(ii)
If
rank
(
λ E − A 1
)
= n
for some
λ ∈ C
and hence
m > n x
, then ther e exist arbitrarily smooth
inhomogeneities ˜
f for which the corr esponding DAE is not solvable.
R emark 2.8.
If (
E , A 1
) is not r egular , it is still possible that the IVP associated with the DDAE
(1.15)
has a unique solution (in the sense of [98]). In this case , the DDAE is called noncausal and under
some technical assumptions [98] pro vides an algor ithm to transform
(3.1a)
such that the trans-
formed pencil ( ˜
E , ˜
A ) is r egular . H o wever , such a process adds additional r estrictions on the histor y
function. F or more details, w e refer to [52, 97] and the for thcoming section 3.1. ♣
A standar d approach in studying linear differential equations is to intr oduce an equivalence r elation
on the system matrices that allo ws to characterize all solutions . In terms of matrix pencils, w e say
that (
E , A 1
) and (
˜︁
E , ˜︁
A 1
) ar e (strongly) equivalent , in symbols (
E , A 1
)
∼
(
˜︁
E , ˜︁
A 1
), if and only if there
exists nonsingular matrices S ∈ GL m ( F ) and T ∈ GL n x ( F ) such that
˜︁
E = S E T and ˜︁
A 1 = S A 1 T .
The canonical form for this equivalence r elation is the Kronecker canonical form [82, Cha. XII, § 4]
(assuming
F = C
). From the canonica l form it is easy to determine whether the matr ix pencil is
r egular , yielding a special form of the Kronecker canonical form that is kno wn as the W eierstraß
canonical form [82, Cha. XII, Thm. 3]. M ore pr ecisely , we have the follo wing characterization of
r egular ity (which is sometimes r eferred to as the quasi-W eierstraß form [31]).
Theor em 2.9
(Quasi-W eierstraß form)
.
The matrix pencil (
E , A 1
)
∈ (︁ F m × n x )︁ 2
is r egular if and only
if m = n x and there e xist matrices S , T ∈ GL n x ( F ) such that
S E T = [︄ I n x ,d 0
0 N ]︄ and S A 1 T = [︄ J 0
0 I n x ,a ]︄ , (2.6)
wher e N ∈ F n x ,a × n x ,a is a nilpotent matrix and J ∈ F n x ,d × n x ,d .
R emark 2.10.
If
F = C
, then the W eierstraß canonical form can be obtained b y choosing
S
and
T
in
Theor em 2.9 such that
N
and
J
ar e in J or dan canonical for m. If
F = C
, then the J ordan canonical form

24 CHAPTER 2. DIFFERENTIAL-ALGEBRAIC EQ U A TIONS AND PRELIMINAR Y RESUL TS
of
J
may not exist and we cannot use the W eierstraß canonical form. F or the analysis of the DAE
(2.5)
this is not important since the relev ant feature of the (quasi-)W eierstraß form is the decoupling
of the matrix pencil (see the forthcoming discussion). ♣
N ote that the numbers
n x ,d
and
n x ,a
in Theor em 2.9 are independent of the specific choice of the
matrices
S
and
T
(see for instance [31, 127]). In mor e detail, consider matrices
S i , T i ∈ GL n x
(
F
) for
i = 1, 2 that transform the r egular pencil ( E , A 1 ) into quasi-W eierstraß form, that is
S i E T i = [︄ I n i 0
0 N i ]︄ and S i A 1 T i = [︄ J i 0
0 I ˜
n i ]︄ for i = 1, 2,
wher e for i = 1, 2 the matr ix N i ∈ F ˜
n i × ˜
n i is nilpotent and J i ∈ F n i × n i . Then
det( λ E − A 1 ) = det( S i ) − 1 det (︄ λ [︄ I n i 0
0 N i ]︄ − [︄ J i 0
0 I ˜
n i ]︄)︄ det( T i ) − 1
= det( S i ) − 1 det( T i ) − 1 det( λ I n i − J i ) det( λ N i − I ˜
n i ) for i = 1, 2.
S ince
N i
is nilpotent, we have
det
(
λ N i − I ˜
n i
)
=
(
−
1)
˜
n i
for any
λ ∈ F
and
i =
1
,
2. With the setting
c i : = ( − 1) ˜
n i det( S i ) − 1 det( T i ) − 1 ∈ F \ {0} we obtain
det( λ E − A 1 ) = c i det( λ I n i − J i )
and thus n 1 = n 2 and ˜
n 1 = n − n 1 = n − n 2 = ˜
n 2 .
The matrices S and T in Theorem 2.9 can be obtained fr om the so-called W ong sequences [220]
V 0 : = F n x , V i + 1 : = A − 1
1 ( E V i ) : = { x ∈ F n | A 1 x ∈ E V i }, for i ∈ N , (2.7a)
W 0 : = {0}, W i + 1 : = E − 1 ( A 1 W i ) : = { x ∈ F n | E x ∈ A 1 W i }, for i ∈ N , (2.7b)
wher e i n this context
E − 1
and
A − 1
1
denote the pr eimage of
E
and
A 1
, respectively . N ote that the
sequences ar e nested, i.e., V i + 1 ⊆ V i and W i ⊆ W i + 1 and thus ther e exists a number k ∈ N such that
V : = V k = V k + j and W : = W k = W k + j for all j ∈ N . (2.8)
F oll o wi ng [31], the regularity of (
E , A 1
) implies
dim
(
V
)
= n x ,d
and
dim
(
W
)
= n x ,a
and for any matrices
V ∈ F n x × n x ,d and W ∈ F n x × n x ,a that satisfy im( V ) = V and im( W ) = W , the matrices
S : = [︂ E V A 1 W ]︂ − 1 and T : = [︂ V W ]︂ (2.9)
transform (
E , A
) into quasi-W eierstraß form
(2.6)
. As a consequence (cf. [31, R emark 2.7]), we obtain
A 1 V = E V J and E W = A 1 W N . (2.10)
The next r esult sho ws that the converse direction is also true , i.e., that if
S , T ∈ GL n x
(
F
) transform the
matrix pencil (
E , A 1
) into quasi-W eierstraß form, then
S , T
ar e of the form
(2.9)
with
im
(
V
)
= V
and
im( W ) = W .

2.1. CL ASSICAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DAES 25
Pr oposition 2.11.
Consider a r egular matrix pencil (
E , A 1
) and matrices
S i , T i ∈ GL n x
(
F
) for
i = 1, 2 that tr ansform ( E , A 1 ) into quasi-W eierstraß form, that is
S i E T i = [︄ I n x ,d 0
0 N i ]︄ and S i A 1 T i = [︄ J i 0
0 I n x ,a ]︄ for i = 1, 2, (2.11)
wher e
N 1 , N 2 ∈ F n x ,a × n x ,a
ar e nilpotent and
J 1 , J 2 ∈ F n x ,d × n x ,d
. Then ther e exist matrices
P ∈ GL n x ,d
(
F
)
and Q ∈ GL n x ,a ( F ) such that
S 2 = [︄ P − 1 0
0 Q − 1 ]︄ S 1 , T 2 = T 1 [︄ P 0
0 Q ]︄ , J 2 = P − 1 J 1 P , and N 2 = Q − 1 N 1 Q .
Proof.
P artition
S i = [︂ X i Y i ]︂ − 1
and
T i = [︂ V i W i ]︂
with
X i , V i ∈ F n x × n x ,d
and
Y i , W i ∈ F n x × n x ,a
. W e
obser ve for i = 1, 2
[︂ E V i E W i ]︂ = E T i = S − 1
i [︄ I n x ,d 0
0 N i ]︄ = [︂ X i Y i ]︂ [︄ I n x ,d 0
0 N i ]︄ = [︂ X i Y i N i ]︂ and
[︂ A 1 V i A 1 W i ]︂ = A 1 T i = S − 1
i [︄ J i 0
0 I n x ,a ]︄ = [︂ X i Y i ]︂ [︄ J i 0
0 I n x ,a ]︄ = [︂ X i J i Y i ]︂
and ther efore
S − 1
i = [︂ E V i A 1 W i ]︂
and
A 1 V i = E V i J i
. W e immediately obser ve
A 1 im
(
V i
)
⊆ E im
(
V i
).
Thus [31, Proposition 2.13] together with
n x ,d = dim
(
im
(
V i
)) implies
im
(
V 1
)
= im
(
V 2
). Hen ce ther e
exists a matrix
P ∈ GL n x ,d
(
F
) with
V 2 = V 1 P
. S ince
T 1
is nonsingular , there exist matrices
˜︁
P ∈ F n x × n x ,d
and Q ∈ F n x × n x ,a with
T 2 = T 1 [︄ P ˜︁
P
0 Q ]︄ and S − 1
2 = S − 1
1 [︄ P ˜︁
P
0 Q ]︄ .
I n par ticular , we have Q ∈ GL n x ,a ( F ). W e obtain
S − 1
1 [︄ P ˜︁
P N 2
0 Q N 2 ]︄ = S − 1
1 [︄ P ˜︁
P
0 Q ]︄ [︄ I n x ,d 0
0 N 2 ]︄ = S − 1
2 [︄ I n x ,d 0
0 N 2 ]︄ = E T 2
= E T 1 [︄ P ˜︁
P
0 Q ]︄ = S − 1
1 [︄ I n x ,d 0
0 N 1 ]︄ [︄ P ˜︁
P
0 Q ]︄ = S − 1
1 [︄ P ˜︁
P
() N 1 Q ]︄ .
H ence
˜︁
P = ˜︁
P N 2
and
N 2 = Q − 1 N 1 Q
. As a consequence of the first equation and the fact that
N 2
is
nilpotent, we deduce ˜︁
P = 0. I t remains to sho w that J 2 = P − 1 J 1 P holds . This follows fr om
S − 1
1 [︄ P J 2 0
0 Q ]︄ = S − 2
2 [︄ J 2 0
0 I n x ,a ]︄ = E T 2 = E T 1 [︄ P 0
0 Q ]︄
= S − 1
1 [︄ J 1 0
0 I n x ,a ]︄ [︄ P 0
0 Q ]︄ = S − 1
1 [︄ J 1 P 0
0 Q ]︄ . ■
As a consequence of Pr oposition 2.11, the index of nilpotency of
N
in the quasi-W eierstraß form
(4.5)
is independent of the choice of the matrices
S
and
T
, which motivates the follo wing definition
(cf. [127]).

26 CHAPTER 2. DIFFERENTIAL-ALGEBRAIC EQ U A TIONS AND PRELIMINAR Y RESUL TS
Definition 2.12
(I ndex of a regular matrix pencil)
.
Let (
E , A 1
) be a r egular matr ix pencil and let
N ∈ F n x ,a × n x ,a
denote the nilpotent matrix with index of nilpotency
ν
of the quasi-W eierstraß form
from Theor em 2.9. Then the number
ind( E , A 1 ) : = ⎧
⎨
⎩
ν , if n x ,a > 0,
0, other wise,
is called the index of the pencil ( E , A 1 ).
Theor em 2.9 allows us to decouple the DAE
(2.5)
. I n more detail, let
S , T ∈ GL n x
(
F
) be matrices that
transform the pencil (
E , A 1
) in quasi-W eierstraß form
(2.6)
. Sinc e both matrices are nonsingular , we
obtain a one-to-one corr espondence between solutions of (2.3a) and solutions of
˙
v = J v + ˜
g , (2.12a)
N ˙
w = w + ˜
h , (2.12b)
with [︄ v
w ]︄ : = T − 1 x and [︄ ˜
g
˜
h ]︄ : = S ˜
f .
While
(2.12a)
is a standar d ordinary differential equation (ODE) in
v
that can be solved with the
Duhamel integr al, the so-called fast subsystem (2.12b) has the solution
w = −
ν − 1
∑︂
k = 0
N k ˜
h ( k ) (2.13)
and hence the function
˜
h
must be
ν
times continuously differ entiable for a classical solution to exist
(cf. [127]). I n addition, a consistent initial value
w
(0) must satisfy equation
(2.13)
. Similar to [200],
we define the matrices
A diff : = T [︄ J 0
0 0 ]︄ T − 1 , A con : = T [︄ I n x ,d 0
0 0 ]︄ T − 1 ,
C 0 : = T [︄ I n x ,d 0
0 0 ]︄ S , C k : = − T [︄ 0 0
0 N k − 1 ]︄ S
(2.14)
for
k =
1
, . . . , ind
(
E , A 1
). As a consequence of Pr oposition 2.11 we notice that the matrices defined in
(2.14) do not depend on the choice of the matrices S and T , see also [205].
Lemma 2.13. Assume that the matrix pencil ( E , A 1 ) is regular . Then the matrices A diff , A con , and
C k
for
k =
0
,
1
, . . . , ind
(
E , A 1
) defined in
(2.14)
do not depend on the matrices
S , T
that tr ansfor m
( E , A 1 ) into quasi-W eierstr aß form (2. 6) .
Proof.
Consider matrices
S i , T i ∈ GL n x
(
F
) for
i =
1
,
2 that transform (
E , A 1
) into quasi-W eierstraß
form, i.e., that satisfy
(2.11)
. Accor ding to Proposition 2.11 ther e exist matrices
P ∈ GL n x ,d
(
F
) and
Q ∈ GL n x ,a ( F ) such that
S 2 = [︄ P − 1 0
0 Q − 1 ]︄ S 1 , T 2 = T 1 [︄ P 0
0 Q ]︄ , J 2 = P − 1 J 1 P , and N 2 = Q − 1 N 1 Q .

2.1. CL ASSICAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DAES 27
Thus
T 2 [︄ J 2 0
0 0 ]︄ T − 1
2 = T 1 [︄ P 0
0 Q ]︄ [︄ J 2 0
0 0 ]︄ [︄ P − 1 0
0 Q − 1 ]︄ T − 1
1 = T 1 [︄ P J 2 P − 1 0
0 0 ]︄ T − 1
1 = T 1 [︄ J 1 0
0 0 ]︄ T − 1
1 .
The proof for the other matrices follo ws similarly . ■
Pr oposition 2.14.
Assume that the mat rix pair (
E , A 1
) is r egular and
˜
f
satisfies Assu mption 2.5.
Then any classical solution x of (2.5) fullfills the so called underlying OD E
˙
x = A diff x +
ind( E , A 1 )
∑︂
k = 0
C k ˜
f ( k ) . (2.15)
Conversely , let
x
be a classical solution of
(2.15)
. Then
x
is a solution of
(2.5)
if and only if ther e
exists s ∈ [0, t f ) such t hat x ( s ) satisfies
x ( s ) = A con x ( s ) +
ind( E , A 1 )
∑︂
k = 1
C k ˜
f ( k − 1) ( s ). (2.16)
I n this case (2.16) is true for all s ∈ [0, t f ) .
Proof.
Let
x
be a classical solution of
(2.5)
and
S , T ∈ GL n x
(
F
) be matrices that satisfy
(2.6)
of the
quasi-W eierstraß form and set ν : = ind( E , A 1 ). Differ entiation of (2.13) yields
˙
x = T [︄ ˙
v
˙
w ]︄ = T [︄ J v + ˜
g
− ∑︁ ν − 1
k = 0 N k ˜
h ( k + 1) ]︄
= T [︄ J 0
0 0 ]︄ [︄ v
w ]︄ + T [︄ I n x ,d 0
0 0 ]︄ [︄ ˜
g
˜
h ]︄ −
ν
∑︂
k = 1
T [︄ 0 0
0 N k − 1 ]︄ [︄ ˜
g ( k )
˜
h ( k ) ]︄
= A diff x +
ν
∑︂
k = 0
C k ˜
f ( k ) .
Conv er sely , let x be a classical solution of (2.15). Then for any s ∈ [0, t f ) we have
x ( t ) = e A diff ( t − s ) x ( s ) + ∫︂ t
s
e A diff ( t − s − ˜
t ) ν
∑︂
k = 0
C k ˜
f ( k ) (︁ ˜
t )︁ d ˜
t . (2.17)
Scaling (2.17) from the left b y T − 1 we obtain
w ( t ) = w ( s ) −
ν
∑︂
k = 1
N k − 1 ∫︂ t
s
˜
h ( k ) (︁ ˜
t )︁ d ˜
t = w ( s ) −
ν − 1
∑︂
k = 0
N k ˜
h ( k ) ( t ) +
ν − 1
∑︂
k = 0
N k ˜
h ( k ) ( s ).
S uppose no w that for a specific
s ∈
[0
, t f
) the solution
x
satisfies
(2.16)
, or equivalently (by scal-
ing (2.16) from the left with T − 1 )
[︄ v ( s )
w ( s ) ]︄ = [︄ v ( s )
− ∑︁ ν − 1
k = 0 N k ˜
h ( k ) ( s ) ]︄ .
T ogether with
(2.13)
this implies that
x
is a solution of
(2.5)
and it is easy to see that
(2.16)
is satisfied
for all t ∈ [0, t f ). The r emaining direction follo ws immediately from (2.13). ■

28 CHAPTER 2. DIFFERENTIAL-ALGEBRAIC EQ U A TIONS AND PRELIMINAR Y RESUL TS
Setting
s =
0 in the pr evious proposition yields the follo wing requir ement for an initial condition to
be consistent.
Cor ollar y 2.15.
Assume t hat the matrix pair (
E , A 1
) is r egular and the inhomogeneity
˜
f
satisfies
Assumpti on 2.5. Then the initial value
x
(0) is consistent if and only if it satisfies the consistency
condition
x (0) = A con x (0) +
ind( E , A 1 )
∑︂
k = 1
C k ˜
f ( k − 1) (0). (2.18)
I n this case, the IVP (2.3) has a unique solution x ∈ C ∞ ([0, t f ); F n x ) .
2.2 Distributional solutions for inconsistent initial values
As we have clearly seen in the pr evious subsection, a classical solution does not exist for all initial
conditions . Instead, a consistent initial condition has to satisfy the consistency condition
(2.18)
in
Cor oll ary 2.1 5. If the DAE under investigation is a result of applying the method of steps
(2.3)
to
the DDAE
(1.13)
, then it is a–prior i not clear , if the initial value is consistent. U nfor tunately , one
cannot sho w that the initial value is always consistent. W e have alr eady seen a counterexample in
Example 1.5. Thus, a gener al solution framework for DDAEs has to be able to deal with inconsistent
initial values .
I nconsistent initial values appear also in other application areas , for instance when an electr ical
cir cuit is switched at a cer tain time [215]. As a consequence, a number of differ ent approaches in
the time and fr equency domain have been proposed to deal with inconsistent initial values . F or an
o ver view we r efer to [205]. All approaches have in common, that jumps or even Dir ac impulses may
occur in the solution, and hence a distributional solution space seems appropriate. F ollo wing [192],
the space of test function
C ∞
0 ( R ; R ) : = { f ∈ C ∞ ( R ; R ) | supp f is bounded}
with
supp
(
f
)
: = { x ∈ R | f ( x ) = 0}
, can be equipped with a locally convex topology (see [117, § 12]),
thus making it a topological space . The set of all linear and continuous maps from
C ∞
0
(
R
;
R
) into
the r eal numbers
D : = { f : C ∞
0 ( R ; R ) → R | f is linear and continuous}, (2.19)
i.e ., the topological dual space of
C ∞
0
(
R
;
R
), is called the space of distributions . Since the test
functions ar e smooth and have compact support, we can define for any locally integrable function
f ∈ L 1,loc a distribution via
f D : C ∞
0 ( R ; R ) → R , ϕ ↦→ ∫︂ ∞
−∞
f ( t ) ϕ ( t )d t .
Consequently , the space L 1,loc can be embedded into D via the injective homomorphism
L 1,loc → D , f ↦→ f D . (2.20)

2.2. DISTRIBUTIONAL SOL UTIONS FOR INCONSISTENT INITIAL V AL UES 29
T o descr ibe the DAE
(2.5)
in a (yet to define) suitable distributional solution space , we need a
derivative in D . F ollowing [117, § 1 9], we define the distributional derivative
d
d t : D → D , f ↦→ (︃ d
d t f : C ∞
0 ( R ; R ) → R , ϕ ↦→ − f (︃ d
d t ϕ )︃)︃ . (2 .21)
S ince
d
d t ϕ ∈ C ∞
0
(
R
;
R
), this is indeed well-defined (see [117, S atz 19.1] for mor e details) and we im-
mediately obser ve , that distr ibutions ar e arbitrarily often differentiable . F or notational convenience ,
we write
˙
f : = d
d t f and f ( k ) : = (︃ d
d t )︃ k
f for f ∈ D .
N ote that we use the symbol
d
d t
for the distributional derivative and the standar d der ivative . This is
consistent, since for any differ entiable (and thus locally integrable) function f : R → R , we have
(︃ d
d t f )︃ D = d
d t (︁ f D )︁ .
Example 2.16. C onsider for Ω ⊆ R the indicator function
1 Ω : R → R , t ↦→ ⎧
⎨
⎩
1, if t ∈ Ω ,
0, other wise.
F or any s ∈ R and ϕ ∈ C ∞
0 ( R ; R ) we obtain
(︃ d
d t (︁(︁ 1 [ s , ∞ ) )︁ D )︁ )︃ ( ϕ ) = − ∫︂ ∞
−∞
1 [ s , ∞ ) ( t ) d
d t ϕ ( t )d t = ϕ ( s ).
The distribution δ s : = d
d t (︁(︁ 1 [ s , ∞ ) )︁ D )︁ is called the Dirac impulse at s . ♠
A generalization of D to vector -valued functions is straightforward, b y defining
D k : = {︃ f = [︂ f 1 . . . f k ]︂ T     f i ∈ D for i = 1, . . . , k }︃
for any k ∈ N . The multiplication of f ∈ D k with a matr ix M ∈ R p × k is then defined via
M f : C ∞
0 ( R ; R ) → R , ϕ ↦→ M f ( ϕ ),
such that the DAE
(2.5)
can be interpr eted as an equation in
D m
with
x ∈ D n x
and
˜
f ∈ D m
. H o wever ,
embedding the DAE into a distributional framework does not r esolve the issue of inconsistent initial
conditions, since we can not evaluate distributions at a point
t 0 ∈ R
. But even if w e restrict the space
of distributions such that the pointwise evalutation at certain points is well-defined, one can sho w ,
see [205], that solutions do not exist for arbitrary initial values . F or instance, th e trivial DAE
x =
0
posses the unique solution x = 0 also in the distributional sense.
I nstead, we assume that the DAE
(2.5)
only holds on [0
, ∞
) (instead of the r eal axis) and the past,
i.e ., the behavior in the interval (
∞ ,
0), is prescribed as an initial trajectory . H o wever , this r e-
quir es u s to define a distributional r estriction to the inter val [0
, ∞
) and this is not possible for

30 CHAPTER 2. DIFFERENTIAL-ALGEBRAIC EQ U A TIONS AND PRELIMINAR Y RESUL TS
general distributions [203, Lemma 2.2.3]. This problem can be r esolved b y consider ing the space of
impulsive-smooth distributions [83] or b y the slightly bigger space of piecewise-smooth distribu-
tions [203]. The latter is also suitable for studying the DDAE
(1.15)
, ther efore we will use this space
in the follo wing as the underlying solution space for (2.5) and (1.15).
Definition 2.17 (Piecwise-smooth distributions) .
(i)
A function
α : R → R
is called piecewise-smooth if, and only if, there exists a family of r eal
numbers
{ t i ∈ R | i ∈ Z }
with
t i < t i + 1
for all
i ∈ Z
and
t ± k → ±∞
as
k → ∞
and smooth
functions α i ∈ C ∞ ( R ; R ) such that
α = ∑︂
i ∈ Z
1 [ t i , t i + 1 ) α i .
The space of piecewise-smooth functions is defined as
C ∞
p w ( R ; R ) : = { α : R → R | α is piecewise-smooth}.
(ii) The space of piecewise-smooth distributions is defined as
D p w C ∞ : = {︄ α D + ∑︂
s ∈ S
D s     
α ∈ C ∞
p w ( R ; R ), S is a discr ete set, and
D s ∈ span{ δ s , ˙
δ s , ¨
δ s , . . .} for s ∈ S }︄ ,
i.e ., a piecewise- smooth distribution is the sum of a piecewise-smooth function and linear
combinations of Dirac impulses and their derivativ es at finitely many time instants in each
compact inter val.
F or piecewise-smooth distributions the restriction
D p w C ∞ × P ( R ) → D pw C ∞ , ( f = α D + ∑︂
s ∈ S
D s , Ω ) ↦→ f Ω : = ( 1 Ω α ) D + ∑︂
s ∈ S ∩ Ω
D s
is well defined. M or eov er , one can sho w (cf. [203, 204]) that each piecewise-smooth distri bution
f ∈ D p w C ∞
posses a derivative and an anti-derivative in
D p w C ∞
. It is important to note that the
r estr iction operator and the distributional derivative operator do not commute . Instead, we have
the follo wing result.
Lemma 2.18
( [204, Proposition 12])
.
F or all
−∞ ≤ t 1 ≤ t 2 ≤ ∞
and
f = α D + ∑︁ s ∈ S D s ∈ D p w C ∞
we have
d
d t (︂ f [ t 1 , t 2 ) )︂ = (︃ d
d t f )︃ [ t 1 , t 2 ) + f ( t −
1 ) δ t 1 − f ( t −
2 ) δ t 2 ,
d
d t (︂ f ( t 1 , t 2 ) )︂ = (︃ d
d t f )︃ ( t 1 , t 2 ) + f ( t +
1 ) δ t 1 − f ( t −
2 ) δ t 2 ,
d
d t (︂ f ( t 1 , t 2 ] )︂ = (︃ d
d t f )︃ [ t 1 , t 2 ) + f ( t +
1 ) δ t 1 − f ( t +
2 ) δ t 2 ,
d
d t (︂ f [ t 1 , t 2 ] )︂ = (︃ d
d t f )︃ [ t 1 , t 2 ] + f ( t −
1 ) δ t 1 − f ( t +
2 ) δ t 2 ,

2.2. DISTRIBUTIONAL SOL UTIONS FOR INCONSISTENT INITIAL V AL UES 31
wher e δ ±∞ = 0 and the left and right sided evaluation at a point t ∈ R ar e defined as
f ( t − ) : = lim
h ↘ 0 α ( t − h ) and f ( t + ) : = lim
h ↘ 0 α ( t + h ) = α ( t ).
F or further results on the space of piecewise-smooth distributions and its r elation to other distr i-
butional solution concepts for DAEs we r efer to [203, 204]. It is no w possible to state an existence
and uniqueness r esult for regular DAEs with possible inconsistent initial values . M ore pr ecisely , we
can interpr et the DAE
(2.5)
in the space of piecewise-smooth distributions as the initial trajectory
problem (ITP)
x ( −∞ , t 0 ) = x 0
( −∞ , t 0 ) ,
( E ˙
x ) [ t 0 , ∞ ) = ( A 1 x + ˜
f ) [ t 0 , ∞ ) , (2.22)
with initial trajectory x 0 ∈ D n x
p w C ∞ and inhomogeneity ˜
f ∈ D m
p w C ∞ and arbitrar y t 0 ∈ R .
Theor em 2.19
( [203, Theor em 3.5 .2])
.
Consider the ITP
(2.22)
with initial tr ajector y
x 0 ∈ D n x
p w C ∞
and inhomogeneity
˜
f ∈ D m
p w C ∞
. If the matrix pair (
E , A 1
) is r egular , then the ITP
(2.22)
has a
unique solution x ∈ D n x
p w C ∞ .
I n the study of DDAE s it turns out that even in the L TI setting, the DDAE
(1.15)
may contain higher -
or der differential equations and thus it is important to study also higher -order DAEs. T o this end,
consider a polynomial matrix P ( s ) ∈ R [ s ] m × n x , i.e.,
P ( s ) =
p
∑︂
j = 0
P j s j with matrices P i ∈ F m × n x for j = 0, 1, . . . , p .
F or a given polynomial matrix
P
(
s
)
∈ R
[
s
]
m × n x
we consider the gener alization of the DAE
(2.5)
given
b y the polynomial DAE
P (︃ d
d t )︃ x = f . (2.23)
N otice that the DAE
(2.5)
can be r ecast in the for m
(2.23)
b y introducing the polynomial matrix
P
(
s
)
= E s − A 1
. Vice versa, b y introducing new variables, we can easily r ecast the polynomial DAE
(2.23)
into the matrix form
(2.5)
and thus immediately obtain the follo wing r esu lt (see e .g. the second
part of the proof of Theor em 7 in [209] or the first par t of the proof of C orollar y 9 in [206]).
Cor ollar y 2.20.
F or given polynomial matrix
P
(
s
)
∈ R
[
s
]
n x × n x
, initial trajectory
x 0 ∈ D n x
p w C ∞
,
inhomogeneity ˜
f ∈ D m
p w C ∞ , and initial time point t 0 ∈ R consider the ITP
x ( −∞ , t 0 ) = x 0
( −∞ , t 0 ) ,
(︃ P (︃ d
d t )︃ x )︃ [ t 0 , ∞ ) = ( ˜
f ) [ t 0 , ∞ ) . (2.24)

32 CHAPTER 2. DIFFERENTIAL-ALGEBRAIC EQ U A TIONS AND PRELIMINAR Y RESUL TS
If det( P ( s )) ∈ R [ s ] \ {0} , then the ITP (2.24) has a uni que solution x ∈ D n x
p w C ∞ .
Proof.
Let
P
(
s
)
= ∑︁ p
j = 0 P j s j
. Then a standard companion form linearization of
(2.23)
yields the
DAE
E ˙
z = A z + F (2.25)
with E , A ∈ R p n x × p n x , given b y
E = ⎡
⎢
⎢
⎢
⎢
⎢
⎣
P p 0 · · · 0
0 I n
. . . .
.
.
.
.
. . . . . . . 0
0 · · · 0 I n
⎤
⎥
⎥
⎥
⎥
⎥
⎦
, A = ⎡
⎢
⎢
⎢
⎢
⎣
− P p − 1 − P p − 2 · · · − P 0
I n 0 · · · 0
.
.
. . . . . . . .
.
.
0 · · · I n 0
⎤
⎥
⎥
⎥
⎥
⎦ , z = ⎡
⎢
⎢
⎢
⎢
⎢
⎣
(︂ d
d t )︂ p − 1 x
.
.
.
d
d t x
x
⎤
⎥
⎥
⎥
⎥
⎥
⎦
, F = ⎡
⎢
⎢
⎢
⎢
⎣
f
0
.
.
.
0
⎤
⎥
⎥
⎥
⎥
⎦ .
N ote that ther e exists (cf. [142 ]) unimodular matrix polynomials R ( s ), S ( s ) ∈ R [ s ] p n x × p n x with
R ( s )( s E − A ) S ( s ) = [︄ P ( s ) 0
0 I ( p − 1) n ]︄ .
H ereb y , R and S ar e given as
R ( s ) = ⎡
⎢
⎢
⎢
⎢
⎣
I ℓ s P p + P p − 1 · · · ∑︁ p
j = 1 s j − 1 P j
− I n
. . .
− I n
⎤
⎥
⎥
⎥
⎥
⎦ , S ( s ) = ⎡
⎢
⎢
⎢
⎢
⎢
⎣
s p − 1 I n · · · s I n I n
.
.
. . . . . . .
s I n . . .
I n
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
The proof no w follows b y the obser vation that ther e exists a constant c = 0 with
0 ≡ det( P ( s )) = c det( s E − A )
and application of Theor em 2.19 to (2.25). ■
R emark 2.21.
The initial trajectory
x 0
( −∞ , t 0 )
in
(2.24)
not only specifies the state
x ( −∞ , t 0 )
but also
its (distributional) derivatives and thus pro viding the initial trajectories for the higher-or der differ -
ential operator
P (︂ d
d t )︂
in
(2.23)
. I t should be noted that in general the standar d companion for m
linearization in
(2.25)
may introduce additional smoothness r equirements for the for cing term
f
(cf. [154, 1 87]) and instead a so-called trimmed linearization [47] should be used if we consider a
classical solution concept. One of the main issues with higher-or der differ ential equations is that
ther e is no simple canonical for m under strong equivalence if deg P ≥ 2 [202]. ♣
Although in principle it is possible to rewrite the higher -order DAE
(2.23)
as a first-or der DAE by
introducing new variables, it is sometimes mor e efficient to work dir ectly with the higher-or der
system. Consequently , we need a generalization of Lemma 2.18 to higher -order differ ential operators .
F or simplicity we consider only the restriction to the time intervals (
−∞ ,
0) and [0
, ∞
). Let
f ∈ D n
p w C ∞
.

2.2. DISTRIBUTIONAL SOL UTIONS FOR INCONSISTENT INITIAL V AL UES 33
R epeated application of Lemma 2.18 yield
(︃ d
d t )︃ k (︁ f ( −∞ ,0) )︁ = (︂ f ( k ) )︂ ( −∞ ,0) −
k − 1
∑︂
j = 0
f ( j ) (0 − ) (︃ d
d t )︃ k − 1 − j
δ 0 ,
(︃ d
d t )︃ k (︁ f [0, ∞ ) )︁ = (︂ f ( k ) )︂ [0, ∞ ) +
k − 1
∑︂
j = 0
f ( j ) (0 − ) (︃ d
d t )︃ k − 1 − j
δ 0
for k ∈ N . F or P ( s ) = ∑︁ p
k = 0 P k s k ∈ R [ s ] m × n x define P [0] ( s ) : = P ( s ) and recursively
P [ i ] ( s ) : = 1
s (︂ P [ i − 1] ( s ) − P [ i − 1] (0) )︂ ∈ R [ s ] m × n x ,
i.e .,
P [ i ] = ∑︁ p
k = i P k s k − i
for
i =
0
,
1
, . . . , k
. Then, we have pro ven the follo wing generalization of
Lemma 2.18.
Lemma 2.22. Let f ∈ D n
p w C ∞ and P ( s ) ∈ R [ s ] ℓ × n with deg( P ) = d ≥ 0 . Then
P (︃ d
d t )︃ (︁ F ( −∞ ,0) )︁ = (︃ P (︃ d
d t )︃ F )︃ ( −∞ ,0) −
d − 1
∑︂
j = 0 (︃ P [ j ] (︃ d
d t )︃ F )︃ (0 − ) (︃ d
d t )︃ d − 1 − j
δ 0 ,
P (︃ d
d t )︃ (︁ F [0, ∞ ) )︁ = (︃ P (︃ d
d t )︃ F )︃ [0, ∞ ) +
d − 1
∑︂
j = 0 (︃ P [ j ] (︃ d
d t )︃ F )︃ (0 − ) (︃ d
d t )︃ d − 1 − j
δ 0 .
A crucial difference in the study of the polynomial DAE
(2.23)
in contrast to the DAE
(2.5)
is the fact
that the entries of the polynomial matrices are elements of a ring and not a field. Although the ring
of polynomials is embedded in the field of rational functions and thus it is str aightfor ward to use
concepts such as the rank, w e have to ensure that whenev er we substitute
d
d t
for the indeterminate
s
,
we only operate in th e ring of polynomials. As a consequence, w e cannot use nonsingular matr ices
(as for instance in the quasi-W eierstraß form), but have to r estr ict ourselves to unimodular matrices,
i.e ., polynomial matrices that are nonsing ular and the inverse matrix is also a polynomial matrix
(see also the proof of C orollar y 2.20). The usage of unimodular matrices in the context of DAEs is not
new: see for instance the work [116], where the authors con struct a unimodular matrix to per form
index-r eduction. An impor tant tool to study polynomial matrices is the S mith canonic al form .
Theor em 2.23
(S mith canonical for m, [118, Thm. 1.8.1])
.
Let
P
(
s
)
∈ F
[
s
]
m × n x
. Then there e xists
unimodular matrices S ( s ) ∈ F [ s ] m × m , T ( s ) ∈ F [ s ] n x × n x such that
S ( s ) P ( s ) T ( s ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
p 1 ( s )
. . .
p r ( s )
0
. . .
0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, (2.26)

34 CHAPTER 2. DIFFERENTIAL-ALGEBRAIC EQ U A TIONS AND PRELIMINAR Y RESUL TS
wher e r : = rank F [ s ] ( P ( s )) , p i ( s ) ∈ F [ s ] \ {0} , and p i ( s ) divides p i + 1 ( s ) for i = 1, . . . , r − 1 .
A dir ect consequence of the Smith canonical form is that we can perform a rank r evealing ro w-
compr ession with a unimodular matrix, i.e., for
P
(
s
)
∈ F
[
s
]
m × n x
with
r : = rank F [ s ]
(
P
(
s
)), ther e exists
a unimodular matrix
U
(
s
)
∈ F
[
s
]
m × m
and a polynomial matrix
P 1
(
s
)
∈ F
[
s
]
r × n x
with
rank F [ s ]
(
P 1
(
s
))
=
r such that
U ( s ) P ( s ) = [︄ P 1 ( s )
0 ]︄ .
S imilarly as for D AEs, we can study the pr operties of linear time-invariant DDAEs by analyzing pairs
of matrix polynomials (
P
(
s
)
, Q
(
s
))
∈ (︁ R [ s ] m × n x )︁ 2
. Although ther e is no equivalent to the W eierstraß
canonical form for pairs of matrix polynomials, one can still use the condensed for m approach
from [46] to construct a condensed form for ( P ( s ), Q ( s )).
Theor em 2.24
( [96, Thm. 1])
.
F or any pair of polynomial matrices (
P
(
s
)
, Q
(
s
))
∈
(
R
[
s
]
m × n x
)
2
ther e exists unimodular matrix polynomials U ( s ) ∈ R [ s ] m × m and V ( s ) ∈ R [ s ] n x × n x such that
U ( s ) P ( s ) V ( s ) = ⎡
⎢
⎣ ˆ︂
P 11 ( s ) 0 ˆ︂
P 13 ( s )
0 0 ˆ︂
P 23 ( s )
0 0 ˆ︂
P 33 ( s ) ⎤
⎥
⎦ , (2.27a)
U ( s ) Q ( s ) V ( s ) = ⎡
⎢
⎣ ˆ︁
Q 11 ( s ) ˆ︁
Q 12 ( s ) ˆ︁
Q 13 ( s )
0 0 ˆ︁
Q 23 ( s )
0 0 ˆ︁
Q 33 ( s ) ⎤
⎥
⎦ , (2.27b)
wher e
ˆ︂
P 11
is a nonsingular diagonal matrix,
ˆ︂
P 23 , ˆ︂
P 33 , ˆ︁
Q 33
ar e block upper triangular matrices
with zero diagonal blocks and ˆ︁
Q 23 is a nonsingular block upper triangular matrix.
2.3 S trangeness-index for nonlinear DAEs
If
F
in
(2.1a)
is nonlinear , then the equation that we have to solve within the method of steps takes
the form of a nonlinear DAE
F ( t , x ( t ), ˙
x ( t ), u ( t )) = 0, (2.28a)
wher e, as befor e,
x
(
t
)
∈ R n x
and
u
(
t
)
∈ R m
denote , respectively , the state and control of the system,
which is posed on the (compact) time interval
I : =
[0
, T
]. By abuse of notation, we use
F
in this
section to denote the DAE (2.28a) and not the DDAE (1.13). The function
F : I × D x × D ˙
x × D u → R m
with open sets
D x , D ˙
x ⊆ R n x
,
D u ⊂ R n u
is assumed to be sufficiently smooth. The DAE
(2.28a)
is
equipped with the initial condition
x (0) = x 0 . (2.28b)
for some
x 0 ∈ R n x
. As for linear DAEs it is well-kno wn that in general we cannot expect a unique
solution if
m = n x
(cf. [127]). W e therefor e restrict ourselves to the case
m = n x
, since one of the
main goals within this thesis is to establish existence and uniqueness-r esults.

2.3. STRANGENESS-INDEX FOR NONLINEAR DAES 35
Definition 2.25.
A function
x ∈ C 1
(
I , R n x
) is called a (classical) solution of
(2.28a)
if
x
satisfies
(2.28a)
pointwise . An initial value
x 0 ∈ R n x
is called consistent if for a given control
u
, the associated
IVP
(2.28)
has at least one solution. The DAE
(2.28a)
is called r egular , if for ever y sufficiently smooth
input
u
ther e exists a consistent initial value and for ever y consistent initial value , the solution of
the ITP (2.28) is unique .
The control problem
(2.28a)
is often studied in the behavior framewor k [172] , see for instance [55,
125]. H ereb y , a new variable
ξ =
[
x , u
] is introduced that includes the state and contr ol var iable such
that the problem is r educed to the analysis of an underdetermined DAE [125], i. e., the meaning of
the variables is not distinguished any mor e. One big advantage of this formalism is that the analysis
determines the free v ar iables in the system, which might not be the original control v ariables, and
hence need to be r einterpreted. Since our main goal is to study the IVP
(2.28)
with a pr escr ibed
input function u this viewpoint is not possible . F or given u we can study the r estr icted problem
˜︁
F ( t , x ( t ), ˙
x ( t )) = 0, x ( t 0 ) = x 0 , (2.29)
with ˜︁
F ( t , x , ˙
x ) = F ( t , x , ˙
x , u ).
If the partial derivative
∂
∂ ˙
x ˜︁
F
is singular , then the solution
x
of
(2.29)
may depend on derivatives of
˜︁
F
.
The difficulties arising with these differentations ar e classified b y so called index concepts (cf. [153]
for a sur vey). In this paper , we make use of the str angeness index concept [127], which is – roughly
speaking – a generalization of the differ entiation index [42] to under- and o verdetermined systems .
The advantage of the str angeness index is that it preserves the algebraic constr aints in the system,
which in turn prev ents numer ical methods to drift away from the solution manifold [126]. The
strangeness index is based on the deriv ative array [51] of level ℓ , defined as
˜︁
D ℓ (︁ t , x , η )︁ : = ⎡
⎢
⎢
⎢
⎢
⎢
⎣
˜︁
F ( t , x , ˙
x )
d
d t ˜︁
F ( t , x , ˙
x )
.
.
.
(︂ d
d t )︂ ℓ ˜︁
F ( t , x , ˙
x )
⎤
⎥
⎥
⎥
⎥
⎥
⎦ ∈ R ( ℓ + 1) n x with η : = ⎡
⎢
⎢
⎢
⎢
⎣
˙
x
¨
x
.
.
.
x ( ℓ + 1)
⎤
⎥
⎥
⎥
⎥
⎦ . (2.30)
S ince it is a-pr iori not clear , that a solution exists, w e need to assume that the set
˜︂
M ℓ : = {︂ (︁ t , x , η )︁ ∈ R ( ℓ + 2) n x + 1    ˜︁
D ℓ (︁ t , x , η )︁ = 0 }︂
is nonempty . Similarly as in the theory for linear DAEs, we are inter ested in deter mining all algebraic
constraints . I n pr incipal, the number of algebraic constr aints may var y due to the nonlinearity of
˜︁
F
. T o exclude this case we have to impose some constant r ank assumptions. F ollo wing [124 ], we
introduce the J acobians
˜︁
E ℓ ( t , x , ˙
x , . . . , x ( ℓ + 1) ) : = [︂ ∂ ˜︁
D ℓ
∂ ˙
x . . . ∂ ˜︁
D ℓ
∂ x ( ℓ + 1) ]︂ (︂ t , x , ˙
x , . . . , x ( ℓ + 1) )︂ ∈ R ( ℓ + 1) n x × ( ℓ + 1) n x ,
˜︂
A ℓ ( t , x , ˙
x , . . . , x ( ℓ + 1) ) : = − [︂ ∂ ˜︁
D ℓ
∂ x 0 . . . 0 ]︂ (︂ t , x , ˙
x , . . . , x ( ℓ + 1) )︂ ∈ R ( ℓ + 1) n x × ( ℓ + 1) n x .
I n order to determine all algebraic equations that ar e encoded in (2.28a), we assume the follo wing.

36 CHAPTER 2. DIFFERENTIAL-ALGEBRAIC EQ U A TIONS AND PRELIMINAR Y RESUL TS
Assumption 2.26. Ther e exist integers µ and a such that the set
˜︂
M µ : = {︂ (︁ t , x , η )︁ ∈ R ( µ + 2) n x + 1    ˜︁
D µ (︁ t , x , η )︁ = 0 }︂
associated with ˜︁
F is nonempty and such that for ever y ( t 0 , x 0 , η 0 ) ∈ ˜︂
M µ , ther e exists a (sufficiently
small) neighbor hood ˜︂
U . M or eo ver we have rank( ˜︁
E µ ) = ( µ + 1) n x − a on ˜︂
M µ ∩ ˜︂
U .
The constant rank assumption allo ws us to define (via the smooth singular value decomposition [43],
see also [127]) a matrix-valued function
˜︁
Z A
of size (
µ +
1)
n x × a
and pointwise maximal rank that
satisfies ˜︁
Z T
A ˜︁
E µ = 0. The (linearized) algebraic equations ar e thus encoded in the matrix
(︄ ˜︁
Z T
A
∂ ˜︁
D µ
∂ x )︄ ( t , x , η ) ∈ R a × n x . (2.31)
T o ensure that the pr oblem is regular , we need to be able to solve the algebr aic equations for
a
unkno wns, r equir ing that the matrix in (2.31) has rank a . W e thus assume the follo wing.
Assumption 2.27. Let Assu mption 2.26 hold, let ˜︁
Z A be constructed as abo ve, and assume
rank (︄(︄ ˜︁
Z T
A
∂ ˜︁
D µ
∂ x )︄ ( t , x , η ) )︄ = a (2.32)
for all ( t , x , η ) ∈ ˜︂
M µ ∩ ˜︂
U .
I n view of the W eierstraß canonical form (cf. Theor em 2.9), we need to ensure that w e have
d : = n x − a
differ ential equations for the remaining
d
variables . U sing
(2.32)
we deduce the existence of a
smooth matrix function ˜︁
T A of size n x × d with pointwise maximal ran k satisfying
(︄ ˜︁
Z T
A
∂ ˜︁
D µ
∂ x ˜︁
T A )︄ ( t , x , η ) = 0.
The r emaining differential equations must be contained in the original DAE (in contrast to the
algebraic equations , which are contained in the derivative arr ay) and thus we assume the follo wing
to guarantee that w e actually have d differential equations .
Assumption 2.28.
Let Assu mptions 2.26 and 2.27 hold, set
d : = n x − a
, let
˜︁
T A
be as abo ve, and
assume
rank (︃(︃ ∂ ˜︁
F
∂ ˙
x ˜︁
T A )︃ ( t , x , η ) )︃ = d
for all ( t , x , η ) ∈ ˜︂
M µ ∩ ˜︂
U .
T o summar ize the pr evious discussion, we make the follo wing assumption, which for historical
r easons (cf. [124]) is referr ed to as a hypothesis.

2.3. STRANGENESS-INDEX FOR NONLINEAR DAES 37
H ypothesis 2.29 ( [124, Hypothesis 3.2]) . Ther e exist integers µ and a such that the set
˜︂
M µ : = {︂ (︁ t , x , η )︁ ∈ R ( µ + 2) n x + 1    ˜︁
D µ (︁ t , x , η )︁ = 0 }︂
associated with ˜︁
F is nonempty and such that for ever y ( t 0 , x 0 , η 0 ) ∈ ˜︂
M µ , ther e exists a (sufficiently
small) neighbor hood ˜︂
U in which the follo wing properties hold:
(i)
W e have
rank
(
∂
∂η ˜︁
D µ
)
=
(
µ +
1)
n x − a
on
˜︂
M µ ∩ ˜︂
U
such that ther e exists a smooth matrix
function ˜︁
Z A of size ( µ + 1) n x × a and pointwise maximal rank that satisfies ˜︁
Z T
A ∂
∂η ˜︁
D µ = 0 .
(ii)
W e have
rank
(
˜︁
Z T
A ∂
∂ x ˜︁
D µ
)
= a
on
˜︂
M µ ∩ ˜︂
U
such that ther e exists a smooth matrix function
˜︁
T A
of size n x × d with d : = n x − a and pointwise maximal rank, satisfying ˜︁
Z T
A (︂ ∂
∂ x ˜︁
D µ )︂ ˜︁
T A = 0 .
(iii)
W e have
rank
(
∂ ˜︁
F
∂ ˙
x ˜︁
T A
)
= d
on
˜︂
M µ ∩ ˜︂
U
such that ther e exists a smooth matrix function
˜︁
Z D
of
size n x × d and pointwise maxi mal r ank, satisfying rank( ˜︁
Z T
D ∂ ˜︁
F
∂ ˙
x ˜︁
T A ) = d .
Definition 2.30.
The smallest possible
µ
for which H ypothesis 2.29 is satisfied is called strangeness
index of the DAE
(2.28a)
. If H ypothesis 2.29 is satisfied with
µ =
0, then the DAE
(2.28a)
is called
str angeness -fr ee .
The quantities
a
and
d
in H ypothesis 2.29 are , respectively , the numbers of algebraic and differ ential
equations contained in the DAE
(2.29)
. U sing the matrix functions
˜︁
Z D
and
˜︁
Z A
, the DAE
(2.29)
can
(locally) be r efor mulated as
0 = ˜︁
D ( t , x , ˙
x ) : = (︂ ˜︁
Z T
D ˜︁
F )︂ ( t , x , ˙
x ), (2.33a)
0 = ˜︁
A ( t , x ) : = (︂ ˜︁
Z T
A ˜︁
D µ )︂ ( t , x ), (2.33b)
which itself is strangeness-fr ee and every solu tion of
(2.29)
also solves
(2.33)
. H ereb y we call
(2.33a)
the differ ential par t of
(2.29)
and
(2.33b)
the algebr aic par t . N ote that although
˜︁
Z A
and
˜︁
D µ
may
depend on derivatives of
x
it can be sho wn (cf. [124]) that their product only depends on
t
and
x
.
U nfor tunately , a solution of
(2.33)
is not necessarily a solution of
(2.29)
. H o wever , if we assume in
addition, that H ypothesis 2.29 is satisfied with characteristic values
µ , a , d
and
µ +
1
, a , d
, then for
every initial value
x µ + 1,0 ∈ M µ + 1
ther e exists a unique solution of
(2.33)
and this solution (locally)
solves
(2.29)
(see [127, Theor em 4.1 3]). As a direct consequence , an initial value
x 0
is consistent if
and only if it is contained in the consistency set
( t 0 , x 0 ) ∈ ˜︁
M : = {︂ ( t , x ) ∈ R n x + 1    ˜︁
A ( t , x ) = 0 }︂ . (2.34)
If state tr ansfor mations ar e allow ed, then the implicit function theorem allo ws to (locally) rewrite
the strangeness-fr ee DAE (2.33) as
˙
ξ = ˜︂
L ( t , ξ ), ζ = ˜︁
R ( t , ξ ) (2.35)
with ξ ( t ) ∈ R d and ζ ( t ) ∈ R a . F or the detailed der ivation we r efer to [127, Cha. 4.1]. Let x = T ( t , ξ , ζ )
denote the transformation for the state . Then, the or dinar y differ ential equation (ODE)
˙
x = ˜︁ f ( t , x ) : = T (︂ t , ˜︂
L ( t , ξ ), (︂ ∂
∂ξ ˜︁
R )︂ ( t , ξ ) ˜︂
L ( t , ξ ) + (︂ ∂
∂ t ˜︁
R )︂ ( t , ξ ) )︂ , (2.36)

38 CHAPTER 2. DIFFERENTIAL-ALGEBRAIC EQ U A TIONS AND PRELIMINAR Y RESUL TS
is called the under l ying ODE for the DAE
(2.29)
and is the basis of the differ entiation index [4 2],
which is defined as µ + 1 if ∂
∂ ˙
x ˜︁
F is singular and 0 other wise [127, Cor . 3.46].
R emark 2.31.
I n ter ms of the L TI DAE
(2.5)
, the strangeness-index
µ
and the index of the matrix
pencil ν (see Definition 2.12) satisfy
ν = ⎧
⎨
⎩
0, if E is nonsingular,
µ + 1, other wise.
With the definition abo ve, the differ entiation index is thus a generalization of the index of the matrix
pencil to nonlinear systems . ♣
If we want to solv e the DAE
(2.28a)
numerically , we ar e not only interested in the existenc e of
solutions but also that the solution of the initial value problem
(2.29)
is unique and depends
continuously on the data. F or DAEs, the so-called well-posedness can be be formulated as follo ws
[127, Theor em 4. 12].
Theor em 2.32.
Let
˜︁
F
as in
(2.29)
be sufficiently smooth and satisfy Hypothesis 2.29. Let
x ⋆ ∈
C 1
(
I , R n x
) be a sufficiently smooth solution of
(2.28)
. Let the (nonlinear) oper ator
˜︂
F : D → Y
,
D ⊆ Z open, be defined by
˜︂
F ( x )( t ) = [︄ ˙
ξ − ˜︂
L ( t , ξ ( t ))
ζ − ˜︁
R ( t , ξ ( t )) ]︄ , (2.37)
with the Banach spaces
Z : = {︂ z ∈ C (︁ I , R n x )︁    ξ ∈ C 1 ( I , R d ), ξ ( t 0 ) = 0 }︂ , Y : = C (︁ I , R n x )︁
accor ding to (2.35) . Then x ⋆ is a regular solution of the str angeness-fr ee problem
˜︂
F ( x ) = 0
in the follo wing sense. Ther e exists a neighborhood
U x ⊆ Z
of
x ⋆
and a neighbor hood
V ⊆ Y
of
the origin such that for every f ∈ V the equat ion
˜︂
F ( x ) = f
has a unique solution
x ∈ U x
that depends continuously on
f
. I n par ticular ,
x ⋆
is the unique
solution in U x belonging to f = 0 .
I n order to apply the theory to the or iginal equation
(2.28a)
we have to ensur e that the characteristic
values
µ
,
a
, and
d
do not depend on the chosen input
u
. A simple way to guarantee this , is to ensure
that the rank assumptions in H ypothesis 2.29 hold for all sufficiently smooth input functions. The

2.3. STRANGENESS-INDEX FOR NONLINEAR DAES 39
derivative array (2.30) with explicit dependency on u takes the form
D ℓ (︂ t , x , η , u , ˙
u , . . . , u ( ℓ ) )︂ : = ⎡
⎢
⎢
⎢
⎢
⎢
⎣
F ( t , x , ˙
x , u )
d
d t F ( t , x , ˙
x , u )
.
.
.
(︂ d
d t )︂ ℓ F ( t , x , ˙
x , u )
⎤
⎥
⎥
⎥
⎥
⎥
⎦ ∈ R ( ℓ + 1) n x with η : = ⎡
⎢
⎢
⎣
˙
x
.
.
.
x ( ℓ + 1)
⎤
⎥
⎥
⎦ .
H ypothesis 2.33.
Ther e exist integers
µ
and
a
, and matrix functions
Z A
(
·
)
∈ R ( µ + 1) n x × a
,
T A
(
·
)
∈
R n x × d
, and
Z D
(
·
)
∈ R n x × d
with pointwise maximal r ank and
d : = n − a
such that for every suffi-
ciently smooth u the set
M µ : = {︂(︂ t , x , η , u , . . . , u ( µ ) )︂ ∈ R ( µ + 2) n x + ( µ + 1) m + 1    D µ (︂ t , x , η , u , . . . , u ( µ ) )︂ = 0 }︂
associated with
F
is nonempty and such that for every (
t 0 , x 0 , η 0 , u 0 , . . . , u ( µ )
0
)
∈ M µ
, ther e exists a
(sufficiently small) neighborhood U in which the follo wing properties hold:
(i) W e have rank( ∂
∂η D µ ) = ( µ + 1) n x − a and Z T
A ∂
∂η D µ = 0 on M µ ∩ U .
(ii) W e have r ank( Z T
A ∂
∂ x D µ ) = a and Z T
A (︂ ∂
∂ x D µ )︂ T A = 0 on M µ ∩ U .
(iii) W e have r ank( ∂ F
∂ ˙
x T A ) = d and rank( Z T
D ∂ F
∂ ˙
x T A ) = d on M µ ∩ U .
R emark 2.34.
N ote that similarly as in H ypothesis 2.29 the existence of the matr ix functions
Z A
,
T A
,
and
Z D
in H ypothesis 2.33 follo ws from the constant rank assumptions and a smooth version of the
singular value decomposition as in [127, Thm. 3.9 and Thm. 4.3]. ♣
Example 2.35.
I t is easy to see that the mass-spring-damper system
(1.2)
in section 1.1.1 with
M >
0
satisfies H ypothesis 2.33 with
µ =
0. The equations for the pendulum
(1.3)
ar e in Hessenb erg-form
and ther efore satisfy H ypothesis 2.33 with strangeness index µ = 2 [127, Thm. 4.23]. ♠
F ol lo w ing the analysis in [124] that leads to the strangeness-fr ee formulation
(2.33)
we observe that
the functions
D
and
A
may depend on
u
and its derivatives . Due to the local chara cter of H ypothe-
sis 2.33 we can assume that
D
does not depend on derivatives of
u
. In any case , Hypothesis 2.33
yields the (local) r efor mulation
0 = D ( t , x , ˙
x , u ) : = (︂ Z T
D F )︂ ( t , x , ˙
x , u ), (2.38a)
0 = A (︂ t , x , u , ˙
u , . . . , u ( µ ) )︂ : = (︂ Z T
A D µ )︂ (︂ t , x , u , ˙
u , . . . , u ( µ ) )︂ , (2.38b)
which itself is strangeness-fr ee. The corr esponding explicit for m
(2.35)
and the underlying ODE
(2.36) ther efore take the form
˙
ξ = L ( t , ξ , u ), ζ = R ( t , ξ , u , ˙
u , . . . , u ( µ ) ) (2.39)
and
˙
x = f (︂ t , x , u , . . . , u ( µ + 1) )︂ . (2.40)

40 CHAPTER 2. DIFFERENTIAL-ALGEBRAIC EQ U A TIONS AND PRELIMINAR Y RESUL TS
Clearly , if a system satisfies H ypothesis 2.33, then it also satisfies H ypothesis 2.29 (with given
u
) and
thus all pr evious results hold as well.
R emark 2.36.
Let us emphasize that although derivatives of
u
up to or der
µ
, r espectively
µ +
1
appear in the algebraic equation
(2.38b)
, the explicit algebraic equation
(2.39)
, and the underlying
ODE (2.40), r espectively , we may have
∂
∂ u ( ℓ ) f (︂ t , x , u , . . . , u ( µ + 1) )︂ ≡ 0
for some
ℓ ∈ {
1
, . . . , µ +
1
}
, i.e ., the underlying ODE
(2.40)
may not necessarily depend on all deriva-
tives of u up to or der µ + 1. ♣

3
Distributional solutions fo r linea r time-inva riant DD AEs
As outlined in the introduction (cf. section 1.2), one major aspect of this thesis is the development
of general existence and uniqueness r esults for delay differ ential-algebr ai c equations (DDAEs), and
a first starting point is to consider the initial value problem (IVP) for linear time-invariant (L TI)
problems of the form (1.15). R ecall that the IVP is given as
E ˙
x ( t ) = A 1 x ( t ) + A 2 x ( t − τ ) + f ( t ), for t ∈ I : = [0, t f ), (3.1a)
x ( t ) = φ ( t ), for t ∈ [ − τ , 0 ], (3.1b)
wher e
E , A 1 , A 2 ∈ F m × n x
ar e matr ices o ver the field
F ∈ { R , C }
,
f : I → F m
is the inhomogeneity , and
φ :
[
− τ ,
0]
→ F n x
is the histor y function or initial trajectory . F or the ease of presentation we intr oduce
the shift oper ator ( σ τ x )( t ) : = x ( t − τ ) for τ > 0 and thus, we can write
E ˙
x = A 1 x + A 2 σ τ x + f in [0, t f ) (3.2)
instead of
(3.1a)
. The Examples 1.4 and 1.5 demonstrate that w e cannot expect the existence of
a classical or even continuous solution for the IVP
(3.1)
. Instead, we start our analysis with the
distributional solution concept from section 2.2.
M ost of the r esults in this section are obtained together with S tephan T renn (U niversity of Gr oningen)
and published in [206, 2 07].
3.1 Distributional shift operator and delay -regularity
I n o r der to interpret
(3.1)
within the space of piecewise-smooth distributions (cf. Definition 2.17),
we need to define a distributional analogue of the time delay: F or
τ >
0 we define the distributional
shift oper ator
σ τ : D → D , f ↦→ (︁ C ∞
0 ( R ; R ) → R , ϕ ↦→ f ( ϕ ( · + τ )) )︁ , (3.3)
wher e
D
denotes the space of distributions as defined in
(2.19)
. N ote that for any continuous
function f ∈ C ( R ; R ) and any ϕ ∈ C ∞
0 ( R ; R ) we have
(︁ σ τ f )︁ D ( ϕ ) = ∫︂ ∞
−∞
f ( t − τ ) ϕ ( t )d t = ∫︂ ∞
−∞
f ( t ) ϕ ( t + τ )d t = (︁ σ τ (︁ f D )︁)︁ ( ϕ ),
41

42 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
and thus
(︁ σ τ f )︁ D = σ τ (︁ f D )︁
. Mor eo ver , it is easy to see that
σ τ
is a linear operator and
σ τ f ∈ D p w C ∞
for any f ∈ D pw C ∞ .
Lemma 3.1.
The distributional shift oper ator
σ τ
defined in
(3.3)
and the distributional derivative
d
d t defined in (2.21) commute in D , i.e., d
d t ◦ σ τ = σ τ ◦ d
d t .
Proof. Let f ∈ D and ϕ ∈ C ∞
0 ( R ; R ). Then we obtain
(︃ d
d t ◦ σ τ )︃ ( f )( ϕ ) = (︃ d
d t f )︃ (︁ ϕ ( · + τ ) )︁ = − f (︃ d
d t ϕ ( · + τ ) )︃
= (︁ σ τ (︁ − f )︁)︁ (︃ d
d t ϕ )︃ = (︃ σ τ ◦ d
d t )︃ ( f )( ϕ ),
wher e we have used that the derivative and the shift commute in C ∞
0 ( R ; R ). ■
Lemma 3.2. Let Ω ⊆ R , β ∈ C ∞ ( R ; R ) , and f ∈ D pw C ∞ . Then
σ τ (︁ 1 Ω )︁ = 1 Ω + τ , σ τ (︁ 1 Ω β )︁ = 1 Ω + τ σ τ β , and σ τ (︁ f Ω )︁ = (︁ σ τ f )︁ Ω + τ ,
wher e Ω + τ : = { ω + τ | ω ∈ Ω } .
Proof. Let t ∈ R . Then
(︁ σ τ (︁ 1 Ω β )︁)︁ ( t ) = (︁ 1 Ω β )︁ ( t − τ ) = ⎧
⎨
⎩
β ( t − τ ), if t − τ ∈ Ω ,
0, other wise ,
= ⎧
⎨
⎩
β ( t − τ ), if t ∈ Ω + τ ,
0, other wise , = (︁ 1 Ω + τ σ τ β )︁ ( t )
and thus
σ τ (︁ 1 Ω β )︁ = 1 Ω + τ σ τ β
. The first asser tion follo ws b y choosing
β ≡
1. F or the remaining
assertion let
f = α D + ∑︁ s ∈ S D s
with
α ∈ C ∞
p w
(
R
;
R
), discr ete set
S
and
D s ∈ span{ δ s , ˙
δ s , ¨
δ s , . . .}
for
s ∈ S
.
Lemma 3.1 together with the alr eady pro ven identities, and the definition of the Dirac impulse
(cf. Example 2.16) imply σ τ D s = D s + τ for s ∈ S . W e ther efore conclude
σ τ (︁ f Ω )︁ = σ τ (︄ (︁ 1 Ω α )︁ D + ∑︂
s ∈ S ∩ Ω
D s )︄ = (︁ 1 Ω + τ σ τ α )︁ D + ∑︂
s ∈ S ∩ ( Ω + τ )
D s + τ = (︁ σ τ f )︁ Ω + τ . ■
As in Section 2.2 for differ enti al-algebr aic equations (DAEs), we can no w i nterpr et the DDAE
(3.1)
,
r espectively
(3.2)
in the space of piecewise-smooth distributions as the initial trajectory problem
(ITP)
x ( −∞ ,0) = x 0
( −∞ ,0) ,
( E ˙
x ) [0, ∞ ) = ( A 1 x + A 2 σ τ x + f ) [0, ∞ ) , (3.4)
r espectively the distribu tional DDAE
E ˙
x = A 1 x + A 2 σ τ x + f , (3.5)
with initial trajectory x 0 ∈ D n x
p w C ∞ and inhomogeneity f ∈ D m
p w C ∞ .

3.1. DISTRIBUTIONAL SHIFT OPERA TOR AND DELA Y -REGULARIT Y 43
Definition 3.3.
Consider the ITP
(3.4)
with
f ∈ D m
p w C ∞
. An initial tr ajector y
x 0 ∈ D n x
p w C ∞
is called
feasible for the ITP
(3.4)
, if ther e exists
x ∈ D n x
p w C ∞
that satisfies
(3.4)
. In this case ,
x
is called a
(distributional) solution of
(3.4)
. The ITP
(3.4)
is called solvable if ther e exists a feasible initial
trajectory x 0 ∈ D n x
p w C ∞ for the ITP (3.4).
R emark 3.4.
Defining
f ITP : = f [0, ∞ ) + (︂ E ˙
x 0 − A 1 x 0 − A 2 σ τ x 0 )︂ ( −∞ ,0)
, then it is straightforward to see
that every solution of the ITP (3.4) is also a solution of the distr ibutional DDAE
E ˙
x = A 1 x + A 2 σ τ x + f ITP . (3.6)
Conv e rsely , let
x ∈ D n x
p w C ∞
satisfy
(3.6)
and define
x 0 : = x
. Then
x 0
is feasible and thus the ITP
(3.4)
is solvable . F or a similar discussion for DAEs we r efer to [178 , 205]. ♣
If the matrix pair (
E , A 1
) is r egular , then we can use Theor em 2.19 to establish existence and unique-
ness of solutions of the ITP
(3.4)
via integration on successive time intervals [
i τ ,
(
i +
1)
τ
), which is
also r eferred to as method of steps (cf. chapter 2 and the forthcoming section 4.1).
Theor em 3.5.
Consider the ITP
(3.4)
with
x 0 ∈ D n x
p w C ∞
and
f ∈ D m
p w C ∞
. If the matrix pair (
E , A 1
)
is r egular , then the ITP (3.4) has a unique distributional solution x ∈ D n x
p w C ∞ .
Proof. A pplying the method of steps to (3.4) r esults in the sequence of DAE ITPs
x i
( −∞ ,( i − 1) τ ) = x i − 1
( −∞ ,( i − 1) τ ) ,
(︂ E ˙
x i )︂ [( i − 1) τ , ∞ ) = (︂ A 1 x i + ˜
f i )︂ [( i − 1) τ , ∞ ) , (3.7)
with
˜
f i : = A 2 x i − 1 + f
and
i ∈ N
. Theor em 2.19 implies r e cursively the existence of a unique solution
x i ∈ D n x
p w C ∞
of
(3.7)
for each
i ∈ N
. In particular , for each
i ∈ N 0
ther e exists
α i ∈ C ∞
p w
(
R
;
R
), a discr ete
set
S i ⊆ R
and distributions
D i
s ∈ span{ δ s , ˙
δ s , ¨
δ s , . . .}
for each
s ∈ S i
such that the
j
th component
x i
j
of x i is given b y
x i
j = α i
j D + ∑︂
s ∈ S i
j
D i
j ; s .
W e sho w that
x : = x 0
( −∞ ,0) + ∑︁ ∞
i = 1 x i
[( i − 1) τ , i τ )
is the solution of
(3.4)
. First note that for the
j
th compo-
nent the set (︂ S 0
j ∩ ( −∞ , 0) )︂ ∪ ⋃︂
i ∈ N (︂ S i
j ∩ [( i − 1) τ , i τ ) )︂
is discr ete and (
α 0
j
)
( −∞ ,0) + ∑︁ ∞
i = 1
(
α i
j
)
[( i − 1) τ , i τ ) ∈ C ∞
p w
(
R
;
R
), which implies
x ∈ D n x
p w C ∞
. Furthermore ,
b y constr uction we have x ( −∞ ,0) = x 0
( −∞ ,0) . F or i ∈ N , Lemma 2.18 implies
( E ˙
x ) [( i − 1) τ , i τ ) = (︄ E d
d t (︄ (︂ x 0 )︂ ( −∞ ,0) + ∞
∑︂
k = 1 (︂ x k
[( k − 1) τ , k τ ) )︂ )︄)︄ [( i − 1) τ , i τ )
= E (︂ ˙
x i
[( i − 1) τ , i τ ) + (︂ x i (( i − 1) τ − ) − x i − 1 (( i − 1) τ − ) )︂ δ ( i − 1) τ )︂ .

44 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
U sing x i
( −∞ ,( i − 1) τ ) = x i − 1
( −∞ ,( i − 1) τ ) and (3.7) we obtain
( E ˙
x ) [( i − 1) τ , i τ ) = E ˙
x i
[( i − 1) τ , i τ )
= (︂ A 1 x i + ˜
f i )︂ [( i − 1) τ , i τ )
= A 1 x i
[( i − 1) τ , i τ ) + A 2 x i − 1
[( i − 1) τ , i τ ) + f [( i − 1) τ , i τ )
= A 1 x i
[( i − 1) τ , i τ ) + A 2 σ τ x i
[( i − 1) τ , i τ ) + f [( i − 1) τ , i τ )
= (︁ A 1 x + A 2 σ τ x + f )︁ [( i − 1) τ , i τ ) .
Thus,
x
is a solution of the ITP
(3.4)
. S ince
x [( i − 1) τ , i τ ) = x i
[( i − 1) τ , i τ )
for each
i ∈ N
and
x i
is the unique
solution of (3.7), we conclude that x is the unique solution of (3.4). ■
R emark 3.6.
The existence and uniqueness of distributional solutions for DDAEs was already hinted
in [53] and [96]. R esu lts for stronger solution concepts ar e presented for instance in [13, 69, 98] and
the forthcoming chapter 4, although under much stronger assumptions on the history function and
additional properties of the matrix pair (
E , A 1
). A generalization of Theor em 3.5 to switched DDAE
is pr esented in [ 38]. ♣
S imilarly to Theor em 2.7 we may expect that a singular matrix pencil (
E , A 1
) may r esult in an ITP
that is either not uniquely solvable or not solvable at all. H o wever , as the follo wing example show ,
this is not the case; in addition see [94].
Example 3.7. C onsider the scalar DDAE (3.1a) with ( E , A 1 , A 2 ) = (0, 0, 1), i.e.
0 = σ τ x + f , (3.8)
which clearly has the unique (acausal) solution
x = σ − τ f
for any inhomogeneity
f
, although the
matrix pair (
E , A 1
)
=
(0
,
0) is not r egular . N ote ho wever , that it is not possible to fr eely prescribe the
initial trajectory for x on [ − τ , 0) because it is already fully specified b y f given on [0, τ ). ♠
The example sho ws that b y introducing a time-delay term to a DAE with a singular matr ix pair
(
E , A 1
) we may arrive at a DDAE that is r egular in a cer tain sense . Thus, we need to formalize the
notion of r egular ity for DDAEs. F o llo wing [204], we give the follo wing generalization of r egu larity .
Definition 3.8.
The DDAE
(3.2)
is called delay-r egular , if for all inhomogeneities
f ∈ D m
p w C ∞
with
support in [0
, ∞
) ther e exists a solution
x ∈ D n x
p w C ∞
and each solution for the same
f
is uniquely
determined b y the past, i.e., for two solutions x 1 , x 2 ∈ D n x
p w C ∞ of (3.2), the implication
x 1 ( −∞ ,0) = x 2 ( −∞ ,0) = ⇒ x 1 = x 2
holds . The matr ix triple (
E , A 1 , A 2
) is called delay-r egular if and only if the corresponding DDAE is
delay -regular .
I n order to analyze the existence and uniqueness of solutions of the ITP
(3.4)
, Example 3.7 r eveals
that in some sense the DDAE given b y (
E , A 1 , A 2
)
=
(0
,
0
,
1) with singular matrix pair (
E , A 1
) is

3.1. DISTRIBUTIONAL SHIFT OPERA TOR AND DELA Y -REGULARIT Y 45
equivalent to the DDAE (
ˆ︁
E , ˆ︁
A , ˆ︁
D
)
=
(0
,
1
,
0) with a shifted inhomogeneity
ˆ︁
f : = σ − τ f
, where no w
( ˆ︁
E , ˆ︁
A ) is r egular . W e therefor e want to define a notion of delay -equivalence. U nfor tunately , it is not
sufficient to consider matrix triplets only , since higher -order differ ential equations may be hidden
in the DDAE (cf. [53] and [97]). This fact is illustrated with the follo wing example.
Example 3.9. C onsider the DDAE (3.1a) with
E = ⎡
⎢
⎣
0 1 0
0 0 1
0 0 0 ⎤
⎥
⎦ , A 1 = ⎡
⎢
⎣
0 0 0
0 1 0
0 0 1 ⎤
⎥
⎦ , A 2 = ⎡
⎢
⎣
0 0 0
0 0 0
1 0 0 ⎤
⎥
⎦ , f = ⎡
⎢
⎣
f 1
f 2
f 3
⎤
⎥
⎦ .
Clearly , (
E , A 1
) is not r egular , ho wever differ entiating the last equation twice and plugging in the
first two equations yields
0 = σ τ ¨
x 1 + f 1 + ˙
f 2 + ¨
f 3 .
The same trick as applied in Example 3.7 can be used to shift the time-delay into the inhomogeneity ,
but the r esulting equation cannot be wr itten as a first-or der DAE without increasing the dimension
of the matrices (due to the presence of a secon d derivative). ♠
Example 3.9 motivates to study the mor e general DDAE
P (︃ d
d t )︃ x = Q (︃ d
d t )︃ σ τ x + f , (3.9)
r espectively the ITP
x ( −∞ ,0) = x 0
( −∞ ,0) , (3.10a)
(︃ P (︃ d
d t )︃ x )︃ [0, ∞ ) = (︃ Q (︃ d
d t )︃ σ τ x + f )︃ [0, ∞ )
, (3.10b)
with matrix polynomials
P
(
s
)
, Q
(
s
)
∈ R
[
s
]
m × n x
. N ote that
(3.10a)
not only specifies the initial
trajectory but also its (distributional) der ivatives .
Definition 3.10.
The DDAE
(3.9)
is called delay -regular , if for all inhomogeneities
f ∈ D m
p w C ∞
with
support in [0
, ∞
) ther e exists a solution
x ∈ D n x
p w C ∞
and each solution for the same
f
is uniquely
determined b y the past, i.e., for two solutions x 1 , x 2 ∈ D n x
p w C ∞ of (3.9), the implication
x 1 ( −∞ ,0) = x 2 ( −∞ ,0) = ⇒ x 1 = x 2
holds . The pair of matrix polynomials (
P
(
s
)
, Q
(
s
)) is called delay-r egular if and only if the corre-
sponding DDAE is delay -regular .
Definition 3.11.
An initial trajectory
x 0 ∈ D n x
p w C ∞
is called feasible for the ITP
(3.10)
, if ther e exists
x ∈ D n x
p w C ∞
that satisfies
(3.10)
. I n this case,
x
is called a (distributional) solution of
(3.10)
. The
ITP
(3.10)
is called solvable if ther e exists a consistent initial trajectory
x 0 ∈ D n x
p w C ∞
for the ITP
(3.10)
.
W e first highlight the connection of delay -regularity with the solvability of the ITP (3.10).

46 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
Pr oposition 3.12.
If the DDAE
(3.9)
is delay-r egular , then for each
f ∈ D m
p w C ∞
ther e exists an initial
tr ajector y
x 0 ∈ D n x
p w C ∞
such that the ITP
(3.10)
is uniquely solvable. Conversely , if the ITP
(3.10)
is
uniquely solvable for x 0 = 0 and for any inhomogeneity f ∈ D m
p w C ∞ then (3.9) is delay-regular .
Proof.
Let
f ∈ D m
p w C ∞
with support in [0
, ∞
) and assume that
(3.9)
is delay -regular . Then there exists
a solution
x ∈ D n x
p w C ∞
of
(3.9)
. Setting
x 0 : = x
we immediately obtain a solution of the ITP
(3.10)
.
S uppose no w that for
x 0 ∈ D n x
p w C ∞
the ITP
(3.10)
has two solutions
x 1 , x 2 ∈ D n x
p w C ∞
. Then the differ-
ence
˜︁
x : = x 1 − x 2
satisfies the ITP
(3.10)
with initial trajectory
˜︁
x 0 : =
0 and zero inhomogeneity . Then
˜︁
x satisfies
P (︃ d
d t )︃ ˜︁
x = Q (︃ d
d t )︃ σ τ ˜︁
x + (︃ P (︃ d
d t )︃ ˜︁
x − Q (︃ d
d t )︃ σ τ ˜︁
x )︃ ( −∞ ,0)
.
U sing
(︃ P (︃ d
d t )︃ ˜︁
x − Q (︃ d
d t )︃ σ τ ˜︁
x )︃ ( −∞ ,0) = (︃ P (︃ d
d t )︃ ˜︁
x ( −∞ ,0) − Q (︃ d
d t )︃ σ τ (︁ ˜︁
x ( −∞ ,0) )︁ )︃ ( −∞ ,0) = 0
and the delay -r egularity of (3.9) we conclude ˜︁
x = 0 and thus x 1 = x 2 .
N o w assume that the ITP
(3.10)
with
x 0 =
0 has a unique solution
x
for all
f ∈ D m
p w C ∞
. Then with the
same argument as abo ve it follo ws that
x
solves
(3.9)
with inhomogeneity
f [0, ∞ )
. T o sho w uniqueness
assume that
(3.9)
has two solutions, then the differ ence
˜︁
x
satisfies
˜︁
x ( −∞ ,0) =
0 and ther efore solves
the ITP (3.10) with x 0 = 0 and f = 0. H ence ˜︁
x must coincide with the trivial solution of (3.10). ■
I t is impor tant to note the follo wing for delay-r egularity :
(i) Causality with r espect to the inhomogeneity f is not assumed.
(ii) Existence of a solution for all initial tr ajectories is not assumed.
(iii)
U nique solvability of the ITP with zero initial tr ajector y is only a sufficient condition for delay -
r egular ity . In particular , delay-r egul arity does not imply in general that
x 0 =
0 is a feasible
initial trajectory for all inhomogeneities.
I n fact, the second and third point is a consequence from the first point: because of the possi-
ble acausality the curr ent inhomogeneity may deter mine the past (initial) state , see for instance
Example 3.7.
R emark 3.13.
I n reality , a dependence on the futur e is not possible , and therefor e one may question
the utility of the notion of delay -regularity . H o wever , besides its mathematical relev ance, this
notion may also be useful in practice if the futur e value of the inhomogeneity can be interpr eted
as a pr ediction of that future value . Additional applications ar e hybr id numerical-experimental
systems [212], see also Chapter 5, and boundar y value problems for DDAEs. ♣
R emark 3.14.
Although the choice of
t 0 =
0 in Definition 3.10, r espectively Definition 3.8, seems
arbitrary , it co vers the situation that the support of
f
is in [
t 0 , ∞
) for some
t 0 ∈ R
. T o see this,

3.1. DISTRIBUTIONAL SHIFT OPERA TOR AND DELA Y -REGULARIT Y 47
let (
P
(
s
)
, Q
(
s
))
∈ (︁ R
[
s
]
m × n x )︁ 2
be a delay -regular pair of matrix polynomials and define
g : = σ − t 0 f
.
U sing Lemma 3.2 we obtain
g ( −∞ ,0) = (︂ σ − t 0 f )︂ ( −∞ ,0) = σ − t 0 (︂ f ( −∞ , t 0 ) )︂ = 0
implying the existence of z ∈ D n x
p w C ∞ that satisfies
P (︃ d
d t )︃ z = Q (︃ d
d t )︃ σ τ z + g .
Setting x : = σ t 0 z , Lemma 3.1 implies
P (︃ d
d t )︃ x = σ t 0 (︃ P (︃ d
d t )︃ z )︃ = σ t 0 (︃ Q (︃ d
d t )︃ σ τ z + g )︃ = Q (︃ d
d t )︃ σ τ x + f .
If
˜︁
x ∈ D n x
p w C ∞
is another solution of
(3.9)
satisfying
˜︁
x ( −∞ , t 0 ) = x ( −∞ , t 0 )
, then similar arguments as
befor e sho w
x = ˜︁
x
. U sing Pr oposition 3.12 we immediately conclude that the same arguments apply
to the initial time point in the ITP (3.10). ♣
As a consequence of Theor em 3.5, we obtain the follo wing sufficient condition for delay -regularity
of the general DDAE (3.9) (cf. C orollar y 2.20).
Theor em 3.15.
Consider the ITP
(3.10)
with
m = n x
and
det
(
P
(
s
))
≡
0 . Then for any past tr a-
jector y
x 0 ∈ D n x
p w C ∞
and any inhomogeneity
f ∈ D m
p w C ∞
, ther e exists a unique solution
x ∈ D n x
p w C ∞
of (3.10) . I n par ticular , the DDAE (3.9) is delay-regular .
Proof.
The r esult follows as a consequence of C orollar y 2.20 and Theorem 3.5. F or the sake of
completeness, we pr esent the details her e as well. Let
P
(
s
)
= ∑︁ p
j = 0 P j s j
and
Q
(
s
)
= ∑︁ q
j = 0 Q j s j
. S ince
adding zero terms to
P
(
s
) does not alter the determinant of
P
(
s
), we may assume without loss of
general p = q + 1. Then a standard compa nion form linearization of (3.9) yiel ds the DDAE
E ˙
z = A z + D σ τ z + F (3.11)
with E , A , D ∈ R m + ( p − 1) n x × p n x , given b y
E = ⎡
⎢
⎢
⎢
⎢
⎢
⎣
P p 0 · · · 0
0 I n x
. . . .
.
.
.
.
. . . . . . . 0
0 · · · 0 I n x
⎤
⎥
⎥
⎥
⎥
⎥
⎦
, A = ⎡
⎢
⎢
⎢
⎢
⎣
− P p − 1 − P p − 2 · · · − P 0
I n x 0 · · · 0
.
.
. . . . . . . .
.
.
0 · · · I n x 0
⎤
⎥
⎥
⎥
⎥
⎦ , z = ⎡
⎢
⎢
⎢
⎢
⎢
⎣
(︂ d
d t )︂ p − 1 x
.
.
.
d
d t x
x
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
D = ⎡
⎢
⎢
⎢
⎢
⎣
Q p − 1 · · · Q 0
0 · · · 0
.
.
. . . . .
.
.
0 · · · 0
⎤
⎥
⎥
⎥
⎥
⎦ , F = ⎡
⎢
⎢
⎢
⎢
⎣
f
0
.
.
.
0
⎤
⎥
⎥
⎥
⎥
⎦ .

48 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
N ote that ther e exist (cf. [142]) unimodular matr ix polynomials
R
(
s
)
∈ R
[
s
]
m + ( p − 1) n x × m + ( p − 1) n x
and
S ( s ) ∈ R [ s ] p n x × p n x with
R ( s )( s E − A ) S ( s ) = [︄ P ( s ) 0
0 I ( p − 1) n x ]︄ .
The proof no w follows b y the obser vation that ther e exists a constant c = 0 with
0 ≡ det( P ( s )) = c det( s E − A )
and application of Theor em 3.5 to (3.11). ■
If
det
(
P
(
s
))
≡
0 then we cannot apply Theor em 3.15. I nstead, we want to use the condensed
form
(2.27)
from Theor em 2.24: there exist unimodular matrices
U
(
s
)
∈ R
[
s
]
m × m
,
V
(
s
)
∈ R
[
s
]
n x × n x
such that
U ( s ) P ( s ) V ( s ) = ⎡
⎢
⎣ ˆ︂
P 11 ( s ) 0 ˆ︂
P 13 ( s )
0 0 ˆ︂
P 23 ( s )
0 0 ˆ︂
P 33 ( s ) ⎤
⎥
⎦ and U ( s ) Q ( s ) V ( s ) = ⎡
⎢
⎣ ˆ︁
Q 11 ( s ) ˆ︁
Q 12 ( s ) ˆ︁
Q 13 ( s )
0 0 ˆ︁
Q 23 ( s )
0 0 ˆ︁
Q 33 ( s ) ⎤
⎥
⎦ ,
wher e
ˆ︂
P 11
is a nonsingular diagonal matrix,
ˆ︂
P 23 , ˆ︂
P 33 , ˆ︁
Q 33
ar e block upper tr iangular matrices
with zero diagonal blocks and
ˆ︁
Q 23
is a nonsingular block upper triangular matrix. W e therefor e
have to ensur e that the transformation of (
P
(
s
)
, Q
(
s
)) with unimodular matrices does not affect
delay -regularity .
Pr oposition 3.16.
Consider a pair of matrix polynomials (
P
(
s
)
, Q
(
s
))
∈
(
R
[
s
]
m × n x
)
2
and unimod-
ular matrices U ( s ) ∈ R [ s ] m × m and V ( s ) ∈ R [ s ] n x × n x . Let
ˆ︂
P ( s ) : = U ( s ) P ( s ) V ( s ) and ˆ︁
Q ( s ) : = U ( s ) Q ( s ), V ( s ).
Then ( P ( s ), Q ( s )) is delay-r egular if, and only if ( ˆ︂
P ( s ), ˆ︁
Q ( s )) is delay-r egular .
Proof.
First note that it is sufficient to sho w one direction. Let
ˆ︁
f ∈ D m
p w C ∞
with support in [0
, ∞
) and
consider
ˆ︂
P (︃ d
d t )︃ ˆ︁
x = ˆ︁
Q (︃ d
d t )︃ σ τ ˆ︁
x + ˆ︁
f . (3.12)
Define f : = U (︂ d
d t )︂ − 1 ˆ︁
f and obser ve that
f ( −∞ ,0) = (︃ U (︃ d
d t )︃ − 1 ˆ︁
f )︃ ( −∞ ,0) = (︃ U (︃ d
d t )︃ − 1 ˆ︁
f ( −∞ ,0) )︃ ( −∞ ,0) = 0.
Delay -r egular ity of
(3.9)
thus implies the existence of a solution
x ∈ D n x
p w C ∞
of
(3.9)
. The choice
ˆ︁
x = V (︂ d
d t )︂ − 1 x together with
ˆ︂
P (︃ d
d t )︃ ˆ︁
x = U (︃ d
d t )︃ P (︃ d
d t )︃ V (︃ d
d t )︃ x = U (︃ d
d t )︃ (︃ Q (︃ d
d t )︃ σ τ x + f )︃ = ˜︁
Q (︃ d
d t )︃ σ τ ˆ︁
x + ˆ︁
f .

3.1. DISTRIBUTIONAL SHIFT OPERA TOR AND DELA Y -REGULARIT Y 49
sho ws existence of a solution for
(3.12)
. Assume now th at
ˆ︁
x 1 , ˆ︁
x 2 ∈ D n x
p w C ∞
ar e solutions of
(3.12)
for
the same
ˆ︁
f ∈ D m
p w C ∞
satisfying (
ˆ︁
x 1
)
( −∞ ,0) =
(
ˆ︁
x 2
)
( −∞ ,0)
. F or
i =
1
,
2 define
x i : = V (︂ d
d t )︂ ˆ︁
x i
and obser ve
that
P (︃ d
d t )︃ x i = U (︃ d
d t )︃ − 1 ˜︂
P (︃ d
d t )︃ ˆ︁
x i = U (︃ d
d t )︃ − 1 (︃ ˜︁
Q (︃ d
d t )︃ σ τ ˆ︁
x i + ˆ︁
f )︃ = Q (︃ d
d t )︃ σ τ x i + U (︃ d
d t )︃ − 1 ˆ︁
f .
Delay - r egular ity thus implies x 1 = x 2 and since V ( s ) is unimodular we conclude ˆ︁
x 1 = ˆ︁
x 2 . ■
Theor em 3.17.
Consider (
P
(
s
)
, Q
(
s
))
∈ (︁ R [ s ] m × n x )︁ 2
. Let
U
(
s
)
∈ R
[
s
]
m × m
and
V
(
s
)
∈ R
[
s
]
n x × n x
be the unimodular matrices from Theor em 2.24. Define
ˆ︂
P
(
s
)
: = U
(
s
)
P
(
s
)
V
(
s
)
∈ R
[
s
]
m × n x
and
ˆ︁
Q ( s ) : = U ( s ) Q ( s ) V ( s ) ∈ R [ s ] m × n x , i.e.,
ˆ︂
P ( s ) = ⎡
⎢
⎣ ˆ︂
P 11 ( s ) 0 ˆ︂
P 13 ( s )
0 0 ˆ︂
P 23 ( s )
0 0 ˆ︂
P 33 ( s ) ⎤
⎥
⎦ , ˆ︁
Q ( s ) = ⎡
⎢
⎣ ˆ︁
Q 11 ( s ) ˆ︁
Q 12 ( s ) ˆ︁
Q 13 ( s )
0 0 ˆ︁
Q 23 ( s )
0 0 ˆ︁
Q 33 ( s ) ⎤
⎥
⎦ . (3.13)
Then the follo wing statements ar e t rue.
(i) The pair ( ˆ︂
P 23 ( s ), ˆ︁
Q 23 ( s )) is delay-regular .
(ii)
The pair (
ˆ︂
P
(
s
)
, ˆ︁
Q
(
s
)) is delay-r egular if and only if the second column and the thir d block
ro w are not pr esent .
(iii) The pair ( P ( s ), Q ( s )) is delay-r egular if and only if ( ˆ︂
P ( s ), ˆ︁
Q ( s )) is delay-r egular .
Befor e we pr esent the proof of Theorem 3.17 let us r evisit Example 3.9.
Example 3.18. C onsider the DDAE from Example 3.9, i.e .,
P ( s ) = ⎡
⎢
⎣
0 s 0
0 − 1 s
0 0 − 1 ⎤
⎥
⎦ and Q ( s ) = ⎡
⎢
⎣
0 0 0
0 0 0
1 0 0 ⎤
⎥
⎦ .
W e obtain
⎡
⎢
⎣
0 1 s
0 0 1
1 s s 2 ⎤
⎥
⎦ P ( s ) ⎡
⎢
⎣
0 0 1
1 0 0
0 1 0 ⎤
⎥
⎦ = ⎡
⎢
⎣ − 1 0 0
0 − 1 0
0 0 0 ⎤
⎥
⎦ ,
⎡
⎢
⎣
0 1 s
0 0 1
1 s s 2 ⎤
⎥
⎦ Q ( s ) ⎡
⎢
⎣
0 0 1
1 0 0
0 1 0 ⎤
⎥
⎦ = ⎡
⎢
⎣
0 0 s
0 0 1
0 0 s 2 ⎤
⎥
⎦ .
Clearly , the second block column and the thir d block-ro w in
(3.13)
ar e not present, such that
Theor em 3.17 implies that ( P ( s ), Q ( s )) is delay - r egular . ♠
F or the proof of Theor em 3.17 we first need the follo wing technical r esult, which generalizes the
findings from Example 3.7.

50 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
Lemma 3.19.
Let
Q
(
s
)
∈ R
[
s
]
n x × n x
satisfy
det
(
Q
(
s
))
≡
0 . Then the pair (0
, Q
(
s
)) is delay-r egular .
I n par ticular , for any
f ∈ D n x
p w C ∞
with support in [0
, ∞
) ther e exists
x ∈ D n x
p w C ∞
with
x ( −∞ , − τ ) =
0
that satisfies the DDAE
0 = Q (︃ d
d t )︃ σ τ x + f . (3.14)
Proof.
Let
f ∈ D n x
p w C ∞
with support in [0
, ∞
). Theor em 3 .15 implies that ther e exists a unique
solution x ∈ D n x
p w C ∞ of the ITP
x ( −∞ , − τ ) = 0,
(︃ Q (︃ d
d t )︃ x )︃ [ − τ , ∞ ) = − (︁ σ − τ f )︁ [ − τ , ∞ ) , (3.15)
see also R emark 3.14. Lemmata 3.1 and 3.2 yield
(︃ Q (︃ d
d t )︃ σ τ x )︃ ( −∞ ,0) = σ τ (︃ Q (︃ d
d t )︃ x )︃ ( −∞ , − τ ) = σ τ (︃ Q (︃ d
d t )︃ x ( −∞ , − τ ) )︃ ( −∞ , − τ ) = 0,
and (︃ Q (︃ d
d t )︃ σ τ x + f )︃ [0, ∞ ) = σ τ (︃ Q (︃ d
d t )︃ x + σ − τ f )︃ [ − τ , ∞ ) = 0,
sho wing that
x
satisfies the DDAE
(3.14)
. Suppose no w that
x 1 , x 2 ∈ D n x
p w C ∞
solve
(3.14)
and satisfy
( x 1 ) ( −∞ ,0) = ( x 2 ) ( −∞ ,0) . Then the difference ˜︁
x : = x 1 − x 2 satisfies
0 = Q (︃ d
d t )︃ σ τ ˜︁
x
and thus also the ITP
(3.15)
with
f =
0. Theor em 3. 15 implies
˜︁
x =
0, which completes the proof.
■
Proof of Theor em 3.17.
(i) S ince ˆ︂
P 23 ( s ) and ˆ︁
Q 23 ( s ) are block upper triangular , we write (omitting s )
ˆ︂
P 23 = ⎡
⎢
⎢
⎢
⎢
⎢
⎣
0 ˜︂
P 1,2 · · · ˜︂
P 1, k
. . . . . . .
.
.
. . . ˜︂
P k − 1, k
0
⎤
⎥
⎥
⎥
⎥
⎥
⎦
, ˆ︁
Q 23 = ⎡
⎢
⎢
⎣ ˜︁
Q 1,1 · · · ˜︁
Q 1, k
. . . .
.
.
˜︁
Q k , k
⎤
⎥
⎥
⎦ , z = ⎡
⎢
⎢
⎣
z 1
.
.
.
z k
⎤
⎥
⎥
⎦ , g = ⎡
⎢
⎢
⎣
g 1
.
.
.
g k
⎤
⎥
⎥
⎦ ,
and study the DDAE
ˆ︂
P 23 (︃ d
d t )︃ z = ˆ︁
Q 23 (︃ d
d t )︃ σ τ z + g
with
g ( −∞ ,0) =
0. S ince
ˆ︁
Q 23
(
s
) is nonsingular , we conclude that
˜︁
Q i , i
(
s
) is nonsingular for all
i =
1
, . . . , k
. In particular , Lemma 3.19 implies that (0
, ˜︁
Q k , k
(
s
)) is delay -regular , i.e., ther e exists
z k with (︁ z k )︁ ( −∞ , − τ ) = 0 satisfying
0 = ˜︁
Q k , k (︃ d
d t )︃ σ τ z k + g k .

3.1. DISTRIBUTIONAL SHIFT OPERA TOR AND DELA Y -REGULARIT Y 51
S ubstituting z k into the ( k − 1)th block equation yields the DDAE
0 = ˜︁
Q k − 1, k − 1 (︃ d
d t )︃ σ τ z k − 1 + ˜︁
g k − 1
with ˜︁
g k − 1 : = g k − 1 − ˜︂
P k − 1, k (︂ d
d t )︂ z k + ˜︁
Q k − 1, k (︂ d
d t )︂ σ τ z k . H aving
(︁ ˜︁
g k − 1 )︁ ( −∞ , − τ ) = − (︃ ˜︂
P k − 1, k (︃ d
d t )︃ (︁ z k )︁ ( −∞ , − τ ) + ˜︁
Q k − 1, k (︃ d
d t )︃ σ τ (︂ (︁ z k )︁ ( −∞ , − 2 τ ) )︂ )︃ ( −∞ , − τ ) = 0,
we conclude with the same line of arguing that
(︄[︄ 0 ˜︂
P k − 1, k ( s )
0 0 ]︄ , [︄ ˜︁
Q k − 1, k − 1 ( s ) ˜︁
Q k − 1, k ( s )
0 ˜︁
Q k , k ( s ) ]︄)︄
is delay -regular . Repeating this process sho ws that ( ˆ︂
P 2,3 ( s ), ˆ︁
Q 2,3 ( s )) is delay -r egular .
(ii) F or ˆ
f T = [︂ ˆ
f T
1 ˆ
f T
2 ˆ
f T
3 ]︂ consider the DDAE
ˆ︂
P (︃ d
d t )︃ ˆ
x = ˆ︁
Q (︃ d
d t )︃ σ τ ˆ
x + ˆ
f . (3.16)
S ince ( ˆ︂
P 23 ( s ), ˆ︁
Q 23 ( s )) is delay -r egular , there exists ˆ
x 3 satisfying
ˆ︂
P 23 (︃ d
d t )︃ ˆ
x 3 = ˆ︁
Q 23 (︃ d
d t )︃ σ τ ˆ
x 3 + ˆ
f 2 . (3.17)
Assume first that ( ˆ︂
P ( s ), ˆ︁
Q ( s )) is delay -regular . Setting
ˆ
f 3 : = ˆ︂
P 33 (︃ d
d t )︃ ˆ
x 3 − ˆ︁
Q 33 (︃ d
d t )︃ ˆ
x 3 + 1 [0, ∞ ) ⎡
⎢
⎢
⎣
1
.
.
.
1
⎤
⎥
⎥
⎦
yields 0
= 1 [0, ∞ )
, which is true only if the third block r o w is not pr esent. In addition, for any
ˆ
x 2
,
and any ˆ
x 0
1 the ITP
( ˆ
x 1 ) ( −∞ ,0) = ( ˆ
x 0
1 ) ( −∞ ,0) ,
(︃ ˆ︂
P 11 (︃ d
d t )︃ ˆ
x 1 )︃ [0, ∞ ) = (︃ ˆ︁
Q 11 (︃ d
d t )︃ σ τ ˆ
x 1 + g )︃ [0, ∞ )
(3.18)
with
g : = ˆ
f 1 − ˆ︂
P 13 (︂ d
d t )︂ ˆ
x 3 + ˆ︁
Q 12 (︂ d
d t )︂ σ τ ˆ
x 2 + ˆ︁
Q 13 (︂ d
d t )︂ σ τ ˆ
x 3
has a unique solution (cf. Theo-
r em 3.15 ). I n par ticular we can choose (
ˆ
x 2
)
[0, ∞ )
arbitrarily and thus the delay -r egularity of
( ˆ︂
P ( s ), ˆ︁
Q ( s )) implies that the second block column is not pr esent. Conversely , assume
ˆ︂
P ( s ) = [︄ ˆ︂
P 11 ( s ) ˆ︂
P 13 ( s )
0 ˆ︂
P 23 ( s ) ]︄ , ˆ︁
Q ( s ) = [︄ ˆ︁
Q 11 ( s ) ˆ︁
Q 13 ( s )
0 ˆ︁
Q 23 ( s ) ]︄ .
W e have alr eady established that there exists
ˆ
x 3
solving
(3.17)
. Additionally , the ITP
(3.18)
has
a unique solution for any initial trajectory ˆ
x 0
1 and thus ( ˆ︂
P ( s ), ˆ︁
Q ( s )) is delay -regular .
(iii) This is a consequence of Pr oposition 3.16. ■

52 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
Theor em 3.20.
The pair (
P
(
s
)
, Q
(
s
)) is delay-r egular if and only if
m = n x
and ther e exists
s , ω ∈ C
with
det( P ( s ) − ω Q ( s )) = 0. (3.19)
I n par ticular , if
m < n x
then the ITP
(3.10)
possesses a nontrivial solution
x ∈ D n x
p w C ∞
with
f =
0
and
x ( −∞ ,0) =
0 . If instead
m ≥ n x
and
rank
(
P
(
s
)
− ω Q
(
s
))
< m
for all
s , ω ∈ C
then ther e exists an
f ∈ D m
p w C ∞ for which the ITP (3.10) has no solution.
Proof.
Theor em 3.17 i mplies that
(3.9)
is delay -regular if an d only if the second block column and
the thir d block ro w in (2.27) do not appear . W e obtain
det ( P ( s ) − ω Q ( s ) ) = c det ( ˆ︂
P ( s ) − ω ˆ︁
Q ( s ) )
= c det (︁ ˆ︂
P 11 ( s ) − ω ˆ︁
Q 11 ( s ) )︁ det (︁ ˆ︂
P 23 ( s ) − ω ˆ︁
Q 23 ( s ) )︁ ,
with
c : = det
(
U
(
s
))
− 1 det
(
V
(
s
))
− 1 =
0. U sing the notation from the proof of Theor em 3.17 we obtain
det (︁ ˆ︂
P 23 ( s ) − ω ˆ︁
Q 23 ( s ) )︁ =
k
∏︂
i = 1
det (︁ − ω ˜︁
Q i , i ( s ) )︁ = ( − ω ) ρ k
∏︂
i = 1
det( ˜︁
Q i , i ( s )) ≡ 0,
since
˜︁
Q i , i
(
s
) is nonsingular for
i =
1
, . . . , k
. H ereb y ,
ρ
denotes the dimension of the squar e ma-
trix
ˆ︁
Q 23
(
s
). The nonsingular ity of
ˆ︂
P 11
(
s
) thus sho ws that the delay -regularity of (
P
(
s
)
, Q
(
s
)) implies
(3.19)
. F o r the converse dir ection we observe that
(3.19)
immediately implies that the second block
column cannot be pr esent. Fr om
n x = m
we infer that also the thir d block ro w cannot be pr esent,
such that Theor em 3.17 implies delay-r egularity of (
P
(
s
)
, Q
(
s
)). F or the remaining asser tions notice
that
m < n x
implies that the second block column appears in
(2.27)
. On the other hand, if
m ≥ n x
and
rank R [ s , ω ] ( P ( s ) − ω Q ( s )) < m ,
then we conclude that the thir d block-ro w in
(2.27)
is pr esent. The result follo ws from the proof of
Theor em 3.17 (ii). ■
A pplying Theorem 3.20 to the DDAE (3.2) we obtain the follo wing corollar y .
Cor ollar y 3.21.
The triplet (
E , A 1 , A 2
) is delay-r egular if and only if
m = n x
and ther e exists
s , ω ∈ C
with
det( s E − A 1 − ω A 2 ) = 0. (3.20)
I n par ticular , if
m < n x
then the ITP
(3.4)
possesses a nontrivial solution
x ∈ D n x
p w C ∞
with
f =
0
and
x ( −∞ ,0) =
0 . If instead
m ≥ n x
and
rank
(
P
(
s
)
− ω Q
(
s
))
< m
for all
s , ω ∈ C
then ther e exists an
f ∈ D m
p w C ∞ for which the ITP (3.4) has no solution.
R emark 3.22.
The statements of Theor em 3.20 and Corollary 3.21 are a generalization of Theo-
r em 2.7 to DDAEs. ♣

3.2. INTERL UDE: FE EDBA CK REGUL ARIZA TION OF DAES WITH DEL A Y 53
Let us emphasize that for delay -regularity we r equire existence of solutions for all
f ∈ D m
p w C ∞
.
Consequently , the ITP
(3.10)
may possess a unique solution even if (
P
(
s
)
, Q
(
s
)) is not delay -regular .
Consider for instance the DDAE
0 = Q (︃ d
d t )︃ σ τ x + [︄ f
˙
f ]︄ (3.21)
with
Q
(
s
)
: = [︁ 1
s ]︁
and
f ∈ D p w C ∞
. Theor e m 3.20 immediately implies that (0
, Q
(
s
)) is not delay -
r egular . S till , the unique solution of (3.21) is given b y x = − σ − τ f .
R emark 3.23.
Equation (3.21) contradicts [96, C or . 2 and Cor . 3], which claim that whenever a
solution of the ITP
(3.10)
exists it is unique only if
m = n x
and
(3.19)
holds . N ote that in view of the
condensed form
(2.27)
, uniqueness is r el ated to the (non-)existence of the second block column,
while existence is r elated to the (non-)existence of the third block ro w . ♣
3.2 Inter lude: F eedback r egular ization of DAE s with delay
A standar d control concept uses feedback , i.e., the control law depends on the curr ent state or output
of the system (see for instance Section 1.1.2). F or linear time-invariant systems of the form
E ˙
x = A 1 x + B u + f ,
y = C x (3.22)
wher e
B ∈ R m × n u
,
C ∈ R n y × n x
,
u
is the
n u
-dimensional input, and
y
is the
n y
-dimensional output, a
simple feedback law takes the form
u = F y = F C x (3.23)
for some feedback matrix F ∈ R n u × n y . The closed-loop system is thus given b y the DAE
E ˙
x = (︁ A 1 + B F C )︁ x + f , (3.24)
sho wing that the feedback can be used to alter system properties . F or instance, suppose that
(
E , A 1
)
∈ (︁ R n x × n x )︁ 2
is singular . W e say that is is possible to r egularize
(3.22)
, if there exists some
F ∈ R n u × n y such that the pencil ( E , A 1 + B F C ) is regular . I n fact the following r esult from [76] holds .
Lemma 3.24. Consider the DAE (3.22) with m = n x . There e xists F ∈ R n u × n y such that the closed-
loop system ( E , A 1 + B F C ) is regular if and only if
rank (︂[︂ λ E − A 1 B ]︂)︂ = rank (︄[︄ λ E − A 1
C T ]︄)︄ = n x
for some λ ∈ C .
Although instantaneous feedback is a convenient theor etical approach, it is usuall y not imple-
mentable , in par ticular , when the signals have to be measured first, and some calculations h ave to

54 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
be carried out, thus resulting in an intrinsically necessary time delay . W e therefor e are inter ested
whether we can r egularize (3.22) with a delayed feedback
u = F σ τ y = F C σ τ x , (3.25)
i.e ., if we can find
F ∈ R n u × n y
such that (
s E − A 1 , B C F
) is delay -regular . As impor tant consequence
of Theor em 3.20, respectively C orollar y 3.21, we obtain the follo wing result.
Theor em 3.25.
F or
m = n x
consider the descriptor system
(3.22)
. There e xists
F ∈ R n u × n y
such
that (
E , A 1 + B F C
) is r egular if and only if there e xists
ˆ︁
F ∈ R n u × n y
such that
( s E − A 1 , B ˆ︁
F C )
is
delay-r egular .
Proof. Assume first that ther e exists F ∈ R n u × n y such that ( E , A 1 + B F C ) is regular . Then we have
n x = rank R [ s ] ( s E − A 1 − B F C ) ≤ rank R [ s , ω ] ( s E − A 1 − ω B F C ) ≤ n x ,
i. e., (
s E − A 1 , B F C
) is delay -regular . On the other hand, assume the existence of
ˆ︁
F ∈ R n u × n y
such
that ( s E − A 1 , B ˆ︁
F C ) is delay -r egular . Then there exists ˆ︁
ω ∈ R such that
det( s E − A 1 − ˆ︁
ω B ˆ︁
F C ) ≡ 0.
The choice F = ˆ︁
ω ˆ︁
F guarantees that ( E , A 1 + B F C ) is r egular . ■
The proof of Theor em 3.25 details that for any feedback matrix
F ∈ R n u × n y
that r enders the pencil
(
E , A 1 + B F C
) r egular , also the tr iplet (
E , A 1 , B F C
) is delay -regular . The converse dir ection is ho wever
not true, as w e can see from the follo wing example:
Example 3.26.
Consider the scalar DAE
(3.22)
with
E =
0,
A 1 =
1,
B =
1, and
C =
1. F or
ˆ︁
F = −
1
the pair of matrix polynomials (
s E − A 1 , B ˆ︁
F C
)
=
(
−
1
, −
1) is delay -regular . H o wever , the pencil
( E , A 1 + B ˆ︁
F C ) = (0, 0) is not regular . ♠
The r eason for the behavior in Example 3.26 is due to the fact that the limit
τ →
0 (implying
ω →
1)
may be singular , as for example pointed out in [214] in ter ms of stability of a neutral delay differ e ntial
equation (DDE).
Cor ollar y 3.27.
F or
m = n x
consider the descriptor system
(3.22)
. Then ther e exists a feedback
matrix F ∈ R n u × n y such that ( s E − A 1 , B F C ) is delay-r egular if and only if
rank [︂ λ E − A 1 B ]︂ = rank [︄ λ E − A 1
C ]︄ = n x
for some λ ∈ C .
Proof. The pr oof follo ws from Theor em 3.25 and [76, Thm. 2]. ■

3.3. DEL A Y -EQ UIV ALE NCE AND THE COMPRESS-AND-SHIFT AL GORITHM 55
3.3 Delay -equivalence and the compr ess-and-shift algorithm
I n the proof of the delay -regularity of (
ˆ︂
P 23
(
s
)
, ˆ︁
Q 23
) in Theor em 3.17 (i), we used Lemma 3.19 to
construct a solution of the associated ITP. The key idea in the proof of Lemma 3.19 is to shift the
equations to obtain a transformed DDAE (
˜︂
P
(
s
)
, ˜︁
Q
(
s
)) with
det
(
˜︂
P
(
s
))
≡
0. T o for malize this idea, we
introduce the notion of delay -equivalence .
Definition 3.28.
T wo pairs of matrix polynomials (
P
(
s
)
, Q
(
s
))
,
(
˜︂
P
(
s
)
, ˜︁
Q
(
s
))
∈
(
R
[
s
]
m × n x
)
2
ar e called
delay-equivalent if and only if ther e exists a bijective map
T : D m
p w C ∞ → D m
p w C ∞
such that for all
( x , f ) ∈ D n x
p w C ∞ × D m
p w C ∞ and ˜︁
f : = T f the equivalence
P (︃ d
d t )︃ x = Q (︃ d
d t )︃ σ τ x + f ⇐ ⇒ ˜︂
P (︃ d
d t )︃ x = ˜︁
Q (︃ d
d t )︃ σ τ x + ˜︁
f
holds . I n this case we wri te (
P
(
s
)
, Q
(
s
))
d
∼
(
˜︂
P
(
s
)
, ˜︁
Q
(
s
)). W e say that two DDAEs are delay -equivalent
if the associated matrix polynomials are delay -equivalent.
I t is easy to verify that delay -equivalence is indeed an equivalence r elation. W e note that delay -
equivalence is a property of a distributional DDAE and delay -regularity is a property of an ITP.
H ence we first have to establish a r elation between delay -equivalence and delay -r egular ity .
Pr oposition 3.29.
Consider pairs of matrix polynomials (
P
(
s
)
, Q
(
s
))
,
(
˜︂
P
(
s
)
, ˜︁
Q
(
s
))
∈
(
R
[
s
]
n x × n x
)
2
and assume ( P ( s ), Q ( s )) d
∼ ( ˜︂
P ( s ), ˜︁
Q ( s )) . T hen the pair ( P ( s ), Q ( s )) is delay-regular if and only if
the pair ( ˜︂
P ( s ), ˜︁
Q ( s )) is delay-r egular .
Proof.
First, let (
˜︂
P
(
s
)
, ˜︁
Q
(
s
)) be delay -regular . Assume that (
P
(
s
)
, Q
(
s
)) is not delay -regular . Let
(
U
(
s
)
, V
(
s
)) denote the matrices from Theor em 2 .24 that transform (
P
(
s
)
, Q
(
s
)) to the condensed
form (2.27), i.e.,
U ( s ) P ( s ) V ( s ) = ⎡
⎢
⎣ ˆ︂
P 11 ( s ) 0 ˆ︂
P 13 ( s )
0 0 ˆ︂
P 23 ( s )
0 0 ˆ︂
P 33 ( s ) ⎤
⎥
⎦ and U ( s ) Q ( s ) V ( s ) = ⎡
⎢
⎣ ˆ︁
Q 11 ( s ) ˆ︁
Q 12 ( s ) ˆ︁
Q 13 ( s )
0 0 ˆ︁
Q 23 ( s )
0 0 ˆ︁
Q 33 ( s ) ⎤
⎥
⎦ ,
wher e
ˆ︂
P 11
(
s
) is a nonsingular diagonal matrix,
ˆ︂
P 23
(
s
)
, ˆ︂
P 33
(
s
)
, ˆ︁
Q 33
(
s
) ar e block upper tr iangular
matrices with zero diagonal blocks and
ˆ︁
Q 23
(
s
) is a nonsingular block upper triangular matrix. N ote
that
m = n x
implies that the second block column and the thir d block ro w are both pr esent. Let
˜
f ∈ D n x
p w C ∞
with support in [0
, ∞
). Since (
˜︂
P
(
s
)
, ˜︁
Q
(
s
)) is delay -reg ular , there exists
˜
x ∈ D n x
p w C ∞
solving
the DDAE
˜︂
P (︃ d
d t )︃ ˜
x = ˜︁
Q (︃ d
d t )︃ σ τ ˜
x + ˜
f . (3.26)
B y assumption ˜
x is a solution of
P (︃ d
d t )︃ ˜
x = Q (︃ d
d t )︃ σ τ ˜
x + T − 1 ˜
f .

56 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
Define
x : = ⎡
⎢
⎣
x 1
x 2
x 3
⎤
⎥
⎦ : = V (︃ d
d t )︃ − 1
˜
x and ⎡
⎢
⎣
f 1
f 2
f 3
⎤
⎥
⎦ : = U (︃ d
d t )︃ T − 1 ˜
f .
Define ˆ
x 2 : = x 2 + 1 [1, ∞ ) and let ˆ
x 1 be the unique solution (cf. Theor em 3.15) of the ITP
(︁ ˆ
x 1 )︁ ( −∞ ,0) = (︁ x 1 )︁ ( −∞ ,0) ,
(︃ ˆ︂
P 11 (︃ d
d t )︃ ˆ
x 1 )︃ [0, ∞ ) = (︃ ˆ︁
Q 11 (︃ d
d t )︃ σ τ ˆ
x 1 + ˆ︁
Q 12 (︃ d
d t )︃ σ τ ˆ
x 2 + ˆ︁
Q 13 (︃ d
d t )︃ σ τ x 3 − ˆ︂
P 13 (︃ d
d t )︃ x 3 + f 1 )︃ [0, ∞ )
.
Define
ˇ
x : = V (︃ d
d t )︃ ⎡
⎢
⎣
ˆ
x 1
ˆ
x 2
x 3
⎤
⎥
⎦ .
B y constr uction (and Lemma 2.22) we obtain ˇ
x ( −∞ ,0) = ˆ
x ( −∞ ,0) , ˇ
x = ˆ
x , and
P (︃ d
d t )︃ ˇ
x = Q (︃ d
d t )︃ σ τ ˇ
x + T − 1 ˜
f .
Delay -equivalence implies that
ˇ
x
is another solution of the ITP
(3.26)
contradicting the delay -
r egular ity of (
˜︂
P
(
s
)
, ˜︁
Q
(
s
)). Thus (
P
(
s
)
, Q
(
s
)) is delay -regular . Inter changing the roles of (
˜︂
P
(
s
)
, ˜︁
Q
(
s
))
and ( P ( s ), Q ( s )) shows the conv erse direction. ■
Analyzing the situation in Lemma 3.19, respectiv ely i n Theor em 3.17 (ii) sho wcases that we shift
equations if they depend solely on delayed variables . If the equations are not y et in such a form, one
first has to transform them, for instance with a rank-r evealing decomposition of
P
(
s
).
Lemma 3.30.
F or (
P
(
s
)
, Q
(
s
))
∈
(
R
[
s
]
m × n x
)
2
and inhomogeneity
f ∈ D m
p w C ∞
choose a unimodular
matrix U ( s ) = [︂ U 1 ( s )
U 2 ( s ) ]︂ ∈ R [ s ] m × m such that
U ( s ) P ( s ) = : [︄ P 1 ( s )
0 ]︄ , U ( s ) Q ( s ) = : [︄ Q 1 ( s )
Q 2 ( s ) ]︄ ,
wher e
P 1
(
s
)
, Q 1
(
s
)
∈ R
[
s
]
k × n x
with
rank R [ s ] (︁ P 1 ( s ) )︁ = k
. Then the DDAE
(3.9)
is delay-equivalent
to the partially time-shifted DDAE
P 1 (︃ d
d t )︃ x = Q 1 (︃ d
d t )︃ σ τ x + f 1 ,
Q 2 (︃ d
d t )︃ x = − σ − τ f 2
(3.27)
with
[︂ f 1
f 2 ]︂
:
= U
(
d
d t
)
f
. In particular , (
P
(
s
)
, Q
(
s
)) is delay-r egular if and only if
(︂[︂ P 1 ( s )
Q 2 ( s ) ]︂ , [︁ Q 1 ( s )
0 ]︁ )︂
is
delay-r egular .

3.3. DEL A Y -EQ UIV ALE NCE AND THE COMPRESS-AND-SHIFT AL GORITHM 57
Proof.
Assume that (
x , f
)
∈ D n x
p w C ∞ × D m
p w C ∞
satisfies the DDAE
(3.9)
. M ultiplication of
(3.9)
from
the left with U (︂ d
d t )︂ sho ws that (︂ x , U (︂ d
d t )︂ f )︂ solves
P 1 (︃ d
d t )︃ x = Q 1 (︃ d
d t )︃ σ τ x + f 1 ,
0 = Q 2 (︃ d
d t )︃ σ τ x + f 2 .
(3.28)
S ince
U
(
s
) is unimodular we can r everse the transformation such that
(3.28)
is delay -equivalent
to
(3.9)
. Applying a negativ e time-shift
σ − τ
on the second equation and taking into account that
differ entiation and shifting commute (cf. Lemma 3.1) we obtain the DDAE
P 1 (︃ d
d t )︃ x = Q 1 (︃ d
d t )︃ σ τ x + f 1
Q 2 (︃ d
d t )︃ x = ¯
f 2
wher e
¯
f 2 : = − σ − τ f 2
. Clearly , the transformation of the inhomogeneity is r eversible, hence this DDAE
is delay -equivalent to (3.9). ■
The pr evious results suggest to perform a simple compress-and-shift algorithm (see for instance
[53, 206]) to determine whether a DDAE is delay -regular or not: If P ( s ) is rank deficient, perform a
ro w compr ession of P ( s ). If Q 2 ( s ) is rank deficient, then the DDAE is not delay -regular . Otherwise
shift
Q 2
(
s
) and r estar t the algorithm with the transformed matrix polynomials. The details are
outlined in Algorithm 1. Lemma 3.30 ensur es that Algorithm 1 constr ucts a sequence of delay -
equivalent polynomial matrix pairs ( P ν ( s ), Q ν ( s )) ∈ ( R [ s ] m × n x ) 2 .
Lemma 3.31.
Assume that Algorithm 1 terminates after
ν
iter ati ons for the polynomial matrix
pair (
P
(
s
)
, Q
(
s
))
∈ R
[
s
]
n x × n x
. Then the pair of polynomial matrices (
P
(
s
)
, Q
(
s
)) is delay-r egular if
and only if k ν = n x .
Proof.
If
k ν = n x
, then
det
(
P ν
(
s
))
≡
0 and thus Theor em 3.15 implies that (
P ν
(
s
)
, Q ν
(
s
)) is delay -
r egular . U sing Pr oposition 3.29 we conclude that (
P
(
s
)
, Q
(
s
)) is delay -regular . Conv ersely , assume
k ν < n x . Then
rank R [ s , w ] (︁ P ν ( s ) − ω Q ν ( s ) )︁ = r ank R [ s , w ] (︄[︄ P ν ,1 ( s )
0 ]︄ − ω [︄ Q ν ,1 ( s )
0 ]︄)︄ = k ν < n x .
Thus, Theor em 3.20 implies that (
P ν
(
s
)
, Q ν
(
s
)) is not delay -regular , w hich completes the proof.
■
Example 3.32. C onsider the matrix polynomials
P ( s ) = [︄ s 2 0
0 0 ]︄ and Q ( s ) = [︄ 0 s − 1
s 0 ]︄ .

58 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
Algorithm 1 Compr ess-and-shift
I nput: P ( s ), Q ( s ) ∈ R [ s ] m × n x
1: Set ν = 1 and P ν ( s ) : = P ( s ), Q ν ( s ) : = Q ( s ).
2: Choose unimodular
U ν ( s ) = ⎡
⎢
⎣
U ν ,1 ( s )
U ν ,2 ( s )
U ν ,3 ( s ) ⎤
⎥
⎦ ∈ R [ s ] m × m with matrix polynomials ⎧
⎪
⎨
⎪
⎩
U ν ,1 ( s ) ∈ R [ s ] k ν × m ,
U ν ,2 ( s ) ∈ R [ s ] ρ ν × m ,
U ν ,3 ( s ) ∈ R [ s ] m − k ν − ρ ν × m ,
(3.29)
such that
U ν ( s ) P ν ( s ) = : ⎡
⎢
⎣
P ν ,1 ( s )
0
0 ⎤
⎥
⎦ and U ν ( s ) Q ν ( s ) = : ⎡
⎢
⎣
Q ν ,1 ( s )
Q ν ,2 ( s )
0 ⎤
⎥
⎦ ,
wher e
P ν ,1
(
s
)
: = U ν ,1
(
s
)
P ν
(
s
)
∈ R
[
s
]
k ν × n x
and
Q ν ,2
(
s
)
: = U ν ,2
(
s
)
Q ν
(
s
)
∈ R
[
s
]
ρ ν × n x
have full ro w
rank.
3: if ρ ν = 0 then
4: terminate
5: else
6: Set P ν + 1 ( s ) = [︂ P ν ,1 ( s )
Q ν ,2 ( s ) ]︂ and Q ν + 1 ( s ) = [︂ Q ν ,1 ( s )
0 ]︂ .
7: Set ν ← ν + 1.
8: Go to Line 2
9: end if

3.3. DEL A Y -EQ UIV ALE NCE AND THE COMPRESS-AND-SHIFT AL GORITHM 59
A pplying Algor ithm 1 to ( P ( s ), Q ( s )) yields U 1 ( s ) = I with Q 1,2 ( s ) = [︂ s 0 ]︂ in Line 2. W e obtain
P 2 ( s ) = [︄ s 2 0
s 0 ]︄ , Q 2 ( s ) = [︄ 0 s − 1
0 0 ]︄ , U 2 ( s ) = [︄ 0 1
1 − s ]︄ ,
and ρ 2 = 1. The new matrix polynomials
P 3 ( s ) = [︄ s 0
0 s − 1 ]︄ and Q 3 ( s ) = [︄ 0 0
0 0 ]︄
satisfy rank R [ s ] (︁ P 3 ( s ) )︁ = 2 = n x and thus Lemma 3.31 implies that ( P ( s ), Q ( s )) is delay -regular . ♠
Example 3.33. A pplying Algor ithm 1 to the matrix polynomials
P ( s ) = ⎡
⎢
⎣
s 1 0
s 0 s 2
s 2 s 0 ⎤
⎥
⎦ and Q ( s ) = ⎡
⎢
⎣
0 0 s
0 0 0
− s 0 0 ⎤
⎥
⎦
r esults in the sequence
U 1 ( s ) = ⎡
⎢
⎣
1 0 0
0 1 0
− s 0 1 ⎤
⎥
⎦ , U 1 ( s ) P 1 ( s ) = ⎡
⎢
⎣
s 1 0
s 0 s 2
0 0 0 ⎤
⎥
⎦ , U 1 ( s ) Q 1 ( s ) = ⎡
⎢
⎣
0 0 s
0 0 0
− s 0 − s 2 ⎤
⎥
⎦
U 2 ( s ) = ⎡
⎢
⎣
1 0 0
0 1 0
0 1 1 ⎤
⎥
⎦ , U 2 ( s ) P 2 ( s ) = ⎡
⎢
⎣
s 1 0
s 0 s 2
0 0 0 ⎤
⎥
⎦ , U 2 ( s ) Q 2 ( s ) = ⎡
⎢
⎣
0 0 s
0 0 0
0 0 0 ⎤
⎥
⎦ ,
such that Algorithm 1 terminates with
ν =
2. F rom Lemma 3.31 we deduce that (
P
(
s
)
, Q
(
s
)) is not
delay -regular . ♠
The important assumption in Lemma 3.31 is that Algorithm 1 ter minates after a finite number of
steps . W e immediately obser ve that b y construction
rank R [ s ] ( P ν + 1 ( s )) ≥ r ank R [ s ] ( P ν ( s )), (3.30a)
rank R [ s ] ( Q ν + 1 ( s )) ≤ r ank R [ s ] ( Q ν ( s )). (3.30b)
If in each iter ation of Algorithm 1 one of these inequalities is str ict, then Algorithm 1 ter minates after
a finite number of iterations . Or equivalently , Algorithm 1, does not terminate if and only if after
finitely many iterations the r anks in
(3.30)
r emain constant in all fur ther iterations of the algorithm.
The follo wing example sho ws that this indeed can happen.
Example 3.34. C onsider the input data
P ( s ) = [︄ s 1
s 2 s ]︄ , Q ( s ) = [︄ − s − 1
0 0 ]︄ ,

60 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
for Algorithm 1. W e obtain
U 1 ( s ) = [︄ 1 0
− s 1 ]︄ , P 2 ( s ) = [︄ s 1
s 2 s ]︄ = P ( s ), Q 2 ( s ) = [︄ − s − 1
0 0 ]︄ = Q ( s ).
H ence each iteration of Algorithm 1 works with the same pair of polynomial matrices and conse-
quently Algorithm 1 does not terminate. ♠
F rom this example one may conjecture that once the ranks in
(3.30)
r emain constant, they also will
r emain constant in future iter ations so that at least the algor ithm can be terminated with a warning.
U nfor tunately , this is not true as the follo wing example sho ws.
Example 3.35. A pplying Algor ithm 1 to the matrices
P ( s ) = ⎡
⎢
⎣
s 0 0
0 0 1
0 0 0 ⎤
⎥
⎦ and Q ( s ) = ⎡
⎢
⎣
0 0 1
0 1 0
0 0 1 ⎤
⎥
⎦
yields in the first iteration
U 1 ( s ) = ⎡
⎢
⎣
1 0 0
0 1 0
0 0 1 ⎤
⎥
⎦ , P 2 ( s ) = ⎡
⎢
⎣
s 0 0
0 0 1
0 0 1 ⎤
⎥
⎦ Q 2 ( s ) = ⎡
⎢
⎣
0 0 1
0 1 0
0 0 0 ⎤
⎥
⎦ .
N ote that neither of the rank inequalities in (3.30) is strict. Ho wever , we continue with
U 2 ( s ) = ⎡
⎢
⎣
1 0 0
0 0 1
0 1 − 1 ⎤
⎥
⎦ , P 3 ( s ) = ⎡
⎢
⎣
s 0 0
0 0 1
0 1 0 ⎤
⎥
⎦ , Q 3 ( s ) = ⎡
⎢
⎣
0 0 1
0 0 0
0 0 0 ⎤
⎥
⎦
and conclude that ( P ( s ), Q ( s )) is delay -r egular . ♠
Another issue with Algorithm 1 is that the rank-r evealing decomposition
(3.29)
is not unique and
that the non-uniqueness of
U ν
(
s
) may influence the termination of Algorithm 1 as the follo wing
examples illustrates .
Example 3.36. C onsider the matrix polynomials
P ( s ) = [︄ 1 1
1 1 ]︄ , Q ( s ) = [︄ − 1 − 1
0 0 ]︄ .
Picking
U ν
(
s
)
= [︁ 1 0
− 1 1 ]︁
we obtain for all
ν ∈ N
the equality (
P ν
(
s
)
, Q ν
(
s
))
=
(
P
(
s
)
, Q
(
s
)) and conse-
quently ρ ν = 1 for all ν ∈ N . If we use U 1 ( s ) = [︁ 0 1
1 − 1 ]︁ we obtain
P 2 ( s ) = [︄ 1 1
1 1 ]︄ , Q 2 ( s ) = [︄ 0 0
0 0 ]︄
and thus ρ 3 = 0 and thus Algorithm 1 terminates with k 2 = 1 < 2 = n x . ♠

3.3. DEL A Y -EQ UIV ALE NCE AND THE COMPRESS-AND-SHIFT AL GORITHM 61
T o summarize the previous discussion ther e are two major issues with Algorithm 1:
(i) I t may happen that the algorithm does not ter minate and
(ii) the choice of U ν ( s ) may influence the termination of the algorithm.
F rom
(3.30)
we immediately observe that the algorithm fails to ter minate if and only if ther e exists
an index ˜
ν ∈ N such that
rank R [ s ] ( P ν ( s )) = r ank R [ s ] ( P ˜
ν ( s )), (3.31a)
rank R [ s ] ( Q ν ( s )) = r ank R [ s ] ( Q ˜
ν ( s )) (3.31b)
for all
ν ≥ ˜
ν
and
rank R [ s ]
(
P ν
(
s
))
< m
. The follo wing theor em details that this cannot happen if the
DDAE (3.9) is delay -regular .
Theor em 3.37.
A lgori thm 1 terminates for any delay-r egular DDAE
(3.9)
. In particular , the
DDAE (3.9) is delay-equivalent to a DDAE
˜︂
P (︃ d
d t )︃ x = ˜︁
Q (︃ d
d t )︃ σ τ x + ˜︁
f
with det( ˜︂
P ( s )) ≡ 0 .
I n order to pro ve Theorem 3.37 w e obser ve that it suffices to apply Algorithm 1 directly to the
condensed polynomial matrices
(2.27)
. Since we assume that the DDAE
(3.9)
is delay -regular , we
can simplify (2.27) as follo ws.
Lemma 3.38.
Consider a delay-r egular pair of matrix polynomials (
P
(
s
)
, Q
(
s
))
∈
(
R
[
s
]
n x × n x
)
2
.
Ther e exist unimodular matrices U ( s ), V ( s ) ∈ R [ s ] n x × n x such that
U ( s ) P ( s ) V ( s ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
P 1,1 ( s ) P 1,2 ( s ) · · · · · · P 1, k ( s )
0 0 P 2,3 ( s ) · · · P 2, k ( s )
.
.
. .
.
. . . . . . . .
.
.
.
.
. .
.
. . . . P k − 1, k ( s )
0 0 · · · · · · 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, (3.32a)
U ( s ) Q ( s ) V ( s ) = ⎡
⎢
⎢
⎢
⎢
⎣
Q 1,1 ( s ) Q 1,2 ( s ) · · · Q 1, k ( s )
0 Q 2,2 ( s ) · · · Q 2, k ( s )
.
.
. . . . . . . .
.
.
0 · · · 0 Q k , k ( s )
⎤
⎥
⎥
⎥
⎥
⎦ (3.32b)
wher e
P 1,1
(
s
) ,
Q i , i
(
s
) for
i =
2
, . . . , k
ar e nonsingular and the matrices
P i , i + 1
(
s
) have full ro w rank
for i = 2, . . . , k − 1 .
Proof.
The form
(3.32)
follo ws directly fr om Theorem 2.24 and Theor em 3.17 (ii) with nonsingular
blocks
P 1,1
(
s
),
Q i , i
(
s
) for
i =
2
, . . . , k
. Let
j ∈ {
2
, . . . , k −
1
}
denote the largest number such that the

62 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
polynomial matrix
P j , j + 1
(
s
) does not have full ro w rank. Then there exists a unimodular matrix
U j ( s ) such that
U j ( s ) P j , j + 1 ( s ) = [︄ ˆ︂
P j , j + 1 ( s )
0 ]︄
wher e
ˆ︂
P j , j + 1
(
s
) has full ro w rank. S ince
Q j , j
(
s
) is nonsingular , ther e exists a unimodular matr ix
V j ( s ) such that
U j ( s ) Q j , j ( s ) V j ( s ) = [︄ ˘
Q j , j ( s ) ˇ
Q j , j ( s )
0 ˜
Q j , j ( s ) ]︄
with nonsingular matrices
˘
Q j , j
(
s
) and
˜︁
Q j , j
(
s
). F or the corresponding sub block matrices in
(3.32)
we thus obtain
⎡
⎣ U j ( s )
I d ⎤
⎦ ⎡
⎣ 0 P j , j + 1 ( s )
0 ⎤
⎦ ⎡
⎣ V j ( s )
I d ⎤
⎦ = ⎡
⎢
⎢
⎢
⎣
0 0 ˆ︂
P j , j + 1 ( s )
0 0
0 0
⎤
⎥
⎥
⎥
⎦
and
⎡
⎣ U j ( s )
I d ⎤
⎦ ⎡
⎣ Q j , j ( s ) Q j , j + 1 ( s )
Q j + 1, j + 1 ( s ) ⎤
⎦ ⎡
⎣ V j ( s )
I d ⎤
⎦ = ⎡
⎢
⎢
⎢
⎣
˘
Q j , j ( s ) ˇ
Q j , j ( s ) ˇ
Q j , j + 1 ( s )
0 ˜
Q j , j ( s ) ˜
Q j , j + 1 ( s )
0 Q j + 1, j + 1 ( s )
⎤
⎥
⎥
⎥
⎦ ,
wher e clearly the blocks have the desired pr oper ties. Repeating this pr ocedur e for the remaining
rank defectiv e blocks yields the desir ed result. ■
Proof of Theor em 3.37.
I t suffices to sho w that we can ensure that the situation in
(3.31)
cannot
happen. Lemma 3.38 implies that applying Algor ithm 1 to
(3.32)
yield a shift of the last block ro w .
After shifting the last block ro w in
(3.32)
, we observe that the compression step affects only the last
two ro ws. Since P k − 1, k ( s ) has full ro w rank, this implies that ther e exists a unimodular matr ix
ˆ︂
U ( s ) = [︄ ˆ
U 1 ( s ) ˆ
U 2 ( s )
ˆ
U 3 ( s ) ˆ
U 4 ( s ) ]︄ with ˆ︂
U ( s ) [︄ P k − 1, k ( s )
Q k , k ( s ) ]︄ = [︄ 0
ˆ︁
Q k , k ( s ) ]︄ .
S ince
P k − 1, k
(
s
) has full ro w rank and
Q k , k
(
s
) is nonsingular , we conclude that
ˆ
U 1
(
s
) has full ro w
rank implying that
ˆ
U 1
(
s
)
Q k − 1, k − 1
(
s
) is nonsingular . W e can thus repeat the abo ve procedur e with
the submatrices that are obta ined b y remo ving the last block column and last block ro w . Proceeding
iteratively , we conclude that Algorithm 1 terminates after k − 1 shifts. ■
R emark 3.39.
S uppose that the DDAE
(3.1a)
is delay -regular . Then Theor em 3.37 i mplies that
Algorithm 1 constructs a delay -equivalent DDAE
˜︂
P (︃ d
d t )︃ x = ˜︁
Q (︃ d
d t )︃ σ τ x + ˜︁
f

3.3. DEL A Y -EQ UIV ALE NCE AND THE COMPRESS-AND-SHIFT AL GORITHM 63
with
det
(
˜︂
P
(
s
))
≡
0. P er forming a first-order r eformulation as for instance in the proof of Theo-
r em 3.15 y ields a DDAE
E ˙
z = A 1 z + A 2 σ τ z + F
with r egular matr ix pencil ( E , A 1 ). ♣
Let us emphasize that shifting enlarges the set of feasible initial tr ajector ies for the ITP, and thus
solutions of the ITP for the partially time-shifted DDAE
(3.27)
may not be solutions of the original
ITP, see for instance [212, Ex. 4.7].
Lemma 3.40. Let the notation be as in Lemma 3.30 and set
˜︂
P ( s ) : = [︄ P 1 ( s )
Q 2 ( s ) ]︄ , ˜︁
Q ( s ) : = [︄ Q 1 ( s )
0 ]︄ .
F or f ∈ D m
p w C ∞ and x 0 ∈ D n x
p w C ∞ define
f ITP : = f [0, ∞ ) + (︃ P (︃ d
d t )︃ x 0 − Q (︃ d
d t )︃ σ τ x 0 )︃ ( −∞ ,0)
, ˜
f ITP : = ⎡
⎣ U 1 (︂ d
d t )︂ f ITP
− σ − τ U 2 (︂ d
d t )︂ f ITP ⎤
⎦ , ˜
f : = ( ˜
f ITP ) [0, ∞ ) .
Assume that x ∈ D n x
p w C ∞ is a solution of the ITP
x ( −∞ ,0) = x 0
( −∞ ,0) ,
(︃ ˜︂
P (︃ d
d t )︃ x )︃ [0, ∞ ) = (︃ ˜︁
Q (︃ d
d t )︃ σ τ x + ˜
f )︃ [0, ∞ )
.
If x 0 satisfies (︃ U 2 (︃ d
d t )︃ Q (︃ d
d t )︃ σ τ x 0 )︃ [0, τ ) = − (︃ U 2 (︃ d
d t )︃ f ITP )︃ [0, τ )
, (3.33)
then x is a solu tion of the ITP (3.10) .
Proof. F rom the definition of ˜
f we immediately obtain
(︁ ˜
f ITP )︁ [0, ∞ ) = ˜
f = ˜
f [0, ∞ ) .
Further , Lemma 2.22 implies
(︃ U 1 (︃ d
d t )︃ f ITP )︃ ( −∞ ,0) = (︃ P 1 (︃ d
d t )︃ x 0 − Q 1 (︃ d
d t )︃ σ τ x 0 )︃ ( −∞ ,0)
,
(︃ U 2 (︃ d
d t )︃ f ITP )︃ ( −∞ ,0) = − (︃ Q 2 (︃ d
d t )︃ σ τ x 0 )︃ ( −∞ ,0)
,
such that (3.33) yields
(︃ U 2 (︃ d
d t )︃ f ITP )︃ ( −∞ , τ ) = − (︃ Q 2 (︃ d
d t )︃ σ τ x 0 )︃ ( −∞ , τ )
.

64 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
Thus Lemma 3.2 implies
(︁ ˜
f ITP )︁ ( −∞ ,0) = ⎡
⎣ U 1 (︂ d
d t )︂ f ITP
− σ − τ U 2 (︂ d
d t )︂ f ITP ⎤
⎦ ( −∞ ,0) = ⎡
⎣ P 1 (︂ d
d t )︂ x 0 − Q 1 (︂ d
d t )︂ σ τ x 0
Q 2 (︂ d
d t )︂ x 0 ⎤
⎦ ( −∞ ,0)
= (︃ ˜︂
P (︃ d
d t )︃ x − ˜︁
Q (︃ d
d t )︃ σ τ x 0 )︃ ( −∞ ,0)
.
W e conclude that x satisfies
˜︂
P (︃ d
d t )︃ x = ˜︁
Q (︃ d
d t )︃ σ τ x + ˜
f ITP .
Delay - equivalence thus implies that x solves
P (︃ d
d t )︃ x = Q (︃ d
d t )︃ σ τ x + f ITP
and thus is a solution of the ITP (3.10). ■
A pplying Lemma 3.4 0 sever al times yields, together with Theorems 3.15 and 3.37, the follo wing
sufficient condition for an initial trajectory to be consistent.
Theor em 3.41.
Consider the ITP
(3.10)
with delay-r egular pair (
P
(
s
)
, Q
(
s
))
∈
(
R
[
s
]
n x × n x
)
2
, exter -
nal forcing
f ∈ D n x
p w C ∞
, and
x 0 ∈ D n x
p w C ∞
. Let Algorithm 1 applied to (
P
(
s
)
, Q
(
s
)) terminate after
ν ∈ N iter ation s. Define
f ITP,1 : = f [0, ∞ ) + (︃ P (︃ d
d t )︃ x 0 − Q (︃ d
d t )︃ σ τ x 0 )︃ ( −∞ ,0)
,
f ITP, k : = ⎡
⎣ U k ,1 (︂ d
d t )︂ f ITP, k − 1
− σ τ U k ,2 (︂ d
d t )︂ f ITP, k − 1 ⎤
⎦
for k = 2, . . . , ν − 1 . If x 0 satisfies
(︃ U k ,2 (︃ d
d t )︃ f ITP,k )︃ [0, τ ) = − (︃ U k ,2 (︃ d
d t )︃ Q k ( d
d t ) σ τ x 0 )︃ [0, τ )
,
for k = 1, . . . , ν − 1 , then x is a solution of the ITP (3.10) .
I t remains to analyz e the situation what happens with Algorithm 1 when the DDAE is not delay -
r egular . Checking the proof of Theor em 3.37 reveals that Algorithm 1 terminates also in the case that
the second block column in
(2.27)
is pr esent. Thus, the only r eason for non-termination is hidden
in the thir d block ro w in
(2.27)
. A modification of Algor ithm 1 to prev ent non-ter mination must
thus be able to r ecognize this case. W e have already seen in E xample 3.35 that it is not sufficient to
terminate Algorithm 1 whenever the ranks in
(3.30)
do not change from one iter ation to another .
This can also be seen from the proof of Theor em 3.37. H o wever , we obser ve that the image of
Q ν ,2
(
s
)
is differ ent for ever y
ν
in the delay -regular case whenev er the ranks remain const ant. I t thus suffices

3.3. DEL A Y -EQ UIV ALE NCE AND THE COMPRESS-AND-SHIFT AL GORITHM 65
to check that the matrix ⎡
⎢
⎢
⎢
⎢
⎣
Q 1,2 ( s )
Q 2,2 ( s )
.
.
.
Q ν ,2 ( s )
⎤
⎥
⎥
⎥
⎥
⎦
has full ro w rank. A modified version of Algorithm 1 is presented in Algor ithm 2.
Algorithm 2 Compr ess-and-shift (modified)
I nput: P ( s ), Q ( s ) ∈ R [ s ] m × n x
1: Set ν = 1 and P ν ( s ) : = P ( s ), Q ν ( s ) : = Q ( s ).
2: Set K ( s ) : = [] ∈ R [ s ] 0 × n x
3: Choose unimodular matrix
U ν ( s ) = ⎡
⎢
⎣
U ν ,1 ( s )
U ν ,2 ( s )
U ν ,3 ( s ) ⎤
⎥
⎦ ∈ R [ s ] m × m with matrix polynomials ⎧
⎪
⎨
⎪
⎩
U ν ,1 ( s ) ∈ R [ s ] k ν × m ,
U ν ,2 ( s ) ∈ R [ s ] ρ ν × m ,
U ν ,3 ( s ) ∈ R [ s ] m − k ν − ρ ν × m ,
such that
U ν ( s ) P ν ( s ) = : ⎡
⎢
⎣
P ν ,1 ( s )
0
0 ⎤
⎥
⎦ and U ν ( s ) Q ν ( s ) = : ⎡
⎢
⎣
Q ν ,1 ( s )
Q ν ,2 ( s )
0 ⎤
⎥
⎦ ,
wher e
P ν ,1
(
s
)
: = U ν ,1
(
s
)
P ν
(
s
)
∈ R
[
s
]
k ν × n x
and
Q ν ,2
(
s
)
: = U ν ,2
(
s
)
Q ν
(
s
)
∈ R
[
s
]
ρ ν × n x
have full ro w
rank.
4: if ρ ν = 0 then
5: terminate
6: else
7: Set P ν + 1 ( s ) = [︂ P ν ,1 ( s )
Q ν ,2 ( s ) ]︂ and Q ν + 1 ( s ) = [︂ Q ν ,1 ( s )
0 ]︂ .
8: Set K ( s ) = [︂ K ( s )
Q ν ,2 ( s ) ]︂ .
9: if K ( s ) does not have full ro w rank then
10: terminate (not delay -regular)
11: end if
12: Set ν ← ν + 1.
13: Go to Line 3
14: end if
The discussion abo ve yields our final r esult of this chapter .
Theor em 3.42. Algorithm 2 terminates for any DDAE (3.9) .
W e conclude this chapter b y revisiting Example 3.34, wher e Algor ithm 1 failed to terminate. The

66 CHAPTER 3. DISTRIB UTIONAL SOL UTIONS FOR LINEAR TIME-INV ARIANT DDAES
matrix pair in Example 3.34 is given b y
P ( s ) = [︄ s 1
s 2 s ]︄ , Q ( s ) = [︄ − s − 1
0 0 ]︄ .
A pplying Algor ithm 2 to ( P ( s ), Q ( s )) yields in the first iteration
U 1 ( s ) = [︄ 1 0
− s 1 ]︄ , P 2 ( s ) = [︄ s 1
s 2 s ]︄ = P ( s ), Q 2 ( s ) = [︄ − s − 1
0 0 ]︄ = Q ( s ),
and K ( s ) = [︂ s 2 s ]︂ . The previous computations sho w that in the next iteration, we obtain
K ( s ) = [︄ s 2 s
s 2 s ]︄ ,
such that Algorithm 2 terminates with the infor mation that (
P
(
s
)
, Q
(
s
)) is not delay -r egular . This is
the corr ect result, which can be easily verified with Theor em 3.20.

4
Classical solutions and discontinuit y p ropagation
H aving established the existence and uniqueness of solutions for the linear time-invariant (L TI)
delay differ ential-algebraic equation (DDAE)
(1.15)
in a distributional solution space (cf. Theo-
r em 3.20), we no w tur n our attention to a mor e classical solution concept, namely solutions that ar e
continuously differ entiable almost ever ywhere . F ollo wing our analysis in chapter 3 it is sufficient to
focus on delay -regular DDAEs. Invoking Theor em 3.37 and Algorithm 1, we can thus restrict our
analysis to linear DDAEs
E ˙
x ( t ) = A 1 x ( t ) + A 2 x ( t − τ ) + f ( t ) (4.1a)
with r egular matr ix pencil (
E , A 1
)
∈ (︁ F n x × n x )︁ 2
. As before , the DDAE
(4.1a)
is equipped with the initial
trajectory
x ( t ) = φ ( t ) for t ∈ [ − τ , 0 ]. (4.1b)
Alr eady in the case of differential-algebr aic equations (DAEs), i.e., in the case
A 2 =
0, a necessar y
condition for the existence of a classical solution is that
f
is sufficiently smooth (cf. Assumption 2.5).
F or DDAEs this in additional implies that also the histor y function
φ
needs to be sufficiently smooth,
which we assume for the r emainder of this chapter . In summary , we invoke the follo wing assumption
for the upcoming analysis .
Assumption 4.1.
The matrix pair (
E , A 1
)
∈ (︁ F n x × n x )︁ 2
in
(4.1a)
is r egular , i.e., there e xists
λ ∈ F
such
that
det
(
λ E − A 1
)
=
0 . M oreo ver , we assume that the histor y function
φ :
[
− τ ,
0]
→ F n x
and the
inhomogeneity f : I → F n x are infinitely many times continuously differ entiable.
4.1 Continuous solutions and classification
S imilarly as in the proof of Theorem 3.5, w e can apply the DAE theor y to the sequence of DAEs
(2.3)
that arises from applying the method of steps to
(1.13)
. The corresponding sequence of DAEs for
67

68 CHAPTER 4. CLASSICAL SOLUTIONS AND DISCONTINUITY PR OP A GA TION
the initial trajectory problem (4.1) is given b y
E ˙
x [ i ] ( t ) = A 1 x [ i ] ( t ) + ˜
f [ i ] ( t ), t ∈ [0, τ ), (4.2a)
x [ i ] (0) = x [ i − 1] ( τ − ) (4.2b)
with
x [0]
(
t
)
= φ
(
t − τ
) and
˜
f [ i ]
(
t
)
= A 2 x [ i − 1]
(
t
)
+ f
(
t +
(
i −
1)
τ
) for
t ∈
[0
, τ
] and
i ∈ I = {
1
, . . . , M }
.
W e can use the quasi-W eierstraß form (cf. Theor em 2.9) for our analysis: Ther e exist matrices
S , T ∈ GL n x ( F ) such that
S E T = [︄ I n x ,d 0
0 N ]︄ and S A 1 T = [︄ J 0
0 I n x ,a ]︄ (4.3)
with matrix
J ∈ F n x ,d × n x ,d
and nilpotent matrix
N ∈ F n x ,a × n x ,a
. F or the upcoming analysis we introduce
[︄ B d
B a ]︄ : = S A 2 , [︄ B d,1 B d,2
B a,1 B a,2 ]︄ : = S A 2 T , [︄ v
w ]︄ : = T − 1 x , [︄ g
h ]︄ : = S f , and [︄ ψ
η ]︄ : = T − 1 φ , (4.4)
wher e we use the same block dimensions as in (4.3). Applying the matrices S , T to (4.1a) yields
˙
v = J v + B d,1 σ τ v + B d,2 σ τ w + g , (4.5a)
N ˙
w = w + B a,1 σ τ v + B a,2 σ τ w + h . (4.5b)
Example 4.2.
F or the DDAE
(1.24)
in Example 1.5 we dir ectly obser ve that the matrices
S = [︁ 1 0
0 1 ]︁
and
T = [︁ 0 1
1 0 ]︁
transform the associated matrix pair to quasi-W eierstraß form with
n x ,d =
0 and
n x ,a =
2.
The accor ding for m (4.5) is given with the matrices
N = [︄ 0 1
0 0 ]︄ and B a,2 = [︄ 0 0
− 1 0 ]︄ .
♠
E ven with Assumption 4.1 we cannot expect a continuously differ entiable solution, as is illustrated
in the Examples 1.4 and 1.5. This is mainly due to the fact that the identity
lim
t ↘ 0 ˙
x ( t ) = lim
t ↗ 0
˙
φ ( t ),
which can be written in the form
˙
x
(0)
= ˙
φ
(0
−
), is not satisfied in general and this discont inuity in
the first derivative at
t =
0 may propagate o ver time (cf. [26] and Examples 1.4 and 1.5), which is
the r eason for analyzing solutions in the space of piecewise smooth distributions (see section 3.1).
If we ar e interested in solutions that ar e at least continuous, then we can sear ch for a solution in
the space of absolutely continuous functions, i.e ., functions that ar e continuous and differentiable
almost everywhere . Assuming that the histor y function
φ
and the inhomogeneity
f
ar e sufficiently
smooth, we expect discontinuities only at integer multiples of the time delay
τ
and thus consider
the space of piecewise continuously differ entiable functions as solution space. M ore pr ecisely , we
emplo y the follo wing solution concept for the remainder of this chap ter .

4.1. CONTINUOUS SOL UTIONS AND CL ASSIFICA TION 69
Definition 4.3
(Solution concept)
.
Assume that the matrix pair (
E , A 1
) in the DDAE
(2.1a)
is r egular
and the history function
φ
and the inhomogeneity
f
ar e infinitely many times continuously differ-
entiable . W e call
x ∈ C
(
I , F n x
) a solution of
(4.1)
if for all
i ∈ I
the r estr iction
x [ i ]
of
x
as in
(2.2)
is a
solution of
(2.3)
. W e call the histor y function
φ :
[
− τ ,
0]
→ F n x
consistent if the initial value problem
(4.1) has at least one solution.
S ince our sol ution concept is inher ently related to the method of steps we immediately obtain the
follo wing relation betw een the DDAE (4.1a) and the sequence of DAEs (4.2).
Pr oposition 4.4.
Let the DDAE
(4.1a)
satisfy Assumpt ion 4.1. If
x
is a solution of the initial
trajectory problem (ITP)
(4.1)
, then the r estriction
x [ i ]
(
t
)
= x
(
t +
(
i −
1)
τ
) for
i ∈ I
is a solution of
(4.2) . Conversely , if the sequence ( x [ i ] ) i ∈ I is a solution of (4.2) , t hen
x ( t ) = ⎧
⎨
⎩
x [ i ] ( t − ( i − 1) τ ), if ( i − 1) τ ≤ t < i τ for some i ∈ N ,
φ ( t ), otherwise ,
is a solution of (4.1) .
The thr ee introductor y examples, namely Examples 1.3 to 1.5, sho w that solutions of linear DDAEs
may have very different smoothness pr oper ties . Since the standar d classification for delay equa-
tions is only valid for scalar equations, w e pursue the following str ategy : W e first introduce a new
classification, which is based on the worst possible smoothing behavior , and then give an algebraic
characterization of the differ ent types in ter ms of the matrices of the DDAE. T o this end we recall
Pr oposition 2.1 4, which establishes a connection of the DAE
(4.2a)
and the so-called underlying
or dinar y differ ential equati on (ODE) (2.15)
˙
x = A diff x +
ind( E , A 1 )
∑︂
k = 0
C k ˜
f ( k ) (4.6)
via the consistency condition (2.16). Her eb y , the matr ices ar e defined as (see (2.14))
A diff = T [︄ J 0
0 0 ]︄ T − 1 , A con = T [︄ I n x ,d 0
0 0 ]︄ T − 1 , C 0 = T [︄ I n x ,d 0
0 0 ]︄ S , C k = − T [︄ 0 0
0 N k − 1 ]︄ S ,
wher e
S , T ∈ GL n x
(
F
) denote matrices that transform (
E , A 1
) into quasi-W eierstraß form
(2.6)
. I ntro-
ducing the matrices
B k : = C k A 2 for k = 0, . . . , ind( E , A 1 )
allo ws us to re-substitute
˜
f [ i ]
(
t
)
= A 2 x [ i − 1]
(
t
)
+ f
(
t +
(
i −
1)
τ
). This yields the delay differ ential
equation (DDE)
˙
x = A diff x +
ind( E , A 1 )
∑︂
k = 0 (︂ B k σ τ x ( k ) + C k f ( k ) )︂ , (4.7)
which we call the the under lying DDE for the DDAE
(4.1a)
. F rom Corollary 2.15 and the discussion
ther eafter we immediately obser ve the follo wing.

70 CHAPTER 4. CLASSICAL SOLUTIONS AND DISCONTINUITY PR OP A GA TION
Lemma 4.5.
Let
(4.1)
satisfy A ssumption 4.1. A necessar y condition for the histor y function
φ
to
be consistent is that φ satisfies the equation
φ (0) = A con φ (0) +
ind( E , A 1 )
∑︂
k = 1 (︂ B k φ ( k − 1) ( − τ ) + C k f ( k − 1) (0) )︂ . (4.8)
U nfor tunately , as Example 1.5 suggests, this condition is not sufficient for consistency , and even
worse , the consistency of a histor y function depends on the time inter val for which we want to
solve the DDAE. S ince our main goal is to analyze the propagation of primary discontinuities, it
is sufficient to ensur e that a solution exists for some time, which gives rise to the follo wing definition.
Definition 4.6.
Let the ITP
(4.1)
with histor y function
φ :
[
− τ ,
0]
→ F n x
satisfy Assumption 4.1. Then
φ is called admissible for the ITP (4.1) if x [1] (0) = φ (0) i s consistent for the DAE
E ˙
x [1] ( t ) = A 1 x [1] ( t ) + A 2 φ ( t − τ ) + f [1] ( t ) for t ∈ [0, τ ),
i.e .
φ
satisfies
(4.8)
. S imilarly ,
x [0] :
[0
, τ
]
→ F n
x
is called admissible for the sequence of DAEs
(4.2)
if
the DAE E ˙
x [1] ( t ) = A 1 x [1] ( t ) + A 2 x [0] ( t ) + f ( t ) with x [1] (0) = x [0] ( τ ) has a solution on [0, τ ).
Let
φ :
[
− τ ,
0]
→ F n x
be admissible . As a consequence of Assumption 4.1 ther e exists a number
M ∈ N
and a unique sequence (
x [ i ]
)
i ∈ {0,..., M }
that satisfies
(4.2)
(cf. Corollary 2.15). H ence for any
i ∈ {1, . . . , M } we can define the level ℓ i of the primar y discontinuity as
ℓ i : = min
x [0] ∈ C ∞ ([0, τ ], F n x )
x [0] admissible
min
f ∈ C ∞ ( I , F n x ) max {︄ ℓ ∈ N 0     
x [ j ] solves (4.2) for j = 1, . . . , i and
x ( ℓ )
[ i ] (0) = x ( ℓ )
[ i − 1] ( τ − ) }︄ . (4.9)
If for some
j ∈ N
the initial condition
x [ j ]
(0)
= x [ j − 1]
(
τ
) is not consistent and thus no solution of
(4.2)
exists, w e formally set
ℓ i : = −∞
for all
i ≥ j
. N ote that this definition is independent of the specific
choice of the inhomogeneity
f
and the history
φ
and hence ser ves as the worst-case scenario . T o
simplify the computation of the numbers
ℓ i
we observe the follo wing, which is a generalization
of [102, Theor em 7.1 ]
Pr oposition 4.7.
Let the ITP
(4.1)
satisfy Assu mption 4.1. Then the solution
x
of
(4.1)
is continu-
ously differ entiable on [ − τ , τ ) if and only if φ satisfies
˙
φ (0 − ) = A diff φ (0) +
ind( E , A 1 )
∑︂
k = 0 (︂ B k φ ( k ) ( − τ ) + C k f ( k ) (0) )︂ . (4.10)
The solution x of (4.1) is κ times continuously differentiable on [ − τ , τ ) if and only if φ satisfies
φ ( p + 1) (0 − ) = A diff φ ( p ) (0 − ) +
ind( E , A 1 )
∑︂
k = 0 (︂ B k φ ( k + p ) ( − τ ) + C k f ( k + p ) (0) )︂ (4.11)
for p = 0, 1, . . . , κ − 1 .

4.1. CONTINUOUS SOL UTIONS AND CL ASSIFICA TION 71
Proof.
S ince
φ
is admissible , the initial condition
x [1]
(0)
= φ
(0) is consistent and follo wing Corol-
lar y 2.15 the solution
x
exists on [
− τ , τ
). Thus, it is sufficient to check the point
t =
0. U sing
Pr oposition 2.14 we can consider (2.15) and thus obtain
˙
x [1] (0) = A diff x [1] (0) +
ind( E , A 1 )
∑︂
k = 0 (︂ B k x ( k )
[0] (0) + C k f ( k )
[1] (0) )︂
= A diff φ (0) +
ind( E , A 1 )
∑︂
k = 0 (︂ B k φ ( k ) ( − τ ) + C k f ( k )
[1] (0) )︂
and hence
x
is continuously differ entiable on [
− τ , τ
) if and only if
φ
satisfies
(4.10)
. F or arbitrary
κ ∈ N
we invoke P roposition 2.14, which guarantees that the solution
x
exists on the inter val [0
, τ
)
and allo ws us to consider the underlying DDE
(4.7)
instead of the DDAE. S ince the assumption
guarantees that x is sufficiently smooth on [0, τ ) we can differ entiate (4.7) p ∈ N times to obtain
x ( p + 1)
[1] (0) = A diff x ( p )
[1] (0) +
ind( E , A 1 )
∑︂
k = 0 (︂ B k x ( k + p )
[0] (0) + C k f ( k + p )
[1] (0) )︂
= A diff φ ( p ) (0 − ) +
ind( E , A 1 )
∑︂
k = 0 (︂ B k φ ( k + p ) ( − τ ) + C k f ( k + p )
[1] (0) )︂ ,
which implies the assertion. ■
S ince we requir e
φ ∈ C ∞
([
− τ ,
0]
, F n x
) to be admissible we immediately obtain
ℓ 1 ≥
0. On the other
hand assume that we have giv en the values
φ
(0) and
φ ( k )
(
− τ
) for
k =
0
, . . . , ν
such that
φ
is admissible .
Then we can always construct (via H ermite interpolation)
φ
in such a way that
(4.10)
is not satisfied
and hence
ℓ 1 ≤
0, which yields
ℓ 1 =
0. Thus, the questions about propagation of discontinuities can
be r ephrased as whether
• ther e exists k ∈ N with ℓ k > 0 (i.e. the solution becomes smoother), or
• ther e exists k ∈ N with ℓ k = −∞ (i.e. the solution becomes less smooth),
• or if ℓ i = ℓ 1 for all i ∈ N .
W e notice that the smoothing may not start immediately (i.e. we cannot ask for
ℓ 1 =
1), as the
follo wing example suggests .
Example 4.8. C onsider the DDAE given b y F = R , n x = 2, f ≡ 0, τ = 1, and
E = [︄ 1 0
0 0 ]︄ , A = [︄ 0 0
0 1 ]︄ , B = [︄ 0 1
− 1 0 ]︄ , φ ( t ) = [︄ t ,
− 1 ]︄ .
S ince ( E , A ) is already in W eierstraß form, it is easy to see that the DDAE corr esponds to the DDE
˙
v ( t ) = v ( t − 2 τ ) (4.12)
with coupled equation
w
(
t
)
= v
(
t − τ
). Str aightfor ward calculations sho w that
ℓ 1 ≤
0 (using the
specified histor y function
φ
) and
ℓ 1 ≥
0 implying
ℓ 1 =
0. On the other hand,
(4.12)
is a scalar delay
equation and it is well-kno wn that the solution is continuously differ entiable at
t =
2
τ
, thus we
have ℓ 2 ≥ 1. ♠

72 CHAPTER 4. CLASSICAL SOLUTIONS AND DISCONTINUITY PR OP A GA TION
Definition 4.9.
Consider the DDAE
(4.1a)
on the inter val
I =
[0
, M τ
], set
I : = {
1
, . . . , M }
, and suppose
that (4.1) satisfies Assumption 4.1. W e say that (1.13) is of
• smoothing type if ther e exists j ∈ I , j > 1 such that ℓ j = 1 and ℓ i = 0 for i < j ,
• discontinuity invariant type if ℓ i = 0 for all i ∈ I , and
• de-smoothing type if ther e exists j ∈ I , j > 1 such that ℓ j = −∞ and ℓ i = 0 for i < j .
I n the following, w e analyze the DDAE
(4.1a)
in detail and derive conditions for the matrices
E , A 1
,
and
A 2
, from which the type can be determined. Befor e we analyze the general DDAE case w e focus
on the case of ind( E , A 1 ) ≤ 1, i.e ., the system is a pure DDE or N = 0 in (2.12b).
R emark 4.10. The case ind( E , A 1 ) ≤ 1 includes DDEs of the form
˙
ˆ︁
x ( t ) = ˆ︁
A 1 ˆ︁
x ( t ) + ˆ︁
A 2 ˆ︁
x ( t − τ ) + ˆ︁
D ˙
ˆ︁
x ( t − τ ) + ˆ
f ( t ), (4.13)
with arbitrary matrices
ˆ︁
A 1 , ˆ︁
A 2 , ˆ︁
D ∈ F n x × n x
, since
(4.13)
can be r ecast in the for m
(4.1a)
b y introducing
the new state variable x ( t ) = [︂ ˆ
x ( t )
ˆ
x ( t − τ ) ]︂ and
E : = [︄ − I n x − ˆ︁
D
0 0 ]︄ A 1 : = [︄ ˆ︁
A 1 0
0 I n x ]︄ , A 2 : = [︄ ˆ︁
A 2 0
− I n x 0 ]︄ , f : = [︄ ˆ︁
f
0 ]︄ .
♣
If ind( E , A 1 ) = 0, then the matrix E is nonsingular and the DDAE is of the form
˙
x ( t ) = E − 1 A 1 x ( t ) + E − 1 A 2 x ( t − τ ) + E − 1 f ( t ) (4.14)
and the ODE solution formula together with Proposition 4.7 dir ectly implies
ℓ 1 =
1, i.e .,
(4.14)
is of
smoothing type.
Theor em 4.11.
Consider the DDAE
(4.1a)
on the interval
I =
[0
, M τ
] and suppose that Assu mp-
tion 4.1 holds. If
ind
(
E , A 1
)
=
1 , then
(4.1a)
is of smoothing type if and only if
B a,2
in
(4.4)
is
nilpotent with index of nilpotency ν B and furthermor e we have ν B ≤ M − 1 .
Proof.
Let
S , T ∈ GL n
(
F
) be matrices that transform
(4.1a)
into quasi-W eierstraß form
(4.5)
. A pplying
the method of steps yields
˙
v [ i + 1] = J v [ i + 1] + B d,1 v [ i ] + B d,2 v [ i ] + g [ i + 1] and w [ i + 1] = − B a,1 v [ i ] − B a,2 w [ i ] − h [ i + 1] .
S ince ℓ 1 = 0, we have
w [1] ( τ ) = − B a,1 v [0] ( τ ) − B a,2 w [0] ( τ ) − h [1] ( τ )
= − B a,1 v [1] (0) − B a,2 w [1] (0) − h [2] (0) = w [2] (0)
and thus ℓ 2 ≥ 0. By induction we conclude ℓ i ≥ 0 for i ∈ I . M oreo ver , we have
˙
w [ i + 1] = − B a,1 ˙
v [ i ] − B a,2 ˙
w [ i ] − ˙
h [ i + 1]
= − B a,1 (︁ J v [ i ] + B d,1 v [ i − 1] + B d,2 w [ i − 1] + g [ i ] )︁ − B a,2 ˙
w [ i ] − ˙
h [ i + 1]

4.1. CONTINUOUS SOL UTIONS AND CL ASSIFICA TION 73
which implies ˙
w [ i + 1] (0 + ) − ˙
w [ i ] ( τ − ) = B a,2 (︁ ˙
w [ i − 1] ( τ − ) − ˙
w [ i ] (0 + ) )︁ holds . By induction we hav e
˙
w [ i + 1] (0 + ) − ˙
w [ i ] ( τ − ) = ( − 1) i B i
a,2 (︁ ˙
w [1] (0 + ) − ˙
η (0 − ) )︁ for i = 1, . . . , M − 1.
Thus ℓ i + 1 ≥ 1 holds if and only if B i
a,2 = 0. ■
A pplying Theorem 4.11 to the DDAE in Example 4.8 sho ws that this DDAE is of smoothing type, since
it is alr eady in q uasi-W eierstraß form with
B a,2 =
0. Conversely , if the DDAE
(4.1a)
with
ind
(
E , A 1
)
=
1
is of smoothing type , then the index of nilpotency indicates the number of delays present in the
system. Mor e pr ecisely , we have the follo wing result.
Cor ollar y 4.12.
S uppose t hat the DDAE
(4.1a)
satisfies Assumpt ion 4.1 and is of smoothing type
with
ind
(
E , A 1
)
≤
1 . F urthermore let
ν B
denote the index of nilpotency of
B a,2
if
n x ,a >
0 and
ν B =
0
otherwise. Then ther e exists matrices
D k ∈ F n x ,d × n x ,d
(
k =
0
, . . . , ν B
) and an inhomogeneity
ϑ
such
that the solution v of (4.5a) is a solution of the ITP
˙
z ( t ) = J z ( t ) +
ν B
∑︂
k = 0
D k z ( t − ( k + 1) τ ) + ϑ ( t ) for t ∈ [ ν B τ , t f ), (4.15a)
z ( t ) = v ( t ), for t ∈ [ − τ , ν B τ ]. (4.15b)
Proof.
The r esult is trivial for
ind
(
E , A 1
)
=
0, i.e ., assume
ind
(
E , A 1
)
=
1, which implies that
N =
0 in
(4.5). Let ∆ [ t 0 , t 1 ) denote the characteristic function for the interval [ t 0 , t 1 ), i.e.
∆ ( t 0 , t 1 ] ( t ) = ⎧
⎨
⎩
1, if t ∈ [ t 0 , t 1 ),
0, other wise.
Combination of the fast subsystem (4.5b) and t he initial condition yields
( I n x ,a + B a,2 ∆ [ τ , t f ) σ τ ) w = − B a ∆ [0, τ ) σ τ φ − B a,1 ∆ [ τ , t f ) σ τ v − h . (4.16)
B y induction we obtain ( ∆ [ τ , t f ) ( t ) σ τ ) k = ∆ [ k τ , t f ) ( t ) σ k τ and fr om B ν B
a,2 = 0 we deduce
(︄ ν B − 1
∑︂
k = 0
( − 1) k (︂ B a,2 ∆ [ τ , t f ) σ τ )︂ k )︄ (︂ I n x ,a + B a,2 ∆ [ τ , t f ) σ τ )︂ = I n x ,a
such that w in (4.16) is given b y
w =
ν B − 1
∑︂
k = 0
( − 1) k + 1 (︂ B a,2 ∆ [ τ , t f ) σ τ )︂ k (︂ B a ∆ [0, τ ) σ τ φ + B a,1 ∆ [ τ , t f ) σ τ v + h )︂
=
ν B − 1
∑︂
k = 0
( − 1) k + 1 B k
a,2 (︂ B a ∆ [ k τ ,( k + 1) τ ) σ ( k + 1) τ φ + B a,1 ∆ [ k τ , t f ) σ ( k + 1) τ v + ∆ [ k τ , t f ) σ k τ h )︂ .
I nser ting this identity in (4.5a) and introducing for k = 1, . . . , ν B the matrices
D 0 : = B d,1 , D k : = ( − 1) k B d,2 B k − 1
a,2 B a,1

74 CHAPTER 4. CLASSICAL SOLUTIONS AND DISCONTINUITY PR OP A GA TION
implies that the solution v of (4.5a) is a solution of the ITP (4.15), where ϑ is giv en by
ϑ ( t ) : = g ( t ) +
ν B − 1
∑︂
k = 0
( − 1) k + 1 B d,2 B k
a,2 h ( t − ( k + 1) τ ). ■
Example 4.13.
Consider the DDE
(4.13)
in the DDAE formulation given in R emark 4.10. The
matrices S : = [︃ I n x − ˆ︂
A 1 ˆ︁
D
0 I n x ]︃ and T : = [︃ I n x ˆ︁
D
0 I n x ]︃ transform the DDAE to quasi-W eierstraß form given b y
[︄ I n x 0
0 0 ]︄ [︄ ˙
v ( t )
˙
w ( t ) ]︄ = [︄ ˆ︂
A 1 0
0 I n x ]︄ [︄ v ( t )
w ( t ) ]︄ + [︄ ˆ︂
A 2 + ˆ︂
A 1 ˆ︁
D ( ˆ︂
A 2 + ˆ︂
A 1 ˆ︁
D ) ˆ︁
D
− I n x − ˆ︁
D ]︄ [︄ v ( t − τ )
w ( t − τ ) ]︄ + [︄ f ( t )
0 ]︄ .
H ence , the DDE
(4.13)
is of smoothing type if and only if
ˆ︁
D
is nilpotent. I n this case , the correspond-
ing r etarded equation (4.15a) is given b y
˙
z ( t ) = ˆ︂
A 1 z ( t ) + ( ˆ︂
A 2 + ˆ︂
A 1 ˆ︁
D ) z ( t − τ ) +
ν ˆ︁
D − 1
∑︂
k = 1
( − 1) k ( ˆ︂
A 2 + ˆ︂
A 1 ˆ︁
D ) ˆ︁
D k z ( t − ( k + 1) τ ) + g ( t ),
wher e ν ˆ︁
D is the index of nilpotency of ˆ︁
D . ♠
R emark 4.14.
The delay equation
(4.15)
in Cor oll ary 4.12 may be used to deter mine whether the
DDAE
(4.13)
is stable (which can be done for example via DDE-biftool [73, 196]). This pro vides an
alternative way to the theor y outlined in [68, 69] . ♣
F or the analysis of the general DDAE case with arbitr ar y index, we use the follo wing preliminary
r esult.
Pr oposition 4.15.
S uppos e that the ITP
(4.1)
satisfies Assumpti on 4.1 and let
S , T ∈ GL n x
(
F
) be
matrices that tr an sform (
E , A 1
) to quasi-W eierstr aß form
(2.6)
, such that
(4.1a)
is tr ansforme d to
(4.5)
with
x = T [︁ v
w ]︁
. Then for any
m ∈ N
and any
˜
v ∈ F n x ,d
,
˜
w ∈ F n x ,a
ther e exists an admissible
histor y function φ = T − 1 [︁ ψ
η ]︁ that is analytic and satisfies
ψ ( p ) (0 − ) = v ( p ) (0) for p = 0, 1, . . . , m − 1, (4.17a)
η ( p ) (0 − ) = w ( p ) (0) for p = 0, 1, . . . , m − 1, (4.17b)
˜
v = ψ ( m ) (0 − ) − v ( m ) (0), and (4.17c)
˜
w = η ( m ) (0 − ) − w ( m ) (0). (4.17d)
Proof.
Let
m ∈ N
. Pr oposition 4.7 implies that the solution
x
of the ITP
(4.1)
is
m
times continuously
differ entiable on [
− τ , τ
) if and only if
φ
satisfies
(4.11)
for
p =
0
,
1
, . . . , m −
1. M ultiply
(4.11)
from the
left with T − 1 to obtain
ψ ( p + 1) (0 − ) = J ψ ( p ) (0 − ) + B d,1 ψ ( p ) ( − τ ) + B d,2 η ( p ) ( − τ ) + g ( p ) (0), (4.18a)
η ( p + 1) (0 − ) = −
ind( E , A 1 ) − 1
∑︂
k = 0
N k (︂ B a,1 ψ ( k + p + 1) ( − τ ) + B a,2 η ( k + p + 1) ( − τ ) + h ( k + p + 1) (0) )︂ (4.18b)

4.1. CONTINUOUS SOL UTIONS AND CL ASSIFICA TION 75
for
p =
0
, . . . , m −
1. W e then can proceed as follo ws to construct
ψ
and
η
that satisfy the condi-
tions
(4.17)
. Choose any value for
ψ ( p )
(
− τ
) and
η ( p )
(
− τ
) for
p =
0
, . . . , ind
(
E , A
)
+ m
, and compute
η ( p + 1)
(0
−
) for
p =
0
, . . . , m −
2 accor ding to
(4.18b)
. F or an arbitrary
ψ
(0), set
ψ ( p + 1)
(0
−
) accor ding to
(4.18a) for p = 0, . . . , m − 2. Finally , set
ψ ( m ) (0 − ) = ˜
v + (︂ J ψ ( m − 1) (0 − ) + B d,1 η ( m − 1) ( − τ ) + B d,2 η ( m − 1) ( − τ ) + g ( m ) (0) )︂ and
η ( m ) (0 − ) = ˜
w −
ind( E , A 1 ) − 1
∑︂
k = 0
N k (︂ B a,1 ψ ( k + p + 1) ( − τ ) + B a,2 η ( k + p + 1) ( − τ ) + h ( k + p + 1) (0) )︂ .
The desir ed histor y functions are then given via H er mite interpolation. ■
A pplying the method of steps and the solution for mula (2.13) for the fast subsystem yields
w [ i + 1] = −
ind( E , A 1 ) − 1
∑︂
k = 0
N k (︃ d
d t )︃ k (︁ B a,1 v [ i ] + B a,2 w [ i ] + h [ i + 1] )︁ . (4. 19)
S ince Assumption 4.1 implies that all functions are sufficiently smooth we obtain
w [2] (0) − w [1] ( τ − ) =
ind( E , A 1 ) − 1
∑︂
k = 0
N k (︂ B a,1 (︂ ψ ( k ) (0 − ) − v ( k )
[1] (0) )︂ + B a,2 (︂ η ( k ) (0 − ) − w ( k )
[1] (0) )︂)︂
=
ind( E , A 1 ) − 1
∑︂
k = 0
N k B a T [︄ ψ ( k ) (0 − ) − v ( k )
[1] (0)
η ( k ) (0 − ) − w ( k )
[1] (0) ]︄
=
ind( E , A 1 ) − 1
∑︂
k = 1
N k B a T [︄ ψ ( k ) (0 − ) − v ( k )
[1] (0)
η ( k ) (0 − ) − w ( k )
[1] (0) ]︄ ,
wher e the last identity follo ws from the fact the
φ
is assumed to be admissible . Pr oposition 4.15
implies that
(4.1a)
is of de-smoothing type if ther e exists
k ∈ {
1
, . . . , ind
(
E , A 1
)
−
1
}
such that
N k B a =
0.
If we conv ersely assume N B a = 0 then (4.19) is given b y
w [ i + 1] = − B a,1 v [ i ] − B a,2 w [ i ] −
ind( E , A 1 ) − 1
∑︂
k = 0
N k h ( k )
[ i + 1] ,
which implies ℓ i ≥ 0. T ogether with Theorem 4.11, this pr o ves the follo wing theorem.
Theor em 4.16.
Consider the DDAE
(4.1a)
on the interval
I =
[0
, M τ
] and suppose that the associ-
ated ITP
(4.1)
satisfies As sumption 4.1. Let
N
,
B a
and
B a,2
be the matrices that ar e associated with
the quasi-W eierstr aß form (4.5) . Then (4.1a) is of
• smoothing type if N B a = 0 and B a,2 is nilpotent with nilpotency inde x ν B < M ,
• de-smoothing type if ther e exists k ∈ N such that N k B a = 0 , an d
• discontinuity invariant type, otherwise.
Example 4.17.
I ntroducing the shifted variable
z
(
t
)
= x
(
t − τ
) sho ws that the DDAE associated with
x ( t ) = A 2 x ( t − τ ) + D ˙
x ( t − τ ) + f ( t ) (4.20)
is of de-smoothing type if and only if D = 0. ♠

76 CHAPTER 4. CLASSICAL SOLUTIONS AND DISCONTINUITY PR OP A GA TION
As a dir ect consequence of Definition 4.9 and Theorem 4.16 w e can for mulate our main r esult about
the existence and uniqueness of continuous solutions .
Theor em 4.18.
Let the ITP
(4.1)
satisfy Assu mption 4.1. Let
N
and
B a
be the matrices associated
with the quasi-W eierstr aß form as defined in (4.3) and (4.4) .
(i)
If the history function
φ
is admissible, i.e.,
φ
satisfies
(4.8)
, then
N B a =
0 is a sufficient
condition for the existence of a solution (in the sense of Definition 4.3). In this case, the
solution is unique.
(ii)
If
N B a =
0 , then ther e exists an admissible history function
φ
, such that a solution exists
only on the time interval I = [0, τ ) .
R emark 4.19.
Checking the proof of C orollar y 4.12, we immediately infer from Theor em 4.16 that
Cor oll ary 4.12 is also tr ue for an arbitrary index
ind
(
E , A 1
). As a consequence, if the DDAE
(4.1a)
is of smoothing type, then ther e exists a sequence
j k ∈ N
such that
ℓ j k = k
and hence the solution
becomes arbitrarily smooth o ver time justifying the terminology smoothing type . ♣
N ote that N k B a = 0 for some k ∈ N implies
B k + 1 = C k + 1 A 2 = − T [︄ 0 0
0 N k ]︄ S A 2 = − T [︄ 0
N k B a ]︄ = 0,
i.e . the DDAE (4.1a) is of de-smoothing type if B k = 0 for some k ≥ 2. U sing
B k ( I n x − A con ) = − T [︄ 0 0
N k − 1 B a,1 N k − 1 B a,2 ]︄ T − 1 T (︄[︄ I n x ,d 0
0 I n x ,a ]︄ − [︄ I n x ,d 0
0 0 ]︄)︄ T − 1
= − T [︄ 0 0
0 N k − 1 B a,2 ]︄ T − 1
(4.21)
we immediately see that
B a,2
is nilpotent if and only if
B 1
(
I n x − A con
) is nilpotent, which sho ws
that the r esults of Theorem 4.16 can be formulated in terms of the underlying DDE
(4.7)
. As a
consequence of Lemma 2.13 this sho ws that the pr e vious r esults are independent of the particular
choice of the matrices
S , T
used to transform (
E , A 1
) to quasi-W eierstraß form. I n mor e detail, we
have the follo wing two r esu lts .
Cor ollar y 4.20.
Consider the ITP
(4.1)
with associated under lying DDE
(4.7)
on the interval
I = [0, M τ ) and suppose that Assumption 4.1 appl ies. Then (4.1a) is of
• smoothing type if B 2 = 0 and B 1 ( I n x − A con ) is nilpotent with nilpotency index ν B 1 ≤ M ,
• de-smoothing type if B k = 0 for some k ≥ 2 , and
• discontinuity invariant type otherwise.

4.1. CONTINUOUS SOL UTIONS AND CL ASSIFICA TION 77
Theor em 4.21.
Consider the ITP
(4.1)
with associated underlying DDE
(4.7)
on the interval
I =
[0, M τ ) and suppose that Assumption 4.1 appl ies.
(i) If the history function φ is admissible, i.e., φ satisfies (4. 8) , then B 2 = 0 is a sufficient condi-
tion for the existence of a solution (in the sense of Definition 4.3). In this case, the solution is
unique.
(ii)
If
B 2 =
0 , then ther e exists an admissible histor y function
φ
, such that a solution exists only
on the time interval I = [0, τ ) .
A common approach to analyze the (exponen tial) stability of the DDAE
(4.1a)
is to compute the
spectr al absci ssa , which is defined as
α ( E , A 1 , A 2 ) = sup{R e( λ ) | det( λ E − A 1 − exp( − λτ ) A 2 ) = 0}.
S urpr isingly , the condition
α
(
E , A , D
)
<
0 is not sufficient for a DDAE to be exponentially stable [69].
H o wever , based on the new classification we have the follo wing result.
Cor ollar y 4.22.
S uppose that the DDAE
(4.1a)
is not of de-smoothing type. Then the DDAE
(4.1a)
is exponentially stable if and only if α ( E , A 1 , A 2 ) < 0 .
Proof.
S ince the DDAE
(4.1a)
is not of de-smoothing type, we have
N B a =
0. The result follo ws
dir ectly from [69, Proposition 3.4 and Theor em 3.4]. ■
N ote that we r efrain from using the terminology r etarded , neutr al , and advanced in Definition 4.9,
although these terms are widely used in the delay liter ature [26, 27, 98, 102], see section 1.3 for a
definition. The reason is that in the classical definition, a r etarded DDE becomes advanc ed if it is
solved backwar ds in time, an advanced equation becomes r etar ded and a neutral equation stays
neutral. This is no longer true for the classification introduced in Definition 4.9. T o see this, we
introduce the new variable ξ ( t − τ ) = x ( − t ) such that (4.1a) tra nsforms to
E ˙
ξ ( t − τ ) = − A 2 ξ ( t ) − A 1 ξ ( t − τ ) − f ( − t ).
Definition 4.23. C onsider the DDAE (4.1a) and define
E : = [︄ 0 E
0 0 ]︄ ∈ F 2 n x ,2 n x , A 1 : = [︄ − A 2 0
0 I n x ]︄ ∈ F 2 n x ,2 n x , A 2 : = [︄ − A 1 0
− I n x 0 ]︄ ∈ F 2 n x ,2 n x .
Then we call the DDAE
E ˙
ζ ( t ) = A 1 ζ ( t ) + A 2 ζ ( t − τ ) + F ( t ) (4.22)
with F : I → F 2 n x the backwar d system for the DDAE (4.1a).
The matrix pair (
E , A 1
) is r egular if and only if
det
(
A 2
)
=
0. In th is case , we can transform the
backwar d system (4.22) to quasi-W eierstraß form via the matrices
S = [︄ − A − 1
2 0
0 I n x ]︄ and T = I 2 n x .

78 CHAPTER 4. CLASSICAL SOLUTIONS AND DISCONTINUITY PR OP A GA TION
I n par ticular , we have
( S E T )( S A 2 T ) = [︄ 0 − A − 1
2 E
0 0 ]︄ [︄ A − 1
2 A 1 0
− I n x 0 ]︄ = [︄ − A − 1
2 E 0
0 0 ]︄ .
Thus Theor em 4.16 implies that
E =
0 is a necessar y condition for the backwar d system
(4.22)
to be
of smoothing type or discontinuity invariant type, which implies that the DDAE
(4.1a)
cannot be of
de-smoothing type.
Example 4.24. C onsider the DDAE given b y F = R , n x = 2, f ≡ 0, τ = 1, and
E = [︄ 0 1
0 0 ]︄ , A 1 = [︄ 1 0
0 1 ]︄ , A 2 = [︄ 1 1
0 1 ]︄ .
S ince (
E , A 1
) is alr eady in W eierstraß form and
E A 2 =
0, Theor em 4.16 i mplies that the DDAE is of
de-smoothing type. Since E = 0 also the backwar d system is of de-smoothing type. ♠
Let us mention that if
det
(
A 2
)
=
0, then the method of steps
(2.3)
cannot be used to determine the
solution of the backwar d system. Instead, one may use the shift-index concept defined in [98, 99] to
make the pencil ( E , A 1 ) r egular . This can be achieved for instance with Algorithm 2, since
rank R ( s , ω ) ( s E + A 1 + ω A 2 ) = r ank R ( s , ω ) (︄[︄ − A 2 − ω A 1 s E
− ω I n x I n x ]︄)︄
= rank R ( s , ω ) (︄[︄ s ω E − A 2 − ω A 1 s E
0 I n x ]︄)︄
= rank R ( s , ω ) (︄[︄ s ω E − A 2 − ω A 1 0
0 I n x ]︄)︄
= rank R ( s , ω ) (︁ s E − A 1 − ω A 2 )︁ + n x
= 2 n x
implies that ( E , A 1 , A 2 ) is delay -regular accor ding to Theor em 3.20.
4.2 Impact of splicing conditions
I n the previous section algebraic criteria wer e established to check whether a discontinuity in the
derivative of
˙
x
at
t =
0 is smoothed out, is propagated to
t = τ
or is amplified in the sense that
x
becomes discontinuous at
t = τ
. While the definition of the discontinuity invariant type is valid for
all integer multiples of the delay time, the definitions of smoothing type and de-smoothing type
ar e based on single time points and hence the question whether the (de-)smoothing continues is
imminent. F or DDAEs of smoothing type, this can be answered p ositively (see R emark 4.19). F or
DDAEs of de-smoothing type, the question can be rephr ased as follows: If we r estrict the set of
admissible histor y functions such that the splicing condition (cf. [26])
φ ( k ) (0 − ) = x ( k ) (0) for k = 0, . . . , κ (4.23)

4.2. IMP A CT OF SPLICING CONDITIONS 79
is satisfied for some κ ∈ N , is ther e an integer j ∈ N such that the inital condition
x [ j ] (0) = x [ j − 1] ( τ − )
is not consistent for the DAE
(4.2)
? S imilarly , we can ask if for DDAEs of discontinuity invariant
type the smoothness at integer multiples of the delay time stays invariant. Before w e answer these
questions, we note that in or der to check if the splicing condition
(4.23)
is satisfied, it seems that
one has to solve the DDAE
(4.1a)
first. That is how ever not necessar y , since the splicing condition
(4.23) can be checked b y solely investigating the histor y function φ with Pr oposition 4. 7.
Lemma 4.25.
S uppose t hat the DDAE
(4.1a)
is of discontinuity invariant type and the admissible
histor y function φ ∈ C ∞ ([ − τ , 0], F n x ) satisfies the splicing condition (4.23) . Then
x ( k )
[ i ] (0) = x ( k )
[ i − 1] ( τ − ) for all i ∈ N , k = 0, . . . , κ .
Proof.
S ince
(4.1a)
is of discontinuity invariant type, we have
N B a =
0 in
(4.5)
accor ding to Theo-
r em 4.1 6. It suffices to sho w that
x ( j )
[2] (0) = x ( j )
[1] ( τ − ) for all j = 0, . . . , κ .
S ince φ is admissible and the DDAE is of discontinuity invariant type, equation (4.5a) implies that
˙
v [2] (0) − ˙
v [1] ( τ − ) = J (︁ v [2] (0) − v [1] ( τ ) )︁ + B d (︁ x [1] (0) − φ (0) )︁ = 0.
I teratively , we obtain
v ( k + 1)
[2] (0) − v ( k + 1)
[1] ( τ − ) = J (︂ v ( k )
[2] (0) − v ( k )
[1] ( τ − ) )︂ + B d (︂ x ( k )
[1] (0) − φ ( k ) (0) )︂ = 0
for k = 2, . . . , κ . F or the fast system (4.5b) we dir ectly infer
w ( k )
[2] (0) − w ( k )
[1] ( τ − ) = B a (︂ φ ( k ) (0 − ) − x ( k )
[1] (0) )︂ = 0
for k = 0, 1, . . . , κ , which completes the proof. ■
Lemma 4.25 guarantees that the solution of the DDAE is at least as smooth as the initial tr ansition
from the history function to the solution. Conversely , assume that the J ordan canonical form of
B a,2
exists and let
˜
w ∈ F n x ,a \ {
0
}
be an eigenvector of
B a,2
for the eigenvalue
λ =
0. Then Pr oposition 4 .15
implies (with
m = κ +
1) the existence of a histor y function
φ
such that the solution of the ITP
(4.1)
satisfies
w ( κ + 1)
[2] (0) − w ( κ + 1)
[1] ( τ − ) = B a,2 (︂ η ( κ + 1) (0 − ) − w ( κ + 1)
[1] (0) )︂ = λ ˜
w = 0.
Thus, in gener al, we cannot expect the solution of a DDAE of discontinuity invariant type to get any
smoother , which again justifies the terminology . F or DDAEs of de-smoothing type, Example 1.5
might suggest that the solution becomes less and less smooth until it becomes discontinuous . This
is ho wever not necessarily the case as the follo wing example demonstrates .

80 CHAPTER 4. CLASSICAL SOLUTIONS AND DISCONTINUITY PR OP A GA TION
Example 4.26.
S uppose that the ITP
(4.1)
satisfies Assumption 4.1 and additionally satisfies
N B a,2 =
0,
N B a =
0, and
N 2 B a =
0, i.e., the DDAE is of de- smoothing type accor ding to Theorem 4.16.
S uppose that the histor y function φ satisfies (4.10). Then
w [2] (0) − w [1] (0) =
ind( E , A 1 ) − 1
∑︂
k = 0
N k B a (︂ φ ( k ) (0 − ) − x ( k )
[1] (0) )︂
=
1
∑︂
k = 0
N k B a (︂ φ ( k ) (0 − ) − x ( k )
[1] (0) )︂ = 0.
H o wever , we have
˙
v [2]
(0)
− ˙
v [1]
(
τ −
)
=
0 b y the definition of the slo w system
(4.5a)
and b y induction
we infer
w [ i + 1] (0) − w [ i ] ( τ − ) = N B a,1 (︁ ˙
v [ i − 1] ( τ − ) − ˙
v [ i ] (0) )︁ = 0.
Thus the initial condition
x [ i ]
(0)
= x [ i − 1]
(
τ −
) is consistent for
(4.2)
and hence the solution exists for
all t f > 0. ♠
F or a general analysis w e assume that the ITP
(4.1)
satisfies Assumption 4.1, is of de-smoothing
type, and the history function
φ
satisfies the splicing condition
(4.23)
for some
κ ∈ N
. F rom
(4.5a)
we inductively infer
v ( k )
[2] (0) = J v ( k − 1)
[2] (0) + B d x ( k − 1)
[1] (0) + g ( k )
[2] (0) = v ( k )
[1] ( τ − )
for k = 1, . . . , κ + 1. F or the fast subsystem (4.5b), the splicing condition (4.23) implies
w [2] (0) − w [1] ( τ − ) =
ind( E , A 1 ) − 1
∑︂
k = κ + 1
N k B a (︂ φ ( k ) (0 − ) − x ( k )
[1] (0) )︂
and hence a sufficient condition for the initial condition
w [2]
(0)
= w [1]
(
τ −
) to be consistent is to
assume
N k B a =
0 for
k ≥ κ +
1. This is immediately satisfied for
ind
(
E , A 1
)
≤ κ +
1. T o analyze the
next inter val, we compute
w [3] (0) − w [2] ( τ − ) =
κ
∑︂
k = 1
N k B a T [︄ v ( k )
[1] ( τ − ) − v ( k )
[2] (0)
w ( k )
[1] ( τ − ) − w ( k )
[2] (0) ]︄
=
κ
∑︂
k = 1
N k B a,2 (︂ w ( k )
[1] ( τ − ) − w ( k )
[2] (0) )︂ .
The assumption N B a,2 = 0 implies w [3] (0) − w [2] ( τ − ) = 0. U nfortunately , we have
v (2)
[3] (0) − v (2)
[2] ( τ − ) = B d,2 (︁ ˙
w [2] (0) − ˙
w [1] ( τ − ) )︁ ,
and thus cannot sho w that the initial condition
w [4]
(0)
= w [3]
(
τ
) is consistent without posing further
assumptions on the matrices
E , A 1
, and
A 2
. S ince this becomes quite technical, we summarize our
findings only for the case ind( E , A 1 ) ≤ 3.

4.3. COMP ARISON TO THE EXISTING CLASSIFICA TION 81
Theor em 4.27.
S uppose t hat the ITP
(4.1)
satisfies As sumption 4.1 and
ind
(
E , A 1
)
≤
3 . M or eo ver ,
assume
N B a,2 =
0 and
N 2 B a,1 B d,2 =
0 . Then for ever y admissible histor y function
φ
that satisfies
(4.11) for κ = 2 , the ITP (4.1) has a uni que solution.
Proof.
The assumptions on
φ
imply that the splicing condition
(4.23)
is satisfied for
κ =
2 (see
Pr oposition 4 .7). S ince
ind
(
E , A 1
)
≤
3, we have
N 3 =
0. T ogether with
N B a,2 =
0 the pr evious
discussion guarantees that a solution exists on the interval [
− τ ,
3
τ
]. U sing
N B a,2 =
0, we observe
(inductively)
w [ i + 1] (0) − w [ i ] ( τ − ) =
2
∑︂
k = 0
N k B a,1 (︂ v ( k )
[ i − 1] ( τ − ) − v ( k )
[ i ] (0) )︂
= N 2 B a,1 B d,2 (︁ ˙
w [ i − 2] ( τ − ) − ˙
w [ i − 1] (0) )︁ = 0
and thus the initial condition
x [ i + 1]
(0)
= x [ i ]
(
τ −
) is consistent for all
i ∈ N
. The result follo ws from
Cor ollar y 2.15. ■
The assumptions in Theor em 4.27 can also be for mulated in terms of the underlying DDE
(4.7)
and
the matrices defined in (2.14). M ore pr ecisely , (4.21) and
B 0 ( I n x − A con ) = T [︄ B d,1 B d,2
0 0 ]︄ T − 1 T (︄[︄ I n x ,d 0
0 I n x ,a ]︄ − [︄ I n x ,d 0
0 0 ]︄)︄ T − 1 = T [︄ 0 B d,2
0 0 ]︄ T − 1
imply that N B a,2 = 0 and N 2 B a,1 B d,2 = 0 if and only if
B 2 ( I n x − A con ) = 0 and B 3 A con B 0 ( I n x − A con ) = 0,
r espectively .
R emark 4.28.
The proof of Theor em 4.27 sho ws that the result can be fur ther impro ved b y requiring
differ ent splicing conditions for the histor y function
ψ
for the slo w state
v
and for the history
function η of the fast state w . ♣
4.3 Comparison to the existing classification
I n [98 ] the authors r eplace the delayed argument in the DDAE
(4.1a)
with a function parameter
λ : I → F n x and obtain the initial value problem (IVP)
E ˙
x ( t ) = A 1 x ( t ) + A 2 λ ( t ) + f ( t ),
x ( t ) = φ (0), (4.24)
on the time inter val
I
. They call the function parameter
λ
consistent if ther e exists a consistent initial
condition
φ
(0) for the IVP
(4.24)
. Based on the function par ameter
λ
the follo wing classification for
DDAEs [98] is introduced.

82 CHAPTER 4. CLASSICAL SOLUTIONS AND DISCONTINUITY PR OP A GA TION
Definition 4.29.
The DDAE
(4.1a)
is called r etar ded , neutr al , or advanced , if the minimum smooth-
ness r equirement for a consistent function param eter
λ
is that
λ ∈ C
(
I , F n x
),
λ ∈ C 1
(
I , F n x
), or
λ ∈ C k ( I , F n x ) for some k ≥ 2.
T o compare the classification based on pr opagation of pr imar y discontinuities (cf. Definition 4.9)
with the classification given in [98], we need to understand D efinition 4.29 in terms of the quasi-
W eierstraß form.
Pr oposition 4.30.
S uppose t hat the matrix pair (
E , A 1
) in the DDAE
(4.1a)
is r egular and the
inhomogeneity f is sufficiently smooth. Then the DDAE (4.1a) is
• r etar ded if and only if B a = 0 ,
• neutr al i f and only if B a = 0 and N B a = 0 , and
• advanced otherwise,
wher e B a and N ar e the matrices from the quasi-W eierstr aß form (Theor em 2.9) and (4. 5) .
Proof.
The smoothness r equirements for
λ
can be dir ectly seen from the underlying DDE
(4.7)
. W e
have
B 0 = T [︄ I n x ,d 0
0 0 ]︄ S A 2 = T [︄ B d
0 ]︄ and
B k = − T [︄ 0 0
0 N k − 1 ]︄ S A 2 = − T [︄ 0
N k − 1 B a ]︄
for
k =
1
, . . . , ind
(
E , A 1
). H ence
(4.1a)
is r etarded if and only if
N k − 1 B a =
0 for all
k =
1
, . . . , ind
(
E , A 1
),
which is equivalent to
B a =
0. The DDAE is neutral, if
N k − 1 B a =
0 for all
k =
2
, . . . , ind
(
E , A 1
), which
is equivalent to N B a = 0 and other wise advanced. ■
With the char acter ization, we immediately see that the classification intr oduced [98] pro vides an
upper bound for the new definition in the follo wing sense .
Cor ollar y 4.31. Suppose that the ITP (4.1) satisfies A ssumption 4.1.
• If the DDAE (4. 1a) is not advanced, then it is not of de-smoothing type.
• If the DDAE (4. 1a) is advanced, then it is of de-smoothing type.
S ince the classification introduced in this paper is based on the worst-case scenario , the numerical
method described in [98], which is formulated for DDAEs that are not advanced, is safe to use .
R emark 4.32.
The numerical method introduced in [98] is tailored to DDAEs that ar e not advanced
and cannot be used for advanced DDAEs. H ow ever , if it is kno wn that the histor y function satisfies

4.3. COMP ARISON TO THE EXISTING CLASSIFICA TION 83
the splicing condition
(4.23)
for some
κ >
0 , then also advanced DDAEs may be solved (cf. Theo-
r em 4.27 ). Thus, there is a need for numerical integr ation schemes that can handle such situations .
This is subject to further resear ch. ♣

84 CHAPTER 4. CLASSICAL SOLUTIONS AND DISCONTINUITY PR OP A GA TION

5
Nonlinea r DD AEs
H aving established the existence and uniqueness theor y for linear delay differential-algebr aic
equations (DDAEs) in a distributional (chapter 3) and classical solution framework (chapter 4), we
ar e now r eady to turn our attention to nonlinear initial value problems (IVP s) of the for m
0 = F ( t , x ( t ), ˙
x ( t ), x ( t − τ )), t ≥ 0, (5.1a)
x ( t ) = φ ( t ), t ∈ [ − τ , 0 ] (5.1b)
with (nonlinear) function
F : I × D x × D ˙
x × D σ τ x → F m
defined on open sets
D x , D ˙
x , D σ τ x ⊆ R n x
and
time inter val
I =
[
t 0 , t f
). The function
φ :
[
− τ ,
0]
→ R n x
is called initial trajectory or histor y func-
tion. Examples are for instance th e hybrid pendulum-mass-spring-damper system discussed in
section 1.1.1 and the delayed feedback contr ol for the container crane in section 1.1.2. In both cases ,
the complete physical systems can be separated in two components , namely the numer ical and
experimental part for the hybr id testing approach, and the plant and the contr oller for the feedback
control system. The decomposition is illustrated in Figur e 5.1, which is a special case of Figur e 1.1.
As in the pr evious chapters, we fr equently use the shift operator
σ τ
defined via (
σ τ x
)(
t
)
= x
(
t − τ
).
I n par ticular , the DDAE (5.1a) takes the for m
0 = F ( t , x , ˙
x , σ τ x ).
Let us mention that dir ect extensions to time-var iable or state-dependent delays may be possible
via the transformation described in [165, 166]. This is, ho wever , beyond the scope of this thesis and
r equires further investigation. I n general the solution of
(5.1)
depends on derivatives of
F
and
φ
and
thus we make the follo wing assumption for the remainder of this chapter .
Assumption 5.1. The functions F and φ in (5.1) ar e su fficiently smooth.
If
F
is linear and time-independent, then the analysis in chapter 3 r eveals that the DDAE is delay -
r egular if and only if it can be transformed to a DDAE, where the differ ential-algebraic equation
(DAE) that is obtained from the method of steps (cf.
(2.3a)
) is r egular . W e therefor e restrict ourselves
85

86 CHAPTER 5. NONLINEAR DDAES
0 = ˇ
F ( t , x 1 , ˙
x 1 , u 1 ),
y 1 = ˇ
G ( t , x 1 )
0 = ˆ
F ( t , x 2 , ˙
x 2 , u 2 ),
y 2 = ˆ
G ( t , x 2 )
P HYSICAL S YSTEM
K ( t , u 1 , u 2 , y 1 , y 2 )

F igure 5.1 – Decomposition of a physical system into substructures
to this situation — in particular we assume
m = n x
— and analyze the well-posedness of
(5.1)
in
this chapter with a classical solution concept. W e illu strate our theor etical findings for the hybrid
numerical-experimental system introduced in section 1.1.1, and hence first discuss this system in
mor e detail in section 5.1.
The main tool to establish existence and uniqueness r esults for linear time-invariant DDAEs in
Chapter 4 is the W eierstraß canonical form (cf. Theorem 2.9), which allo ws to decouple a linear
time-invariant DAE into a differ ential equation and an algebraic equation. F or nonlinear DAEs,
the separation into differ ential and algebraic equations r equires the implicit function theor em
(cf. [127, Theor em 4.12] ) and hence the analysis of the propag ation of discontinuities in terms of the
original DDAE becomes difficult. Instead, we make use of the fact that t he r esults of Chapter 4 can
also be stated in terms of the underlying delay differ en tial equation (DDE) — see Theor em 4.21.
The two main contributions in this chapter ar e the follo wing:
(i)
W e sho w that the compr ess-and-shift algorithm (Algorithm 1) from Chapter 3 can be applied
to the nonlinear hybrid numerical-exper imental system and terminates with a r egular DDAE
whenever the two subsystems ar e repr esented b y regular DAEs. The details are pr esented in
Lemma 5.10, Theor em 5.1 5 and Theor em 5.17.
(ii)
W e establish existence and uniqueness results for a class of nonlinear DDAEs in Theor em 5.24
and conclude that the hybrid system is solvable whenever the subsystems ar e strangeness-fr ee
(cf. Cor ollar y 5.25).
5.1 H ybr id numerical-exper imental system
F or the general description of the model equations, we assume that w e have already subdivided
the complete model into two sub-models, which later on repr esent the numerical part and the
experimental part. F or an illustration we r efer to Figure 5.1. The first subsystem is described by the
descriptor system
0 = ˇ
F ( t , x 1 , ˙
x 1 , u 1 ), (5.2a)
y 1 = ˇ
G ( t , x 1 ) (5.2b)

5.1. HYBRID NUMERICAL-EXPERIMENT AL SYSTEM 87
with state
x 1
(
t
)
∈ R n x ,1
, input
u 1
(
t
)
∈ R m 1
, and output
y 1
(
t
)
∈ R p 1
. The second subsystem is given b y
0 = ˆ
F ( t , x 2 , ˙
x 2 , u 2 ), (5.3a)
y 2 = ˆ
G ( t , x 2 ) (5.3b)
with
x 2
(
t
)
∈ R n x ,2
,
u 2
(
t
)
∈ R m 2
, and
y 2
(
t
)
∈ R p 2
. The complete model is given b y imposing the
inter connection
0 = K ( t , u 1 , y 1 , u 2 , y 2 ). (5.4)
I n order to solve this r elation for u 1 and u 2 it is common to r equir e that
[︂ ∂ K
∂ u 1
∂ K
∂ u 2 ]︂ ( t , u 1 , y 1 , u 2 , y 2 )
is nonsingular for all (
t , u 1 , y 1 , u 2 , y 2
). F or simplicity , we r estrict ourselves to the case that the
inter connection is given b y
u 1 ( t ) = y 2 ( t ) and u 2 ( t ) = y 1 ( t ). (5.5)
I n par ticular , we assume
m 1 = p 2
and
m 2 = p 1
. The complete model as depicted in Figur e 5.1 is thus
given b y the implicit equation
0 = [︄ ˇ
F ( t , x 1 , ˙
x 1 , ˆ
G ( t , x 2 ))
ˆ
F ( t , x 2 , ˙
x 2 , ˇ
G ( t , x 1 )) ]︄ (5.6)
with initial conditions
x 1 (0) = ζ 1 and x 2 (0) = ζ 2 . (5.7)
Example 5.2.
T o recast the coupled pendulum-mass-spring-damper system from sec tion 1.1.1 in
this form, we first have to tr ansform the systems to first order . By introducing new v ar iables for the
velocities and after r enaming we obtain
ˇ
F ( t , x 1 , ˙
x 1 , u 1 ) = [︄ ˙
x 1,1 − x 1,2
M ˙
x 1,2 + C x 1,2 + K x 1,1 − u 1 ]︄ , ˇ
G ( t , x 1 ) = x 1,1 ,
ˆ
F ( t , x 2 , ˙
x 2 , u 2 ) =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
˙
x 2,1 − x 2,4
˙
x 2,2 − x 2,5
x 2,6 − u 2
m ˙
x 2,4 + 2 x 2,3 x 2,1
m ˙
x 2,5 + 2 x 2,3 ( x 2,2 − u 2 ) + m g
x 2
2,1 + ( x 2,2 − u 2 ) 2 − L 2
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
, ˆ
G ( t , x 2 ) = − 2 x 2,3 ( x 2,2 − x 2,6 ) − m g .
N ote that we have intr oduced the ar tificial variable
x 2,6
to account for the feedthrough, i.e ., the fact
that the for ce
F pendulum
depends on the vertical position of the mass-spring-damper , which itself is
used as input for the mathematical pendulum. ♠
If both subsystems ar e linear time-invariant, then we write
ˇ
F ( t , x 1 , ˙
x 1 , u 1 ) = ˇ
E ˙
x 1 − ˇ
A x 1 − ˇ
B u 1 + ˇ
f ( t ), ˇ
G ( t , x 1 ) = ˇ
C x 1 ,
ˆ
F ( t , x 2 , ˙
x 2 , u 2 ) = ˆ
E ˙
x 2 − ˆ
A x 2 − ˆ
B u 2 + ˆ
f ( t ), ˆ
G ( t , x 2 ) = ˆ
C x 2 , (5.8)

88 CHAPTER 5. NONLINEAR DDAES
with external forcing functions ˇ
f and ˆ
f , such that the complete model (5.6) is given b y
[︄ ˇ
E 0
0 ˆ
E ]︄ [︄ ˙
x 1
˙
x 2 ]︄ = [︄ ˇ
A ˇ
B ˆ
C
ˆ
B ˇ
C ˆ
A ]︄ [︄ x 1
x 2 ]︄ + [︄ ˇ
f
ˆ
f ]︄ . (5.9)
Befor e we continue our discussion, let us emphasize that, in general, th er e is no relation between
the r egular ity of the subsystems
(5.2)
and
(5.3)
and the r egular ity of the coupled system
(5.6)
. Also ,
the index from the subsystems might differ from the index of the coupled system. As an immediate
consequence , the splitting of the system into smaller subsystems is a delicate task that must be
performed carefully .
Example 5.3. C onsider the linear DAE
⎡
⎢
⎣
1 0 0
0 0 0
0 0 0 ⎤
⎥
⎦ ⎡
⎢
⎣
˙
x 1
˙
x 2
˙
x 3
⎤
⎥
⎦ = ⎡
⎢
⎣
0 c 0
c 0 1
0 1 − 1 ⎤
⎥
⎦ ⎡
⎢
⎣
x 1
x 2
x 3
⎤
⎥
⎦ + ⎡
⎢
⎣
f 1
f 2
f 3
⎤
⎥
⎦ (5.10)
with external forcing function
f =
[
f 1 , f 2 , f 3
] and parameter
c ∈ R
. It is easy to see that for any
c ∈ R
the system has differ entiation index
ν =
1. S plitting the system into
z 1 =
[
x 1 , x 2
] and
z 2 = x 3
we
obtain the two subsystems
[︄ 1 0
0 0 ]︄ [︄ ˙
x 1
˙
x 2 ]︄ = [︄ 0 c
c 0 ]︄ [︄ x 1
x 2 ]︄ + [︄ 0
1 ]︄ u 1 + [︄ f 1
f 2 ]︄ (5.11a)
0 = − x 3 + u 2 + f 3 . (5.11b)
The second subsystem
(5.11b)
has differ entiation index
ν =
1. F or the first subsystem
(5.11a)
we
obser ve that for
c =
0 the pencil of the DAE is singular . F or
c =
0 the pencil is r egular with index
ν = 2, which is higher than the index of the coupled system. ♠
Example 5.4. F or i = 1, 2 we consider the subsystems
[︄ 1 0
0 0 ]︄ ˙
x i = [︄ a i 0
0 1 ]︄ x i + [︄ b i ,1 b i ,2
c i ,1 c i ,2 ]︄ u i ,
which ar e already in W eierstraß canonical form
(2.6)
with index
ν =
1. The coupled system with
coupling r elations u 1 = x 2 and u 2 = x 1 is given b y the linear DAE E ˙
x = A x with
E = ⎡
⎢
⎢
⎢
⎣
1 0 0 0
0 0 0 0
0 0 1 0
0 0 0 0
⎤
⎥
⎥
⎥
⎦ , A = ⎡
⎢
⎢
⎢
⎣
a 1 0 b 1,1 b 1,2
0 1 c 1,1 c 1,2
b 2,1 b 2,2 a 2 0
c 2,1 c 2,2 0 1
⎤
⎥
⎥
⎥
⎦ , and x = [︄ ˙
x 1
˙
x 2 ]︄ .
U sing strong equivalence , see section 2.1, we obtain
( E , A ) ∼ ⎛
⎜
⎜
⎜
⎝ ⎡
⎢
⎢
⎢
⎣
1 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
⎤
⎥
⎥
⎥
⎦ , ⎡
⎢
⎢
⎢
⎣
a 1 b 1,1 0 b 1,2
b 2,1 a 2 b 2,2 0
0 c 1,1 1 c 1,2
c 2,1 0 c 2,2 1
⎤
⎥
⎥
⎥
⎦ ⎞
⎟
⎟
⎟
⎠ ,

5.2. THE METHOD OF STEPS 89
and immediately obser ve that ( E , A ) has differ entiation index ν = 1 if and only if c 1,2 c 2,2 = 1. Other -
wise , we obtain
( E , A ) ∼ ⎛
⎜
⎜
⎜
⎝ ⎡
⎢
⎢
⎢
⎣
1 0 0 0
0 1 0 0
0 0 0 0
0 0 0 0
⎤
⎥
⎥
⎥
⎦ , ⎡
⎢
⎢
⎢
⎣
a 1 b 1,1 b 1,2 0
b 2,1 a 2 − c 1,1 b 2,1 − b 21 c 12 0
c 2,1 − c 1,1 c 2,2 0 0
0 0 0 1
⎤
⎥
⎥
⎥
⎦ ⎞
⎟
⎟
⎟
⎠
sho wing that also ν = 2, ν = 3, and ( E , A ) singular are possible . ♠
R emark 5.5.
If both subsystems ar e por t-H amiltonian systems [21], then, under reasonable con-
ditions, the coupled system itself is again a port-H amiltonian system. In this cas e , [15 2, Thm. 4.3]
implies that the differ entiation index of the coupled system is at most ν = 2. ♣
Our standing assumption is that the first model is simulated numerically , while the second model is
tested experimentally . F ollo wing the discussion in section 1.1.1, the transfer system that realiz es
the numerical results in r eal-time within the experiment is delayed, such that the second model
technically acts at a differ ent time point. The hybrid numerical-exper imental model, which we
study in this paper , is thus given b y
0 = [︄ ˇ
F ( t , x 1 ( t ), ˙
x 1 ( t ), ˆ
G ( t − τ , x 2 ( t − τ )))
ˆ
F ( t − τ , x 2 ( t − τ ), ˙
x 2 ( t − τ ), ˇ
G ( t − τ , x 1 ( t − τ ))) ]︄ , (5.12)
which in the linear case simplifies to
[︄ ˇ
E 0
0 0 ]︄ [︄ ˙
x 1 ( t )
˙
x 2 ( t ) ]︄ + [︄ 0 0
0 ˆ
E ]︄ [︄ ˙
x 1 ( t − τ )
˙
x 2 ( t − τ ) ]︄ =
[︄ ˇ
A 0
0 0 ]︄ [︄ x 1 ( t )
x 2 ( t ) ]︄ + [︄ 0 ˇ
B ˆ
C
ˆ
B ˇ
C ˆ
A ]︄ [︄ x 1 ( t − τ )
x 2 ( t − τ ) ]︄ + [︄ ˇ
f ( t )
ˆ
f ( t − τ ) ]︄ .
(5.13)
N ote that if the hybrid model is initialized at time
t 0
, then the numerical simulation star ts at
t 0
, while
the experimental part star ts at
t 0 + τ
. I n par ticular , it is sufficient to prescribe an initial trajectory
solely for the experimental part, i.e., only for x 2 .
5.2 The method of steps
The standar d procedure to solv e initial trajectory problems for delay equations is via successive
integration on the time intervals [(
i −
1)
τ , i τ
) with
i =
1
, . . . , M
, wher e
M ∈ N
is the smallest integer
such that
T ≤ M τ
. This approach is alr eady discussed in chapter 2 and used in chapters 3 and 4
to establish existence and uniqueness r esults for linear time-invariant DDAEs. F or the sake of
pr esentation, the method is r ecalled in detail. F or the DDAE
(5.1a)
we introduce for
i ∈ I : =
{1, . . . , M }
x [ i ] : [0, τ ] → R n x , t ↦→ x ( t + ( i − 1) τ ),
F [ i ] : [0, τ ] × D x × D ˙
x × D σ τ x → R n x , ( t , x , y , z ) ↦→ F ( t + ( i − 1) τ , x , y , z ),
x [0] : [0, τ ] → R n x , t ↦→ φ ( t − τ ).
(5.14)

90 CHAPTER 5. NONLINEAR DDAES
Then we have to solv e for each i ∈ {1 , . . . , M } the DAE
0 = F [ i ] ( t , x [ i ] , ˙
x [ i ] , x [ i − 1] ), t ∈ [0, τ ), (5.15a)
x [ i ] (0) = x [ i − 1] ( τ − ), (5.15b)
with right continuation
x [ i − 1] ( τ − ) : = lim
t ↗ τ x [ i − 1] ( t ).
If
(5.15)
is uniquely solvable (pro vided that the initial value
x [ i − 1]
(
τ −
) is consistent), then we can
construct the solution of
(5.1)
on the successive time intervals [(
i −
1)
τ , i τ
). As outlined in the
introduction we cannot exp ect a smooth transition of the solution betw een these inter vals, see for
instance Examples 1.4 and 1.5. W e ther efore extend the solution concept from D efinition 4.3 to the
nonlinear case .
Definition 5.6
(Solution concept)
.
Assume that
F
in the DDAE
(5.1)
and the initial trajectory
φ
ar e
sufficiently smooth. W e call
x ∈ C
(
I , R n x
) a solution of
(5.1)
if for all
i ∈ I
the r estr iction
x [ i ]
of
x
as
in
(5.14)
is a solution of
(5.15)
. W e call the initial trajectory
φ :
[
− τ ,
0]
→ R n x
consistent if the initial
value problem (5.1) has at least one solution.
W e emphasize that in or der to check if an initial trajectory is consistent, we actually have to compute
a solution of the initial value problem
(5.1)
, see also section 4.1. This is in contrast to the DAE
theor y , where it suffices to compute th e consistency set
(2.34)
. T o account for this issue, w e adopt
Definition 4.6, which ensur es that we can at least ensur e a solution in the inter val [0, τ ).
Definition 5.7
(Admissible initial tr ajector y)
.
The initial trajectory
φ
is called admissible for the
DDAE (5.1a) if the initial condition
x [1] (0) = φ (0)
is consistent for the DAE (5.15) with i = 1.
F oll o wi ng the discussion in section 2.3, consistent initial values ar e characterized b y the consistency
set (2.34). W e therefor e have to assume that the DAE
0 = F [1] ( t , x [1] , ˙
x [1] , φ ( t − τ )) (5.16)
satisfies H ypothesis 2.29. I n order to simplify the discussion, we make the follo wing definition,
which is motivated from the discussion in [98].
Definition 5.8.
The DAE that is obtained from the DDAE
(5.1a)
b y substituting
x
(
t − τ
) with a control
function
u
(
t
) is called the associated DAE for the DDAE
(5.1a)
. W e say that the DDAE
(5.1a)
satisfies
H ypothesis 2.33 if its associated DAE satisfies Hypothesis 2.33.
S uppose no w that the DDAE
(5.1a)
satisfies H ypothesis 2.33 with strangeness index
µ
. Then the
strangeness-fr ee r eformulation for the associated DAE as discussed in (2.38) is given b y
0 = D ( t , x , ˙
x , u ), (5.17a)
0 = A (︂ t , x , u , ˙
u , . . . , u ( µ ) )︂ . (5.17b)

5.2. THE METHOD OF STEPS 91
Although formally , the algebraic equation depends on derivatives of
u
up to or der
µ
, it may happen
that
∂ A
∂ u ( ℓ ) (︂ t , x , u , ˙
u , . . . , u ( µ ) )︂ ≡ 0
for some
ℓ ≤ µ
. F ollo wing the classification in Definition 4.9, respectively , Theorem 4.16 and
Cor ollar y 4.20, it is essential to know the lar gest number s such that
∂ A
∂ u ( s ) (︂ t , x , u , ˙
u , . . . , u ( µ ) )︂ ≡ 0.
Consequently , from this point for war d, we work with
0 = A (︂ t , x , u , ˙
u , . . . , u ( s ) )︂ (5.18)
instead of (5.17b), with the understanding that the algebraic equation does not depend on u , i.e.,
0 = A ( t , x )
if
s = −∞
. Replacing the contr ol input
u
with the delayed argument r esu lts in the differ ence equation
0 = A (︂ t , x , σ τ x , σ τ ˙
x , . . . , σ τ x ( s ) )︂ . (5.19)
S ince the set of consistent initial values is described by
(5.19)
, we immediately obtain the follo wing
r esult.
Lemma 5.9.
Assume that the histor y function
φ
is sufficiently smooth and the DDAE
(5.1a)
satisfies
H yp othesis 2.33 with str angeness index
µ
. Let
(5.19)
denote the differ ence equation that results
from the str angeness-free r eformulation. Then φ is admissible for the DDAE (5.1a) if and only if
0 = A (︂ t , φ (0), φ ( − τ ), ˙
φ ( − τ ), . . . , φ ( s ) ( − τ ) )︂ . (5.20)
Lemma 5.9 r equir es that the DDAE satisfies H ypothesis 2.33, which in turn implies that the associ-
ated DAE is r egular . U nfor tunately , this is only a sufficient condition for the existence of a unique
solution for the IVP
(5.1)
, as discussed in detail in section 3.1, see for instance Theorem 3.17. I t
is easy to see that the associated DAE for the hybrid numerical-experimental model
(5.12)
is not
r egular and therefor e does not satisfy Hypothesis 2.33 and henc e Lemma 5.9 does not apply to
(5.12).
One strategy to r esolve this issue is to find a r efor mulation of the DDAE
(1.13)
b y shifting certain
equations . This is achieved either b y a combined shift-and-derivative array and the so-called
shift index [94, 98], or b y the compress-and-shift algorithm (Algorithm 2) presented in section 3.3.
The latter algorithm ’ s idea is to identify (after a potential transformation of the equations – the
compr ession step), which equations do not depend on the current state but solely on the past state.
These equations ar e then shifted in time, and the procedur e is iterated. Let us emphasize that neither
the shift-and-derivative arr ay approach nor the compr ess-and-shift algorithm is readily av ailable for

92 CHAPTER 5. NONLINEAR DDAES
general nonlinear DDAEs. S till, the special structur e of the hybrid numerical-experimental model
immediately suggests to shift the second block ro w of equations, yielding
0 = F ( t , x , ˙
x , σ τ x ) : = [︄ ˇ
F ( t , x 1 , ˙
x 1 , σ τ ˆ
G ( t , x 2 ))
ˆ
F ( t , x 2 , ˙
x 2 , ˇ
G ( t , x 1 )) ]︄ , (5.21)
with x ( t ) : = [︂ x 1 ( t )
x 2 ( t ) ]︂ ∈ R n x , n x : = n x ,1 + n x ,2 . In the linear case (5.21) simplifies to
[︄ ˇ
E 0
0 ˆ
E ]︄ [︄ ˙
x 1
˙
x 2 ]︄ = [︄ ˇ
A 0
ˆ
B ˇ
C ˆ
A ]︄ [︄ x 1
x 2 ]︄ + [︄ 0 ˇ
B ˆ
C
0 0 ]︄ [︄ σ τ x 1
σ τ x 2 ]︄ + [︄ ˇ
f
ˆ
f ]︄ . (5.22)
W e immediately obtain
det (︄[︄ s ˇ
E − ˇ
A 0
− ˆ
B ˇ
C s ˆ
E − ˆ
A ]︄)︄ = det( s ˇ
E − ˇ
A ) det( s ˆ
E − ˆ
A )
and thus have pro ven the next r esult.
Lemma 5.10.
The matrix pencil of the associated DAE for the linear shifted hybrid numerical-
experimental system
(5.22)
is r egular if and only if the associated DAEs of the li near subsystems
(5.8)
ar e r e gular .
R emark 5.11.
I n the ter minology of [98], the hybrid numer ical-experimental system
(5.13)
has shift
index
κ =
1. In the liter ature , shifting of equations, i.e., systems with shift index
κ >
0, are often
r eferred to as noncausal and hence not physical. The hybrid numer ical-experimental setup details
that the shifting of equations can also occur if the dynamics of the subsystems affect the o verall
dynamic at differ ent time instants. In particular , the requir ement to shift parts of the equations may
be a r esult of how a system is modeled. ♣
Befor e we proceed let us emphasize that shifting of equations potentially enlar ges the solution space
of the IVP for the differ ential equation.
Example 5.12. C onsider the DDAE
˙
x 1 ( t ) = x 2 ( t − τ ) + f ( t ), (5.23a)
0 = x 2 ( t − τ ) − g ( t ). (5.23b)
N otice that the second equation constitutes a r estr iction for the initial trajectory . Indeed, if we
pr escr ibe the initial trajectory
x 1 ( t ) = φ 1 ( t ), x 2 ( t ) = φ 2 ( t ), for t ≤ 0, (5.24)
then a solution cannot exist if
φ 2
(
t
)
= g
(
t + τ
) for
t ∈
[
− τ ,
0]. If
φ 2
(
t
)
= g
(
t + τ
) for
t ∈
[
− τ ,
0], then
the solution of the initial trajectory problem (5.23),(5.24) is given b y
x 1 ( t ) = φ 1 (0) + ∫︂ t
0 g ( s ) + f ( s ) d s , x 2 ( t ) = g ( t + τ ) for t ≥ 0.

5.2. THE METHOD OF STEPS 93
I n par ticular , the solution space for
x 1
is parameterized b y
φ 1
(0) and thus a one-dimensional vector
space . If we , ho wever , r eplace (5.23b) w ith the shifted equation
x 2 ( t ) = g ( t + τ ) (5.25)
and consider the initial trajectory problem
(5.23a)
,
(5.25)
,
(5.24)
, then for any initial trajectory
φ
that
satisfies φ 2 (0) = g ( τ ) the solution of (5.23a),(5.25),(5.24) for t ∈ [0, τ ] is given b y
x 1 ( t ) = φ 1 (0) + ∫︂ t
0 φ 2 ( s − τ ) + f ( s ) d s , x 2 ( t ) = g ( t + τ ),
such that the solution space for x 1 is infinite-dimensional. ♠
R emark 5.13.
The shifted hybrid system
(5.21)
sho wcases, that only an initial trajectory for the
experimental system
ˆ
F
is r equired, which is in agr eement with the discussion after
(5.13)
. This is no
contradiction to E xample 5.12, since the numer ical and experimental part are initialized at differ ent
time points . ♣
If the linear subsystems ar e regular , then Lemma 5.10 together with Theorem 2.19 immediately im-
plies existence and uniqueness of solutions of the initial trajectory problem (ITP) for the DDAE
(5.22)
in the space of piecewise-smooth distributions, see [203] and section 2.2. Let us emphasize that
τ >
0 is a crucial assumption in Lemma 5.10, since Example 5.4 sho wcases that a similar result
cannot be obtained if
τ =
0. U nfor tunately , it is not immediately clear , what the index of the matr ix
pencil of the associated DAE is .
Example 5.14. C onsider the matrix pencil
⎛
⎜
⎜
⎜
⎝ ⎡
⎢
⎢
⎢
⎣
0 1 0 0
0 0 0 0
0 0 0 1
0 0 0 0
⎤
⎥
⎥
⎥
⎦ , ⎡
⎢
⎢
⎢
⎣
1 0 0 0
0 1 0 0
a b 1 0
c d 0 1
⎤
⎥
⎥
⎥
⎦ ⎞
⎟
⎟
⎟
⎠ ∼ ⎛
⎜
⎜
⎜
⎝ ⎡
⎢
⎢
⎢
⎣
0 1 0 0
0 0 0 0
0 0 0 1
0 0 0 0
⎤
⎥
⎥
⎥
⎦ , ⎡
⎢
⎢
⎢
⎣
1 0 0 0
0 1 0 0
0 0 1 0
c 0 0 1
⎤
⎥
⎥
⎥
⎦ ⎞
⎟
⎟
⎟
⎠
of the associated DAE for the hybrid numerical-exper imental system
(5.22)
, wher e both subsystems
have differ entiation index
ν =
2. If
c =
0, then the pencil also has index
ν =
2, other wise the index is
ν = 3. ♠
The index of the shifted hybrid numerical-experi mental model depends on the coupling functions
ˇ
G
and
ˆ
G
. As a direct consequence , H ypothesis 2.33 has to be checked for each example separately ,
since it is not clear a-priori , what the corr esponding strangeness index
µ
is . A notable exception is
pro vided i n the case that both subsystems ar e strangeness-fr ee.
Theor em 5.15.
S uppose that the subsystems
(5.2)
and
(5.3)
ar e strangeness-fr ee, i.e., satisfy Hy-
pothesis 2.33 with char acteristic values
ˇ
µ = ˆ
µ =
0 ,
ˇ
a , ˆ
a , ˇ
d
, and
ˆ
d
, r espectively . If
τ >
0 , then the
shifted hybrid numerical-experimental model
(5.21)
satisfies H ypothesis 2.33 with characteristic
values µ = 0 , a = ˇ
a + ˆ
a , and d = ˇ
d + ˆ
d .

94 CHAPTER 5. NONLINEAR DDAES
Proof.
Let
ˇ
Z A
,
ˇ
T A
,
ˇ
Z D
, and
ˆ
Z A
,
ˆ
T A
,
ˆ
Z D
denote the matrix functions from H ypothesis 2.33 for the
subsystems (5.2) and (5.3), r espectively . Define a : = ˇ
a + ˆ
a and accor dingly
d = n − a = n 1 − ˇ
a + n 2 − ˆ
a = ˇ
d + ˆ
d .
Choose ˆ
T ⋆
A such that [︂ ˆ
T A ˆ
T ⋆
A ]︂ is nonsingular . From H ypothesis 2.33 we deduce that
(︃ ˆ
Z T
A
∂ ˆ
F
∂ x 2
ˆ
T ⋆
A )︃ ( t , x 2 , ˙
x 2 , ˆ
G ( t , x 1 ))
is nonsingular . Define (omitting arguments) the matrix functions
Z A : = [︄ ˇ
Z A 0
0 ˆ
Z A ]︄ , T A : = [︄ ˇ
T A 0
− ˆ
T ⋆
A (︂ ˆ
Z T
A ∂ ˆ
F
∂ x 2
ˆ
T ⋆
A )︂ − 1 ˇ
Z T
A ∂ ˆ
F
∂ u 2
∂ ˇ
G
∂ x 1
ˆ
T A ˆ
T A ]︄ , Z D : = [︄ ˇ
Z D 0
0 ˆ
Z D ]︄ .
W e have to check the differ ent items from H ypothesis 2.33 for the shifted hybr id numerical-
experimental model (5.21). W e notice that ˇ
µ = 0 = ˆ
µ implies ˇ
D µ = ˇ
F and ˆ
D µ = ˆ
F and obser ve
rank (︃ ∂ F
∂ ˙
x )︃ = rank (︄[︄ ∂ ˇ
F
∂ x 1 0
0 ∂ ˆ
F
∂ x 2 ]︄)︄ = rank (︃ ∂ ˇ
F
∂ x 1 )︃ + rank (︃ ∂ ˆ
F
∂ x 2 )︃ = ˇ
a + ˆ
a = a .
W e immediately conclude
(︃ Z T
A
∂ F
∂ ˙
x )︃ ( t , x , ˙
x , σ τ x ) = ⎡
⎣ (︂ ˇ
Z T
A ∂ ˇ
F
∂ ˙
x 1 )︂ (︁ t , x 1 , ˙
x 1 , σ τ ˆ
G ( t , x 2 ) )︁ 0
0 (︂ ˆ
Z T
A ∂ ˆ
F
∂ ˙
x 2 )︂ (︁ t , x 2 , ˙
x 2 , ˇ
G ( t , x 1 ) )︁ ⎤
⎦ = 0
such that the first item from H ypothesis 2.33 is satisfied. F or the second item we obtain (omitting
arguments)
ˆ
a = rank (︃ ˆ
Z T
A
∂ ˆ
F
∂ x 2 )︃ ≤ rank (︂[︂ ˇ
Z T
A ∂ ˆ
F
∂ u 2
∂ ˇ
G
∂ x 1
ˆ
Z T
A ∂ ˆ
F
∂ x 2 ]︂)︂ ≤ ˆ
a
and thus
rank (︃ Z T
A
∂ F
∂ x )︃ = rank (︄[︄ ˇ
Z T
A ∂ ˇ
F
∂ x 1 0
ˇ
Z T
A ∂ ˆ
F
∂ u 2
∂ ˇ
G
∂ x 1
ˆ
Z T
A ∂ ˆ
F
∂ x 2 ]︄)︄
= rank (︃ ˇ
Z T
A
∂ ˇ
F
∂ x 1 )︃ + rank (︃ ˆ
Z T
A
∂ ˆ
F
∂ x 2 )︃ = ˇ
a + ˆ
a = a .
W e conclude
Z T
A
∂ F
∂ z T A = ⎡
⎣ ˇ
Z T
A ∂ ˇ
F
∂ z 1
ˇ
T A 0
ˇ
Z T
A ∂ ˆ
F
∂ u 2
∂ ˇ
G
∂ x 1
ˆ
T A − ˆ
Z T
A ∂ ˆ
F
∂ z 2
ˆ
T ⋆
A (︂ ˆ
Z T
A ∂ ˆ
F
∂ x 2
ˆ
T ⋆
A )︂ − 1 ˇ
Z T
A ∂ ˆ
F
∂ u 2
∂ ˇ
G
∂ x 1
ˆ
T A ˆ
Z T
A ∂ ˆ
F
∂ z 2
ˆ
T A ⎤
⎦ = 0.
S imilarly as before we have
rank (︃ ∂ F
∂ ˙
x T A )︃ = rank (︃ ∂ ˇ
F
∂ ˙
x 1
ˇ
T A )︃ + rank (︃ ∂ ˆ
F
∂ ˙
x 2
ˆ
T A )︃ = ˇ
d + ˆ
d = d .
The proof follo w s from
rank (︃ Z T
D
∂ F
∂ ˙
x T A )︃ = rank (︃ ˇ
Z T
D
∂ ˇ
F
∂ ˙
x 1
ˇ
T A )︃ + rank (︃ ˆ
Z T
D
∂ ˆ
F
∂ ˙
x 2
ˆ
T A )︃ = d . ■

5.2. THE METHOD OF STEPS 95
R emark 5.16.
The assumption
τ >
0 is crucial in Theorem 5.15. In the case
τ =
0, we have alr eady
seen in Example 5.4 that even if both subsystems ar e strangeness-fr ee, the coupled system might
have stran geness-index µ > 0. ♣
I n the case that either of the subsystems is not strangeness-fr ee we can proceed as follo ws. Let
0 = ˇ
D ( t , x 1 , ˙
x 1 , u 1 ), 0 = ˆ
D ( t , x 2 , ˙
x 2 , u 2 ),
0 = ˇ
A (︂ t , x 1 , u 1 , ˙
u 1 , . . . , u ( ˇ
µ )
1 )︂ , 0 = ˆ
A (︂ t , x 2 , u 2 , ˙
u 2 , . . . , u ( ˆ
µ )
2 )︂ ,
˙
x 1 = ˇ
f (︂ t , x 1 , u 1 , ˙
u 1 , . . . , u ( ˇ
µ + 1) )︂ , ˙
x 2 = ˆ
f (︂ t , x 2 , u 2 , ˙
u 2 , . . . , u ( ˆ
µ + 1) )︂
denote the strangeness-fr ee r efor mulations and the underlying ODEs for
(5.2)
and
(5.3)
, r espectively .
R ecall the coupling conditions
u 1 = σ τ (︁ ˆ
G ( t , x 2 ) )︁ and u 2 = ˇ
G ( t , x 1 ),
which we have to differ entiate
ˆ
µ +
1, r espectively
ˇ
µ +
1 times . W e obser ve that in the inter val [0
, τ
)
the coupling condition for
u 1
does not depend on
x 2
but on the histor y
φ 2
. In particular , we obtain
(assuming that ˆ
G is sufficiently smooth)
˙
u 1 = σ τ (︃ ∂ ˆ
G
∂ t ( t , φ 2 ) + ∂ ˆ
G
∂ x 2
( t , φ 2 ) ˙
φ 2 )︃ ,
¨
u 1 = σ τ (︄ ∂ 2 ˆ
G
∂ t 2 ( t , φ ) + 2 ∂ 2 ˆ
G
∂ t ∂ x 2
( t , φ 2 ) ˙
φ 2 + ∂ 2 ˆ
G
∂ x 2
2
( t , φ 2 ) ˙
φ 2 + ∂ ˆ
G
∂ x 2
( t , φ 2 ) ¨
φ 2 )︄ ,
and similarly for higher derivatives . In particular , there exist functions ˇ
ˇ
D , ˇ
ˇ
A , and ˇ
ˇ
f
0 = ˇ
ˇ
D ( t , x 1 , ˙
x 1 , σ τ φ 2 ),
0 = ˇ
ˇ
A (︂ t , x 1 , σ τ φ 2 , σ τ ˙
φ 2 , . . . , σ τ φ ( ˇ
µ )
2 )︂ ,
˙
x 1 = ˇ
ˇ
f (︂ t , x 1 , σ τ φ 2 , σ τ ˙
φ 2 , . . . , σ τ φ ( ˇ
µ + 1)
2 )︂
for
t ∈
[0
, τ
). Consequently , we can (locally) solve for
x 1
, pro vided that the initial trajectory
φ 2
is
sufficiently smooth and x 1 (0) satisfies the consistency condition
0 = ˇ
ˇ
A (︂ 0, x 1 (0), φ 2 ( − τ ), ˙
φ 2 ( − τ ), . . . , φ ( ˇ
µ )
2 ( − τ ) )︂ .
On the other hand, the input r elation for u 2 implies
˙
u 2 = ∂ ˆ
G
∂ t ( t , x 1 ) + ∂ ˆ
G
∂ x 1
( t , x 1 ) ˙
x 1
= ∂ ˆ
G
∂ t ( t , x 1 ) + ∂ ˆ
G
∂ x 1
( t , x 1 ) ˇ
ˇ
f (︂ t , x 1 , σ τ φ 2 , σ τ ˙
φ 2 , . . . , σ τ φ ( ˇ
µ + 1)
2 )︂ .
N ote that although derivatives of
φ 2
up to or der
ˇ
µ +
1 appear ,
˙
u 2
does not necessarily depend on all
of them (see for instance Example 5.14 and the discussion after D efinition 5.8). In any case , ther e

96 CHAPTER 5. NONLINEAR DDAES
exists functions ˆ
ˆ
D , ˆ
ˆ
A , and ˆ
ˆ
f such that
0 = ˆ
ˆ
D ( t , x 2 , ˙
x 2 , x 1 ),
0 = ˆ
ˆ
A (︂ t , x 2 , x 1 , σ τ φ 2 , σ τ ˙
φ 2 , . . . , σ τ φ ( ˇ
µ + ˆ
µ )
2 )︂ ,
˙
x 2 = ˆ
ˆ
f (︂ t , x 1 , x 2 , σ τ φ 2 , σ τ ˙
φ 2 , . . . , σ τ φ ( ˇ
µ + ˆ
µ + 1)
2 )︂ .
Thus, the underlying delay differential equation for the shifted hybrid numerical-experimental
system (5.21) in [0, τ ) is given b y
[︄ ˙
x 1
˙
x 2 ]︄ = ⎡
⎣
ˇ
ˇ
f (︂ t , x 1 , σ τ φ 2 , σ τ ˙
φ 2 , . . . , σ τ φ ( ˇ
µ + 1)
2 )︂
ˆ
ˆ
f (︂ t , x 1 , x 2 , σ τ φ 2 , σ τ ˙
φ 2 , . . . , σ τ φ ( ˇ
µ + ˆ
µ + 1)
2 )︂ ⎤
⎦ (5.26)
and the differ entiation index is at most ˇ
µ + ˆ
µ + 1 and we have sho wn the follo wing result.
Theor em 5.17.
Assume that the subsystems
(5.2)
and
(5.3)
satisfy H ypothesis 2.33 with strangeness
index
ˇ
µ
,
ˆ
µ
, r espectively . Then the shifted hybrid numerical-experimental system
(5.21)
has a well-
defined differ entiation index, which is at most ˇ
µ + ˆ
µ + 1 .
Example 5.18.
Shifting the equations for the pendulum in the hybrid version of the coupled
pendulum-spring-mass-damper system in
(1.5)
and introducing new variables
v 1 : = ˙
y 1
,
v 2 : = ˙
x 2
,
and v 3 : = ˙
y 2 for the velocities, yields the system
˙
y 1 = v 1 , (5.27a)
˙
x 2 = v 2 , (5.27b)
˙
y 2 = v 3 , (5.27c)
M ˙
v 1 + C v 1 + K y 1 = f ( σ τ y 1 , σ τ y 2 , σ τ λ ), (5.27d)
m ˙
v 2 = − 2 λ x 2 , (5.27e)
m ˙
v 3 = − 2 λ ( y 2 − y 1 ) − m g , (5.27f)
0 = x 2
2 + ( y 2 − y 1 ) 2 − L 2 , (5.27g)
which is a multibody system with forcing term f ( y 1 , y 2 , λ ) = − 2 λ ( y 2 − y 1 ) − m g that solely depends
on delayed variables . S ince multibody systems are sp ecial instances of H essenberg systems, we
conclude from [127, S e c. 4.2] that th e shifted hybrid pendulum-mass-spring-damper system has
strangeness index
µ =
2 and satisfies H ypothesis 2.33 with
a =
3 and
d =
4. The algebraic equations
and the differ ence equations are given b y
0 = x 2
2 + ( y 2 − y 1 ) 2 − L 2 ,
0 = 2 x 2 v 2 + 2( y 2 − y 1 )( v 3 − v 1 ),
0 = 2 v 2
2 + 2( v 2 − v 1 ) 2 − 4
m λ ( x 2
2 + ( y 2 − y 1 ) 2 ) − 2( y 2 − y 1 ) (︃ g + f ( σ τ y 1 , σ τ y 2 , σ τ λ )
M − C
M v 1 − K
M y 1 )︃ .
Let us emphasize that despite the higher index, the algebraic equations do not depend on derivativ es
of
σ τ x
. N ote that also the Lagrange-multiplier is delayed in
(5.27d)
such that this example is not
included in the specific r etarded H essenberg forms as studied in [13]. ♠

5.3. SOL V ABILIT Y OF THE HYBRID MODEL 97
R emark 5.19.
M odels that feature a similar delay structur e as in
(5.21)
and
(5.22)
arise in the
time-discr etization via w aveform r elaxzation [15, 72, 159] or the analysis of semi-explicit time-
integrators [4], see also section 1.1.5. ♣
5.3 Solvability of the hybrid model
I n the previous section, we have established t hat the shifted hybrid numerical-exper imental sys-
tem
(5.21)
can be solved in the interval [0
, τ
) and is r egular in the sense of Theorem 2.32, pro vided
that the subsystems satisfy H ypothesis 2.33 and the histor y function is admissible . The question
that r emains to be answered is whether a solution exists on time intervals [0, T ) with T > τ .
R emark 5.20.
F or the linear time-invariant case , this is discussed in detail in Chapter 3 for a
distributional solution concept and in [50, 52, 53 , 97, 173, 211] for the solution concept as defined
in Definition 5.6. R esults for linear time-varying systems are developed for instance in [98, 99].
M oreo ver , a special class of nonlinear DDAEs is discussed in [13]. ♣
I n view of the method of steps discussed in the previous section, the question that r emains to be
answer ed is, which conditions on the subsystems and the histor y function ensur e that the initial
condition
(5.15b)
is consistent for all
i =
1
, . . . , M
. U nfortunately , the regularity of the subsystems
and an admissible histor y function ar e not sufficient to establish a solution for
t > τ
, see for instance,
the discussion in Chapter 4 and the follo wing example .
Example 5.21. C onsider the r egular DDAE
˙
x ( t ) = y ( t ), 0 = x ( t ) − y ( t − 1).
A pplying the method of steps, equation (5.15) yields
x [ i ] = y [ i − 1] and y [ i ] = ˙
y [ i − 1] . (5.28)
F or the history function
φ
(
t
)
= [︁ 0
t + 1 ]︁
we obtain
x [1]
(
t
)
= t
and
y [1]
(
t
)
=
1, and we deduce that
the history function is admissible. H o wever , the initial value
y [1]
(1)
=
1 is not consistent for the
associated DAE on the interval [1, 2). In particular , the solution exists only on the inter val [0, 1). ♠
The issue in the pr evious example is, as already outlined in chapt er 4, that the equation
z i = ˙
z i − 1
r esults in solutions
z i
that become less smooth for incr easing
i
, and possible discontinuities of the
form
x ( k )
[ i − 1] ( τ − ) = x ( k )
[ i ] (0)
ar e propagated to
x ( k − 1)
[ i ] ( τ − ) = x ( k − 1)
[ i + 1] (0).
The discontinuity propagation leads to the classification intr oduced in Definition 4.9, with a com-
plete characterization pr esented in Theorem 4.16. U nfortunately , the main tool for the proof of
Theor em 4.16 i s the W eierstraß canonical form, and thus in general, we cannot expect to have a
similar r esult for the nonlinear DDAE
(5.1a)
. I nstead, we use the classification given in [98] and

98 CHAPTER 5. NONLINEAR DDAES
make use of the fact, that in the linear case, this classification pro vides an upper bound for the
classification in Definition 4.9 in the sense of Cor ollar y 4.31.
Definition 5.22. Assume that the DDAE (5.1a) satisfies H y pothesis 2.33 and let
˙
x = f (︂ t , x , σ τ x , . . . , σ τ x ( s ) )︂ (5.29)
denote the underlying DDE of the DDAE
(5.1a)
and assume
∂ f
∂σ τ x ( s ) ≡
0. Then
(5.1a)
is called r etar ded ,
neutr al , or advanced if s = 0, s = 1, or s ≥ 2 in (5.29).
Lemma 5.23. Assume that the DDAE (5. 1a) satisfies H ypothesis 2.33 and let
0 = D (︁ t , x , ˙
x , σ τ x )︁ , (5.30a)
0 = A (︂ t , x , σ τ x , σ τ ˙
x , . . . , σ τ x ( s − 1) )︂ (5.30b)
denote the associated strangeness-fr ee reformulation with the convention that either
A
does not
depend on σ τ x ( k ) for any k ∈ N , or
∂ A
∂σ τ x ( s − 1) ≡ 0.
Then
(5.1a)
is r etar ded, neutr al, or advanced, if
∂ A
∂σ τ x ( k ) ≡
0 for all
k ∈ N
,
s =
1 , or
s =
2 , r espectively .
Proof. The pr oof follo ws immediately from r ewr iting (5.30) as in (2.39) and (2.40). ■
Theor em 5.24.
S uppose t hat the DDAE
(5.1a)
is sufficiently smooth, has st r angeness-index
µ
,
satisfies H ypothesis 2.33 with characteristic v al ues
µ , a , d
and
µ +
1
, a , d
, is not advanced, and
the histor y function
φ 0 ∈ C 1
([0
, τ
]
, R n x
) is admissible. Then the initial trajectory problem
(5.1)
is
solvable.
Proof.
S ince the DDAE
(5.1a)
satisfies H ypothesis 2.33 and is not advanced, Lemma 5.23 implies
that the strangeness-fr ee r efor mulation is of the form
0 = D ( t , x , ˙
x , σ τ x ), 0 = A ( t , x , σ τ x ) (5.31)
with the understanding that
A
may not depend on
σ τ x
. Applying the method of steps to
(5.31)
yields
the sequence of initial value problems
0 = D (︁ t + ( i − 1) τ , x [ i ] , ˙
x [ i ] , x [ i − 1] )︁ ,
0 = A (︁ t + ( i − 1) τ , x [ i ] , x [ i − 1] )︁ ,
x [ i ] (0) = x [ i − 1] ( τ − ).
(5.32)
S ince the histor y function is admissible, we can (locally) solve
(5.32)
for
i =
1 and b y [127, Theo-
r em 4.13 ] this solution is also a solution of
(5.15)
. Although this solution is of local natur e it can

5.3. SOL V ABILIT Y OF THE HYBRID MODEL 99
be globalized b y applying the cited theorem again until w e reach the boundary of
M µ
(cf. [127, R e-
mark 4.14]). If we assume that the solution exists on the time interval [0
, τ
) this immediately implies
0 = lim
t ↗ τ A ( t , x [1] ( t ), x [0] ( t )) = A ( τ , x [1] ( τ − ), x [0] ( τ )).
H ence ,
x [1]
(
τ −
) is consistent for the DAE
(5.15)
with
i =
2. The r esult follows iter atively b y repeating
this procedur e. ■
Cor ollar y 5.25.
S uppose t hat the numerical and experimental subsystems
(5.2)
and
(5.3)
both
satisfy H ypothesis 2.33 with
µ =
0 . Then for any
τ >
0 and for any admissible histor y function
φ
, the initial trajectory problem for the shifted hybrid numerical-experimental system
(5.21)
is
solvable.
Proof.
Theor em 5.15 ensures that the shifted hybrid system is strangeness-fr ee . Lemma 5.23 thus
implies that (5.21) is not advanced. The result follo ws from Theorem 5.24. ■
Example 5.26.
Although the system for the pendulum
(1.3)
is not strangeness-fr ee , Example 5.18
sho ws that the shifted hybrid system resulting from coupling the pendulum with the mass-spring-
damper system is not advanced. In particular , Theorem 5.24 ensur es that the associated initial
trajectory problem is solvable . ♠
If the DDAE
(5.1a)
is advanced, then in general w e cannot expect a solution for the ITP
(5.1)
, see
for instance Example 1.5 and the r esults from Chapter 4. H o wever , if the initial trajector y is linked
smoothly to the solution, i.e ., the initial trajector y satisfies the splicing condition
(4.23)
, then we can
expect to establish further results similar to Theorem 4.27. N evertheless, mimicking the strategy fr om
the proof of Theor em 4.27 in the nonlinear case , requir es the use of the underlying DDE
(5.29)
and
thus the implicit function theor em. A detailed analysis of this setting is currently under investigation
and subject to further resear ch.

100 CHAPTER 5. NONLINEAR DDAES

P art II
S tr uctur ed r ealization theor y
101

6
Problem setting and background
F rom a modeling perspective , it is often difficult to describe a physical or chemical system exactly via
differ ential equations and modeling laws usually apply only for ideal settings. Thus, it is desirable to
construct a model in an automated fashion directly fr om data, that may come from some experiment.
The data may be in the form of
• a time series, for instance obtained from a numerical simulation or experiment, or
• transfer function ev aluations, for instance obtained b y a vector network analyzer [11].
The standing assumption in this second part of the thesis is that the data is generated b y a dynamical
system
Σ
, exemplified in Figur e 6.1. H ereb y we do not assume any kno wledge about the system, in
particular , ther e is no realization, i.e ., a descr iption of the internal dynamics, of the system available .
Σ
u y

F igure 6.1 – Input-output mapping of a black-bo x system
Despite the inaccessibility of a description of detailed internal dynamics, ther e may yet be significant
auxiliar y information or at least a basic understanding of ho w the system should behave , allo wing
one to surmise general structural featur es of the underlying dynamical system. F or example, vi-
bration effects ar e naturally associated with subsystems that hav e second-order structur e; inter nal
transport or signal propagation will natur ally be associated with time delays—see T able 6.1 for
further examples.
Example 6.1
(Acoustic tr ansmission, [191])
.
Consider the acoustic tr ansmission example from
section 1.1.4 and suppose we ar e interested in the acoustic pr essure
y
(
t
)
= p
(
ξ 0 , t
) at a fixed point
ξ 0 ∈
(0
, L
) in the duct (see Figur e 1.5), which we view as the output of an abstract yet unkno wn
system that is driven b y the input fluid velocity
u
(
t
), determined by the acoustic driver positioned at
ξ =
0. Instead of using the partial differ ential equ ation (PDE) model
(1.8)
, we simply assume that
the output pr essure depends linearly on the input velocity in a way that is inv ar iant to translation in
103

104 CHAPTER 6. PR OBLEM SET TING AND BA CK GR OUN D
T able 6.1 – Examples for system structures with output mapping y ( t ) = C x ( t )
state space description transfer function
second-or der A 1 ¨
x ( t ) + A 2 ˙
x ( t ) + A 3 x ( t ) = B u ( t ) C (︂ s 2 A 1 + s A 2 + A 3 )︂ − 1 B
state delay A 1 ˙
x ( t ) + A 2 x ( t ) + A 3 x ( t − τ ) = B u ( t ) C (︁ s A 1 + A 2 + e − τ s A 3 )︁ − 1 B
neutral delay A 1 ˙
x ( t ) + A 2 x + A 3 ˙
x ( t − τ ) = B u ( t ) C (︁ s A 1 + A 2 + s e − τ s A 3 )︁ − 1 B
viscoelastic A 1 ¨
x ( t ) + ∫︁ t
0 h ( t − τ ) A 2 ˙
x ( τ )d τ + A 3 x ( t ) = B u ( t ) C (︂ s 2 A 1 + s ˆ
h ( s ) A 2 + A 3 )︂ − 1 B
time , and so the output could be anticipated to involv e some superposition of i nternal states that
ar e lagged in time according to propag ation delays r elated to the distance traveled b y the signal.
Assuming a uniform sound speed
c >
0 throughout the duct, we allo w for a direct pr opagation delay
τ 1 = ξ 0
/
c
between the input and output location and a second propagation delay
τ 2 =
(2
L − ξ 0
)/
c
,
associated with a r eflected signal. A semi-empir ical model for the state evolution of a system that
has these basic featur es could have the form
A 1 x ( t ) + A 2 x ( t − τ 1 ) + A 3 x ( t − τ 2 ) = b u ( t ),
with an output port map given b y
y
(
t
)
= c T x
(
t
). The matr ices
A 1
,
A 2
, and
A 3
, the port maps
associated with the vectors b and c , as well as their dimensions ar e unkno wn. ♠
Throughout this part of the thesis we make the assumption that the system
Σ
in Figur e 6.1 is linear ,
i.e ., there exists a linear time-invariant (L TI) operator
S
with
y = S u
. W e ar e thus interested in
finding a suitable operator (in the sense of section 1.2), that appr oximates the data in some suitable
norm.
Pr oblem 6.2. Construct a structured L TI oper ator ˜︁
S solely from input/output data such that
∥ y − ˜︁
y ∥=∥ S u − ˜︁
S u ∥ ≤ ε ∥ u ∥ (6.1)
for all admissible input signals u , a small par ameter ε ≥ 0 , and suitable norms.
I n order to solve Pr oblem 6.2 we have to addr ess the question what kind of data we assume available
and define pr ecisely , what a structur ed L TI oper ator is . Recall that L TI systems can be r epresented
either in the time domain or in the fr equency domain [6]. The mapping from one domain to the
other is given b y the Laplace-transform for continuous time systems and the Z-tr ansfor m for discr ete
time systems . M oreo ver , the
L ∞
error in the time domain can be bounded b y the
H 2
error in the
fr equency domain v ia
  y − ˜︁
y   L ∞ : = sup
t > 0 ∥ y ( t ) − ˜︁
y ( t ) ∥ ∞ ≤ ∥ S − ˜︁
S ∥ H 2 ∥ u ∥ L 2 . (6.2)
I n fact, for single -input/single-output (SISO) systems, the
H 2
norm is the
L 2
-
L ∞
induced norm of
the underlying convolution operator , i. e .
∥ S − ˜︁
S ∥ H 2
is the smallest number such that
(6.2)
holds for
all inputs u ∈ L 2 [20].

105
I t is well-kno wn (cf. [20] and the refer ences therein) that if the solution oper ator
S
is the convolution
operator of a standar d state-space r ealization, that is (assuming a zero initial condition and no dir ect
feedthrough)
( S u )( t ) = ∫︂ t
0 C exp( A ( t − s )) B u ( s )d s ,
the
H 2
error
∥ S − ˜︁
S ∥ H 2
is minimized if the tran sfer function of
˜︁
S
interpolates the transfer function
of
S
at the mirror images of the poles of
˜︁
S
. Thus our approach to solve Problem 6.2 is to const ruct
˜︁
S
such that it is an interpolant of S in the fr equency domain.
R emark 6.3.
If a state-space description of the dynamical process under investiga tion is kno wn,
one could use model or der reduction (MOR) methods (see the r ecent sur veys and books [6, 8, 17, 28,
29, 106, 177]) to obtain a lo w-dimensional and cheap-to-evaluate surrogate model. Among the many
MOR methods let us mention rational interpolation (formerly kno wn as moment matching) [14, 18]
and
H 2
-optimal interpolation [89] as methods that also interpolate the tran sfer function. N ote that
the
H 2
optimality conditions for structured pr oblems are much mor e involved [22, 70] and to our
kno wledge, ther e exists no general computational str ategy to obtain optimal interpolation points
even if the state-space description is available . Pr eser vation of system structures of the state-space
description is for instance considered in [18, 57, 58, 78, 133, 155, 199]. W e note that almost all of these
approaches r equire an internal description. N otable exceptions ar e pro vided in [71, 184, 189, 191].
♣
R emark 6.4.
F or some model problems , for instance a circuit that involv es a lossless transmission
line [39], it is possible to transform a hyperbolic PDE into a delay equation [61, 139] that is — from a
computational perspective — much easier to solve . See also section 1.1.4 for a detailed example .
Thus even if a state-space description is available , it may be advantageous to choose a differ ent
structure for the surr ogate model than for the original model. N otice that many of these problems
ar e characterized b y slo wly decaying Hankel singular v alu es or K olmogoro v
n
-widths (see [213] for a
connection between the two concepts), which pr events classical MOR methods from succeeding
and thus r equires a special tr eatment [35, 48, 162, 180]. ♣
The second part of the thesis is organized as follo ws: F irst, we pr e cisely state the problem for
r ealizing a delay di ffer ential-algebraic equation (DDAE) from fr eq uency measur ements of a transfer
function in section 6.1 and introduce the term system structure , that allo ws us to not only identify
a DDAE from measur ements, but a larger system class (cf. Problem 6.6 and Problem 6.9). The
framework t o obtain an interpolant of the fr equency domain data is der ived in chapter 7 and can be
understood as a generalization of the Loewner fr amework [150] (see section 6.2 for further details).
If we can access only input/output measur ements in the time domain, we can estimate fr equency
data via the empirical tr an sfer function estimate (ETFE) [137] or the least-squar es transfer function
estimate (lsTFE) [168]. This approach together with the estimation of unkno wn parameters in the
system structure , for instance the delay time τ , is pr esented in chapter 8.
R emark 6.5.
If the data under consider ation is prone to noise one may wish to use a least-squar es
approach instead of interpolation, such as vector fitting [66, 67, 93], or dynamic mode decomposition
[128, 186, 210 ]. W e consider this, ho wever , a second step and thus postpone this to future wor k. ♣

106 CHAPTER 6. PR OBLEM SET TING AND BA CK GR OUN D
Let us mention that most of the content of this part of the thesis is alr eady published in the jour nal
articles [189, 191] and the preprint [77]. The presented r esults ar e j oint work with Christopher
Beattie ( Virginia T ech), Elliot F osong (U niversity of Cambridge), Ser kan Guger cin ( Virginia T ech),
and Philipp Schulze ( TU Berlin).
6.1 Problem setting
R ecall that a L TI DDAE is given b y
E ˙
x ( t ) = A 1 x ( t ) + A 2 x ( t − τ ) + B u ( t ),
y ( t ) = C x ( t ), (6.3)
and — as befor e — we call
x
(
t
)
∈ R n x
,
u
(
t
)
∈ R n u
, and
y
(
t
)
∈ R n y
, the state , input , and output of the
system
(6.3)
, which we assume to be exponentially bounded. In this case the Laplace tr ansform may
be applied to (6.3) and r earranged to ˆ︁
y ( s ) = H ( s ) ˆ︁
u ( s ) with
H ( s ) = C ( s E − A 1 − exp( − τ s ) A 2 ) − 1 B ,
pro vided that
det
(
s E − A 1 − exp
(
− τ s
)
A 2
) is not vanishing identically (i.e ., the DDAE
(6.3)
is delay -
r egular , cf. Theorem 3.20) and the initial condition
x ( t ) = 0 for t ∈ [ − τ , 0 ]
is satisfied. The function
H
:
C → C n y × n u
is called the tr ansfer function of
(6.3)
. Since the tr ansfer
function characterizes the input-output behavior of
(6.3)
, measur ements of
H
seem appropriate to
construct the realization. Mor e pr ecisely , we assume that the follo wing data is available: Suppose
we have 2
n
points in the complex plane , which may be interpreted as complex driving fr equencies,
{ µ 1 , . . . , µ n }
and
{ σ 1 , . . . , σ n }
. In addition to these complex fr equencies, we have the so-called left
tangential dir ection vectors
{ ℓ 1 , . . . , ℓ n }
and the right tangential dir ection vectors
{ r 1 , . . . , r n }
wher e
ℓ i ∈ R p
and
r i ∈ R m
for
i =
1
, . . . , n
. I n the SISO case, i.e .,
n u = n y =
1, these tangential dir ections are
assigned the value one , i.e.,
ℓ i = r i =
1. U nlike projection-based model r eduction, which r eq uir es
access to the state space quantities, data-driven interpolatory model reduction only assumes access
to the action of the transfer function ev aluated at the dr iving fr equencies along the tangential
dir ections, i.e.,
ℓ T
i H ( µ i ) = f T
i and H ( σ i ) r i = g i for i = 1, . . . , n . (6.4)
If the dir ection vectors
ℓ i
and
ℓ j
ar e linearly independent, one can allo w
µ i
to coincide with
µ j
,
and similarly for the
σ i
’ s . H ow ever , for simplicity the only coincidence of interpolation points that
we admit will be between left and right interpolation points, i.e .,
µ i = σ j
. If this is the case for an
index pair (
i , j
), then bitangential derivative data is assumed to be available . Since w e assume that
each of the two sets
{ µ i } n
i = 1
and
{ σ i } n
i = 1
consists of
n
distinct points, if
µ i = σ j
for an index pair (
i , j
),
without loss of generality , we assume
i = j
. Then, the corr esponding bitangential der ivative data is
defined as
ℓ T
i H ′ ( µ i ) r i = θ i ,

6.1. PR OBLEM SET TING 107
wher e
H ′
denotes the derivative of
H
, i.e .,
H ′ : = d
d s H
. F ollo wing [7, 150], we summarize the interpo-
lation data as
left interpolation data: {( µ i , ℓ i , f i ) | µ i ∈ C , ℓ i ∈ C n y , f i ∈ C n u , i = 1, . . . , n },
right interpolation data: {( σ i , r i , g i ) | σ i ∈ C , r i ∈ C n y , g i ∈ C n u , i = 1, . . . , n },
bitangential derivative data: {( i , θ i ) | i ∈ {1, . . . , n } for which µ i = σ i , θ i ∈ C },
(6.5)
with the understanding that the last category may be empty if
{ µ i } n
i = 1 ∩ { σ i } n
i = 1 = ∅
. N ote that in
the case
µ i = σ i
, the compatibility of the conditions
(6.4)
r equires that
f T
i r i = ℓ T
i g i
. F or the ease of
pr esentation in the next sections, we summarize the interpolation data in the matrices
M : = diag( µ 1 , . . . , µ n ) ∈ C n × n , S : = diag( σ 1 , . . . , σ n ) ∈ C n × n ,
L : = [︂ ℓ 1 . . . ℓ n ]︂ ∈ C n y × n , R : = [︂ r 1 . . . r n ]︂ ∈ C n u × n ,
F : = [︂ f 1 . . . f n ]︂ ∈ C n u × n , G : = [︂ g 1 . . . g n ]︂ ∈ C n y × n .
(6.6)
The problem that we ar e inter ested in solving can thus be for mulated as follo ws .
Pr oblem 6.6
(R ealization problem for DDAEs)
.
Giv en the interpol ation data in
(6.5)
, find matrices
˜︁
E , ˜︁
A 1 , ˜︁
A 2 ∈ C n x × n x
,
˜︁
B ∈ C n x × n u
, and
˜︁
C ∈ C n y × n x
and a par ameter
τ ≥
0 , such that the transfer
function
˜︁
H ( s ) = ˜︁
C (︁ s ˜︁
E − ˜︁
A 1 − exp( − τ s ) ˜︁
A 2 )︁ − 1 ˜︁
B (6.7)
satisfies the interpolation conditions
ℓ T
i ˜︁
H ( µ i ) = f T
i and ˜︁
H ( σ i ) r i = g i for i = 1, . . . , n .
If µ i = σ i for any index i , then addition ally ,
ℓ T
i ˜︁
H ′ ( µ i ) r i = θ i
is to be satisfied.
R emark 6.7. One may furthermore ask for the tr ansfer function ˜︁
H in (6.7) to be r eal , i.e., to satisfy
˜︁
H ( s ) = ˜︁
H ( s ) for all s ∈ C , (6.8)
wher e
z
denotes the complex conjugate of
z ∈ C
. A sufficient condition to ensur e a real tr ansfer
function is to ask for the system matrices
˜︁
E , ˜︁
A 1 , ˜︁
A 2 , ˜︁
B , ˜︁
C
to be r eal, which we henceforth refer to as
r eal r ealization problem . ♣
F or
τ =
0, Problem 6.6 r educes to the task of identifying a linear system in generalized state-space
form, a task which is successfully solved via the Loewner framework [150], see the forthcoming
section 6.2. In that sense , the particular str uctur e of the transfer function in
(6.7)
can be seen as a
generalization of the Loewner fr amework.
Depending on the application at hand, it may not be desirable to construct a r ealization that
depends on the past. Instead, one may pr escr ibe a differ ent system stru ctur e. F or instance , a general

108 CHAPTER 6. PR OBLEM SET TING AND BA CK GR OUN D
r esistor -inductor-capacitor (RL C) network may be modeled as differ ential-algebraic equation (DAE)
with integral term [78], given b y
A 1 ˙
x ( t ) + A 2 x ( t ) + A 3 ∫︂ t
0 x ( θ )d θ = B u ( t ), y ( t ) = B T x ( t ). (6.9)
The transfer function associated with (6.9) is giv en by
H ( s ) = B T (︃ s A 1 + A 2 + 1
s A 3 )︃ − 1
B
and we expect better appro ximation properties of the rea lization b y preserving this for m. Further
examples for system structures ar e listed in T able 6.1. Instead of formulating and solving a r ealization
problem for each of these system classes , we ar e interested in a g eneral scheme that is able to
construct a realization for a given system structur e.
R emark 6.8.
I n ter ms of MOR, which aims at producing a computationally inexpensive surr ogate
model of a given dynamical system, the pr eser vation of structure in the r educed or der model (ROM)
often allo ws one to derive a R OM with a smaller state-space dimension
n x
, while maintaining
comparable or at times even better acc uracy than what unstructur ed reduced models produce , see
Section 5 in [18]. Additionally , since the internal structure of models often r eflects core p henomeno-
logical properties, structured models may behave in ways that r emain qualitatively consistent
with the phenomena that ar e being modeled – possibly more so than unstructur ed models hav-
ing higher objective fidelity . In contr ast to the structured r ealization problem (cf. Problem 6.6),
most structure-pr eser ving MOR techniques ar e developed in a projection-based context, thus
assuming access to internal dynamics in the form of differential equations . F or details we r efer
to [18, 57, 58, 78, 133, 155, 199]. N otable exceptions ar e pro vided in [71, 184, 189, 191]. ♣
Although the term system structure can have wide-ranging meanings , for our purposes we will
understand the term to refer to equiv alence classes of systems having realizations associated with a
linearly independent function family { h 1 , h 2 , . . . , h K } that appear as
H ( s ) = C (︄ K
∑︂
k = 1
h k ( s ) A k )︄ − 1
B , (6.10)
wher e
C ∈ R n y × n x
,
A k ∈ R n x × n x
for
k =
1
, . . . , K
,
B ∈ R n x × n u
. W e assume in all that follo ws that the
functions
h k : C → C
ar e meromorphic. F or any given function family , we will r efer to associated
matrix-valued functions having the form
∑︁ K
k = 1 h k
(
s
)
A k
as an affine structur e . B y standar d abuse
of notation, we use
H
(
s
) to denote either the system itself or the tr ansfer function of the system
evaluated at the point
s ∈ C
. The two systems
H
(
s
) and
˜︁
H
(
s
) ar e call ed structur ally equivalent if
H ( s ), ˜︁
H ( s ) ∈ C n y × n u for s ∈ C and if they each have the form
H ( s ) = C (︄ K
∑︂
k = 1
h k ( s ) A k )︄ − 1
B and ˜︁
H ( s ) = ˜︁
C (︄ K
∑︂
k = 1
˜
h k ( s ) ˜︁
A k )︄ − 1
˜︁
B ,
with
span{ h 1 , h 2 , . . . , h K } ≡ span{ ˜
h 1 , ˜
h 2 , . . . , ˜
h K }
. In particular , we allo w different state-space dimen-
sions, i.e ., for
˜︁
C ∈ R n y × n
,
˜︁
A k ∈ R n × n
, and
˜︁
B ∈ R n × n u
the integers
n x
and
n
need not be the same . Given

6.2. THE LOEWNER FRAME WORK 109
an original (full order) system associated with
H
(
s
), we aim to construct a structurally equivalent
system
˜︁
H
(
s
) that interpolates the original system at the driving frequencies , yi elding to the follo wing
generalization of P roblem 6.6.
Pr oblem 6.9
(S tr uctur ed realization problem)
.
Giv en the data in
(6.5)
and a system structur e
associated with the linearly independent function family
{ h 1 , . . . , h K }
, find matrices
˜︁
A k ∈ C n x × n x
,
k = 1, . . . , K , ˜︁
B ∈ C n x × n u , and ˜︁
C ∈ C n y × n x , such that the tr ansfe r function
˜︁
H ( s ) = ˜︁
C (︄ K
∑︂
k = 1
h k ( s ) ˜︁
A k )︄ − 1
˜︁
B (6.11)
satisfies the interpolation conditions
ℓ T
i ˜︁
H ( µ i ) = f T
i and ˜︁
H ( σ i ) r i = g i for i = 1, . . . , n . (6.12a)
If µ i = σ i for any index i , then addition ally ,
ℓ T
i ˜︁
H ′ ( µ i ) r i = θ i (6.12b)
is to be satisfied.
R emark 6.10.
Comparing Problem 6.6 and Problem 6.9, we observe that the coefficient functions
h k
may depend on possibly unkno wn parameters like the time delay
τ
, which also need to be identified.
This may be done b y fitting a realization obtained as solution of Pr oblem 6.9 via least-squares
optimization to additional data (see the forthcoming section 8.3). ♣
6.2 The Loewner framework
A special case of Problem 6.6 and Problem 6.9 is the gener alized state-space realization pr oblem,
which can be obtained b y setting τ = 0 in Problem 6.6, i.e . , b y considering the dynamical system
E ˙
x ( t ) = A 1 x ( t ) + B u ( t ), y ( t ) = C x ( t ) (6.13)
with associated transfer function H ( s ) = C ( s E − A 1 ) − 1 B . This problem is successfully solved b y the
Loewner r ealiz ation framewor k introduced in [150], which uses a Loewner matrix
L ∈ C n × n
and a
shifted Loewner matrix L σ ∈ C n × n , whose entrie s [ L ] i , j and [︁ L σ ]︁ i , j for i , j = 1, . . . , n ar e defined as
[ L ] i , j : = f T
i r j − ℓ T
i g j
µ i − σ j
, and [︁ L σ ]︁ i , j : = µ i f T
i r j − σ j ℓ T
i g j
µ i − σ j
, if µ i = σ j , (6.14a)
[ L ] i , i : = θ i , and [︁ L σ ]︁ i , i : = f T
i r i + µ i θ i , if µ i = σ i . (6.14b)

110 CHAPTER 6. PR OBLEM SET TING AND BA CK GR OUN D
F or SISO systems the definition in (6.14) r educes to
[ L ] i , j = H ( µ i ) − H ( σ j )
µ i − σ j
, and [︁ L σ ]︁ i , j = µ i H ( µ i ) − σ j H ( σ j )
µ i − σ j
, if µ i = σ j ,
[ L ] i , i = H ′ ( µ i ), and [︁ L σ ]︁ i , i = H ( µ i ) + µ i H ′ ( µ i ), if µ i = σ i ,
i.e .,
L
and
L σ
ar e the divided differences matrices corr esponding to the transfer functions
H
(
s
) and
s H ( s ), r espectively .
R emark 6.11. The Loewner matrices satisfy the S ylvester equations
M L − L S = L T G − F T R and M L − L S = M L T G − F T R S ,
with data matrices as defined in (6.6). F or further details we refer t o [150]. ♣
Theor em 6.12
(Loewner r ealiz ation [150])
.
Let the matrices
L
and
L σ
be defined as in
(6.14)
and
assume that det( ˜
s L − L σ ) = 0 for all ˜
s ∈ { µ i } n
i = 1 ∪ { σ i } n
i = 1 . Then the system
− L ˙
˜︁
x ( t ) = − L σ ˜︁
x ( t ) + F T u ( t ), ˜︁
y ( t ) = G ˜︁
x ( t ) (6.15)
with
F , G
as defined in
(6.6)
is a minimal r ealization of an interpolant of the data, i.e., its transfer
function
˜︁
H ( s ) = G ( L σ − s L ) − 1 F T
satisfies the interpolation conditions (6.12) .
Thus in view of Problem 6.9, the Loewner r ealization corresponds to the specific setting
K = 2, h 1 ( s ) ≡ 1, and h 2 ( s ) = − s .
The condition
det
(
˜
s L − L σ
)
=
0 in Theor em 6.12 can be relaxed b y means of the truncated singular
value decomposition (SVD) [6, R emark 3.2.1].
Theor em 6.13 ( [150 , Theor em 5.1]) . Suppose that
rank (︁ ˜
s L − L σ )︁ = rank [︂ L L σ ]︂ = rank [︄ L
L σ ]︄ = : r for all ˜
s ∈ { µ i } ∪ { σ i }. (6.16)
Then a minimal r ealization of an interpolant of t he data is given b y the system
− Y ∗ L X ˙
˜︁
x ( t ) = − Y ∗ L σ X ˜︁
x ( t ) + Y ∗ F T u ( t ),
˜︁
y ( t ) = G X ˜︁
x ( t ), (6.17)
wher e
Y ∈ C n × r
and
X ∈ C n × r
ar e computed from the short SVD
˜
s L − L σ = Y Σ X ∗
for some
˜
s ∈ { µ i } ∪ { σ i }
, wher e
Y ∈ C n × r
and
X ∈ C n × r
have orthonormal columns and
Σ ∈ R r × r
is diagonal
with positive elements.

6.2. THE LOEWNER FRAME WORK 111
R emark 6.14.
I t should be noted that the pencil
s L − L σ
can be singular (i.e .,
det
(
s L − L σ
)
≡
0) while
at the same time the matrices [︂ L L σ ]︂ and [︂ L ∗ L ∗
σ ]︂ have full ro w rank. I n this case, the condition
(6.16) is not satisfied and thus Theor em 6.13 does not apply . Indeed, the matrices
L = ⎡
⎢
⎣
0 1 0
0 0 0
0 0 1 ⎤
⎥
⎦ and L σ = ⎡
⎢
⎣
1 0 0
0 0 1
0 0 0 ⎤
⎥
⎦
have no common (left or right) nullspace . But ev e n if the rank condition
(6.16)
is satisfies and thus
Theor em 6.13 can be applied, the pencil in
(6.17)
may have a high index (cf. Definition 2.12), may be
close to a pencil with high index or may even be close to a singular pencil. In this case , a further
r egular ization is r e quir ed to prevent numerical issues . W e refer to [34] for further details . ♣
The Loewner r ealization framework is an effective and br oadly applicable approach for constructing
rational appr o ximants directly from interpolation data; i t has been extended to paramet ric sys-
tems [10, 115], to realization independent methods for optimal
H 2
appro ximation [19], to bilinear
systems [9], and to switched systems [87]. H o wever , the Loewner framework is only capable of
producing r ational approx imants and, so in particular , it cannot capture the tr anscendental char-
acter of transfer functions for dyna mical systems containing time delays or distributed parameter
subsystems that model convection or tr anspor t (cf. [63]).

112 CHAPTER 6. PR OBLEM SET TING AND BA CK GR OUN D

7
Structured interp olato ry realizations
I n this chapter we pro vide a solution for Problem 6.6 and the mor e general Problem 6.9.
7.1 Interpolation conditions
S uppose we are g iven interpolation data as in
(6.5)
and for the moment assume that we alr eady have
a r ealization of the for m
˜︁
H
(
s
)
= ˜︁
C ˜︂
K
(
s
)
− 1 ˜︁
B
. I f we can impose conditions on
˜︁
C , ˜︁
B
and the matrix
function
˜︂
K
such that
˜︁
H
(
s
)
= ˜︁
C ˜︂
K
(
s
)
− 1 ˜︁
B
satisfies the interpolation conditions
(6.12)
, then we can
r ever t the process and use the condit ions to construct the realization. The follo wing obser vation,
which corr esponds to an equivalent parametrization of the interpolation conditions
(6.12)
, suggests
ho w one might proceed.
Theor em 7.1.
Let
˜︂
K : C → C n × n
be a continuously differ entiable matrix-valued function, which
is nonsingular at
s = µ i
and
s = σ j
for
i , j =
1
, . . . , n
. The realization
˜︁
H
(
s
)
= ˜︁
C ˜︂
K
(
s
)
− 1 ˜︁
B
satisfies
the interpolation conditions (6.12a) if and onl y if
G = ˜︁
C P G and F T = P T
F ˜︁
B , (7.1)
wher e
P G , P F ∈ C n × n
ar e two matrices, whose columns
p i
G : = P G e i
and
p i
F : = P F e i
, r espectively ,
solve the linear systems
˜︂
K ( σ i ) p i
G = ˜︁
B r i and ˜︂
K ( µ i ) T p i
F = ˜︁
C T ℓ i , (7.2)
wher e
e i
is the
i
th column of the
n × n
identity matrix. M or eo ver , if
µ i = σ i
, then
˜︁
H
satisfies in
addition the bitangential interpolation condition (6.12b) , pro vided that
(︂ p i
F )︂ T ˜︂
K ′ ( µ i ) p i
G = − θ i , (7.3)
wher e ˜︂
K ′ denotes the derivative of ˜︂
K .
113

114 CHAPTER 7. STR UCTURED INTERPOL A T OR Y REALIZ A TIONS
Proof.
The transfer function
˜︁
H
(
s
)
= ˜︁
C ˜︂
K
(
s
)
− 1 ˜︁
B
is well-defined at
s = µ i
and
s = σ i
. Assume first
that
(7.1)
and
(7.2)
ar e satisfied. M ultiplying the first equation in
(7.1)
b y
e i
yields
g i = ˜︁
C p i
G
. Then,
using the first equation in
(7.2)
and the fact that
˜︂
K
(
σ i
) is nonsingular , one immediately obtains
g i = ˜︁
H
(
σ i
)
r i
, i.e ., the r ight tangential interpolation holds . Similarly , using the second expr ession in
(7.1)
and the definition of
p i
F
in
(7.2)
, we arrive at
f T
i = ℓ T
i ˜︁
H
(
µ i
); thus
(6.12a)
holds . F or the other
dir ection, we obser ve that if
p i
F
and
p i
G
ar e the unique sol utions of
(7.2)
, then the interpolation
conditions immediately imply (7.1). M or eo ver , if µ i = σ i , then (7.3) yields
ℓ T
i ˜︁
H ′ ( µ i ) r i = − ℓ T
i ˜︁
C ˜︂
K ( µ i ) − 1 ˜︂
K ′ ( µ i ) ˜︂
K ( σ i ) − 1 ˜︁
B r i = − (︂ p i
F )︂ T ˜︂
K ′ ( µ i ) p i
G = θ i . ■
E vidently , in order to satisfy the collected tangent int erpolation conditions
(6.12a)
, we can no w
equivalently r equire the r ealization
˜︁
H
(
s
) to satisfy the conditions of Theor em 7.1. In particular we
need
˜︂
K
(
s
) to be nonsingular at the driving fr equencies
s = µ i
and
s = σ j
. F or
˜︂
K
(
s
)
= ∑︁ K
k = 1 h k
(
s
)
˜︁
A k
,
the other conditions (7.1) and (7.2) can be r ewr itten as
G = ˜︁
C P G , F T = P T
F ˜︁
B , (7.4)
K
∑︂
k = 1 ˜︁
A k P G h k ( S ) = ˜︁
B R ,
K
∑︂
k = 1
h k ( M ) P T
F ˜︁
A k = L T ˜︁
C , (7.5)
wher e we set
h k
(
M
)
: = diag
(
h k
(
µ 1
)
, . . . , h k
(
µ n
)) and
h k
(
S
)
: = diag
(
h k
(
σ 1
)
, . . . , h k
(
σ n
)). T o fulfill
additionally the bitangential interpolation conditions
(6.12b)
for the case
µ i = σ i
, the third condition
of Theor em 7.1 needs to be satisfied.
If the matrices P F and P G ar e nonsingular , then
˜︁
H ( s ) = ˜︁
C ˜︂
K ( s ) − 1 ˜︁
B = G (︂ P T
F ˜︂
K ( s ) P G )︂ − 1 F T ,
and hence the r ealization is unique up to the basis transformation descr ibed b y
P F
and
P G
. I n
this case , the matrices
˜︁
B
and
˜︁
C
ar e given directly b y the data without further computations and
the matrices
P F
and
P G
captur e the non-uniqueness of the realization. In S ection 7.3 we will use
these matrices to tailor the r eal ization to interpolate additional data. In any case , we view equations
(7.4)
and
(7.5)
not as a coupled system but as a stagger ed process. First, fix matr ices
P F , P G
and
determine
˜︁
B
and
˜︁
C
from
(7.4)
. In a second step , use this information to solve
(7.5)
. With this
viewpoint, i.e . , not counting
P F
and
P G
as unkno wns, we hav e
K n 2
unkno wns from the coefficient
matrices
˜︁
A k
and (
n u + n y
)
n
unkno wns from the input and output matrices
˜︁
B
and
˜︁
C
, giving a total
of
K n 2 +
(
n u + n y
)
n
unkno wns. F or these unknowns ,
(7.4)
and
(7.5)
constitute 2
n 2 +
(
n u + n y
)
n
equations, leaving ( K − 2) n 2 degr ees of freedom. In p articular , we can expect a unique solution for
K = 2.
R emark 7.2.
Ther e are (
K −
2)
n 2
degr ees of freedom to solve the structur ed realization problem, and
ther efore the
K =
1 case does not have enough degr ees of freedom to guar antee a solution in general.
T o further examine the case
K =
1, assume for simplicity that
˜︁
H
is a single-input/single-output (SISO)
system, i.e .,
˜︁
B = ˜︁
b ∈ R n
and
˜︁
C T = ˜︁
c ∈ R n
. Then the reduced model has the form
H
(
s
)
= 1
h 1 ( s ) ˜︁
c T ˜︁
A − 1 ˜︁
b
.
Ther efore , the interpolation conditions yield
˜︁
c T ˜︁
A − 1 ˜︁
b = H ( σ i ) h 1 ( σ i ) and ˜︁
c T ˜︁
A − 1 ˜︁
b = H ( µ i ) h 1 ( µ i ), for i = 1, . . . , n . (7.6)

7.1. INTERPOL A TION CONDITIONS 115
S ince ˜︁
c T ˜︁
A − 1 ˜︁
b is constant, for the interpolation problem in (7.6) to have a solution, w e need
H ( σ i ) h 1 ( σ i ) = H ( µ i ) h 1 ( µ i ) = c ,
wher e c is a constant for
i =
1
, . . . , n
. This clearly will not be the case in general and w e cannot expect
to have a solution. Inter estingly , if this condition holds, a solution can be found easily b y setting
˜︁
A =
1,
˜︁
b =
1 and
˜︁
c =
c . Based on these consider ations, we will focus on
K ≥
2 in the r emainder of the
thesis . ♣
R emark 7.3.
The nonsingularity of the matrices
P F
and
P G
is connected to the controllability and
obser vability of the r ealization. A SISO system in standar d state-space for m, i.e.,
˜︂
K
(
s
)
= s I n − ˜︁
A
,
˜︁
B = ˜︁
b ∈ R n
, and
˜︁
C = ˜︁
c ∈ R 1 × n
is called controllable , if
n = rank (︂[︂ ˜︁
b ˜︁
A ˜︁
b · · · ˜︁
A n − 1 ˜︁
b ]︂)︂
. I t i s called
obser vable , if
n = rank (︂[︂ ˜︁
c T ˜︁
A T ˜︁
c T · · · ( ˜︁
A T ) n − 1 ˜︁
c T ]︂)︂
. T o establish the connection between the
matrices
P F
and
P G
to these concepts, w e obser ve that for SISO systems in standar d state-space
form ˜︂
K and its pointwise inverse form a set of commutative matrices . H ence we have
rank (︁ P G )︁ = rank (︂[︂ ˜︂
K ( σ 1 ) − 1 ˜︁
b · · · ˜︂
K ( σ n ) − 1 ˜︁
b ]︂)︂
= rank (︂[︂ ˜︂
K ( σ 1 ) − 1 ˜︁
b ˜︂
K ( σ 1 ) − 1 ˜︂
K ( σ 2 ) − 1 ˜︁
b · · · (︂ ∏︁ n
i = 1 ˜︂
K ( σ i ) − 1 )︂ ˜︁
b ]︂)︂
= rank (︂[︂ ˜︁
b ˜︁
A ˜︁
b · · · ˜︁
A n − 1 ˜︁
b ]︂)︂
such that
P G
is nonsingular if and only if the r ealization is controllable. Similarly ,
P F
is nonsingular
if and only if the r ealization is obser vable. ♣
N ote that the Loewner pencil with
h 1
(
s
)
≡
1 and
h 2
(
s
)
= − s
(cf. section 6.2) satisfies the conditions of
Theor em 7.1 with
˜︂
K
(
s
)
= L σ − s L
, i.e ., the Loewner framework is a special case of Theor em 7.1 with
matrices
P F
and
P G
set to the identity . Indeed, for
µ i = σ j
, the (
i , j
) component of the Loewner
pencil is
e T
i ˜︂
K ( s ) e j = [︁ L σ ]︁ i , j − s [ L ] i , j = (︄ µ i − s
µ i − σ j )︄ f T
i r j + (︄ s − σ j
µ i − σ j )︄ ℓ T
i g j ,
so it immediately follo ws
e T
i ˜︂
K
(
µ i
)
= ℓ T
i G = ℓ T
i ˜︁
C
and
˜︂
K
(
σ j
)
e j = F T r j = ˜︁
B r j
. S i milarly , for the case
µ i = σ i , we obtain
e T
i ˜︂
K ( µ i ) = ℓ T
i ˜︁
C , ˜︂
K ( σ i ) e i = ˜︁
B r i , and e T
i ˜︂
K ′ ( µ i ) e i = − L = − θ i .
F ollo wing the discussion abo ve Remar k 7.2 , we can expect a unique solution of
(7.5)
only for the
special case
K =
2, which we discuss in detail in Section 7.2 and sho w its close relation to the Loewner
framework. I f
K ≥
3, we need a str ategy to deal with the remaining degr ees of freedom. T o this end
we propose two appr oaches, which both pro vide interpolation of fur ther data while maintaining the
dimension of the matrices in the r ealization. The first approach uses additional interpolation points
(Section 7.3.1), while the second one interpolates additional derivative ev aluations of the transfer
functions (Section 7.3.2).

116 CHAPTER 7. STR UCTURED INTERPOL A T OR Y REALIZ A TIONS
7.2 S tr uctur ed Loewner realiza tions for the case K = 2
Setting
P F = P G = I n
gives 2
n 2 +
(
n u + n y
)
n
equations in
(7.4)
and
(7.5)
for the
K n 2 +
(
n u + n y
)
n
unkno wns such that we can expect (under some r egularity) a u nique solution for the case
K =
2. I n
this case ˜︁
B = F T , ˜︁
C = G , and the matrix equations in (7.5) reduce to
h 1 ( M ) ˜︁
A 1 + h 2 ( M ) ˜︁
A 2 = L T G and ˜︁
A 1 h 1 ( S ) + ˜︁
A 2 h 2 ( S ) = F T R .
T o decouple these equations, we multiply the first equation fr om the r ight b y
h 2
(
S
) and the second
equation from the left b y
h 2
(
M
). S ubtracting the r esulting systems yields the Sylv ester-like equation
h 2 ( M ) ˜︁
A 1 h 1 ( S ) − h 1 ( M ) ˜︁
A 1 h 2 ( S ) = h 2 ( M ) F T R − L T G h 2 ( S ). (7.7)
S imilarly , we can eliminate ˜︁
A 1 and obtain
h 1 ( M ) ˜︁
A 2 h 2 ( S ) − h 2 ( M ) ˜︁
A 2 h 1 ( S ) = h 1 ( M ) F T R − L T G h 1 ( S ). (7.8)
R emark 7.4.
If the desir ed model is a generalized state space system as in
(6.13)
, i.e .,
h 1
(
s
)
= s
and
h 2 ( s ) ≡ − 1, then (7.7) and (7.8) are giv en by the S y lvester equations
˜︁
A 1 S − M ˜︁
A 1 = F T R − L T G and ˜︁
A 2 S − M ˜︁
A 2 = M F T R − L T G S , (7.9)
r espectively . U p to a sign factor , these ar e the Sylvester equations that defin e the Loewner matrix
and the shifted Loewner matrix, see 6.11. In particular , if
σ i = µ j
for
i , j =
1
, . . . , n
, then
˜︁
A 1 = − L
and
˜︁
A 2 = − L σ
ar e the unique sol utions of
(7.7)
and
(7.8)
and the Loewner framewor k is a special case of
the general fr amework presented in t his paper . Similarly , the proportional ansatz for the r ealization
of delay systems introduced in [189] is co vered b y our framework. ♣
Those elements of
˜︁
A 1
and
˜︁
A 2
, for which
µ i = σ j
, may be obtained by multiplying
(7.7)
and
(7.8)
from left b y e T
i and from right b y e j yielding
[︁ ˜︁
A 1 ]︁ i , j = h 2 ( µ i ) f T
i r j − ℓ T
i g j h 2 ( σ j )
h 2 ( µ i ) h 1 ( σ j ) − h 1 ( µ i ) h 2 ( σ j ) , [︁ ˜︁
A 2 ]︁ i , j = h 1 ( µ i ) f T
i r j − ℓ T
i g j h 1 ( σ j )
h 1 ( µ i ) h 2 ( σ j ) − h 2 ( µ i ) h 1 ( σ j ) (7.10)
under the generic assumption that
h 1
(
µ i
)
h 2
(
σ j
)
= h 2
(
µ i
)
h 1
(
σ j
). This is satisfied for all possible
choices of interpolation points with
µ i = σ j
if the functions
h 1
and
h 2
satisfy the Haar condition [60],
see also the forthcoming Section 7.3.1. The components for which
µ i = σ i
can be obtained b y
translating the conditions in Theor em 7.1 to the K = 2 case . This yields
h 1 ( µ i )[ ˜︁
A 1 ] i , i + h 2 ( µ i )[ ˜︁
A 2 ] i , i = ℓ T
i g i and h ′
1 ( µ i )[ ˜︁
A 1 ] i , i + h ′
2 ( µ i )[ ˜︁
A 2 ] i , i = − θ i (7.11)
and consequently
[︁ ˜︁
A 1 ]︁ i , i = h 2 ( µ i ) θ i + h ′
2 ( µ i ) ℓ T
i g i
h ′
2 ( µ i ) h 1 ( µ i ) − h ′
1 ( µ i ) h 2 ( µ i ) , [︁ ˜︁
A 2 ]︁ i , i = h 1 ( µ i ) θ i + h ′
1 ( µ i ) ℓ T
i g i
h ′
1 ( µ i ) h 2 ( µ i ) − h ′
2 ( µ i ) h 1 ( µ i ) , (7.12)
for the components with
µ i = σ i
under the generic assumption
h ′
2
(
µ i
)
h 1
(
µ i
)
= h ′
1
(
µ i
)
h 2
(
µ i
). Conse-
quently , we have sho wn the subsequent result.

7.2. STR UCTURED LOEWNER REALIZA TIONS FOR THE CASE K = 2 1 17
Theor em 7.5.
Let
˜︁
A 1
and
˜︁
A 2
be as in
(7.10)
and
(7.12)
wher e the denominators ar e assumed
nonzero . If
det (︁ h 1 ( ˜
s ) ˜︁
A 1 + h 2 ( ˜
s ) ˜︁
A 2 )︁ = 0 for all ˜
s ∈ { µ i } n
i = 1 ∪ { σ i } n
i = 1 , (7.13)
then the tr ansfer function
H
(
s
)
= G (︁ h 1 ( s ) ˜︁
A 1 + h 2 ( s ) ˜︁
A 2 )︁ − 1 F T
satisfies the interpolation conditions
(6.12) .
R emark 7.6. F or the analysis of assumption (7.13) in Theor em 7.5, we obser ve that the function
η : C → C , s ↦→ det (︁ h 1 ( s ) ˜︁
A 1 + h 2 ( s ) ˜︁
A 2 )︁
is meromorphic, since b y definition
h 1
and
h 2
ar e meromorphic. The identity theor em for holomor-
phic functions implies that either
η ≡
0 and hence that
h 1
(
s
)
˜︁
A 1 + h 2
(
s
)
˜︁
A 2
is singular for every
s ∈ C
,
or that set of zeros of
η
has no accumulation point and consequently , is a set of measur e zero , i.e .,
the transfer function H in Theor em 7.5 is defined for almost all s ∈ C . ♣
The matrices
˜︁
A 1
and
˜︁
A 2
have a structur e similar to the Loewner matrix and the shifted Loewner
matrix. This gives rise to the idea that the result of Theor em 7.5 can be obtained from the standar d
Loewner framework using tr ansformed data.
Cor ollar y 7.7.
S uppose t hat
h 2
(
S
) and
h 2
(
M
) ar e nonsingular and that the denominators in
(7.10)
and
(7.12)
ar e nonzero . Construct the Loewner matrix
L
and the shifted Loewner matrix
L σ
for the tr ansformed data
left interpolation data: {︃(︃ h 1 (︁ µ i )︁
h 2 (︁ µ i )︁ , ℓ i
h 2 (︁ µ i )︁ , f i )︃ for i = 1, . . . , n }︃ ,
right interpolation data: {︃(︃ h 1 (︁ σ i )︁
h 2 (︁ σ i )︁ , r i
h 2 (︁ σ i )︁ , g i )︃ for i = 1, . . . , n }︃ ,
bitangential derivative data: {︄(︄ i , h 2 (︁ µ i )︁ θ i + h ′
2 (︁ µ i )︁ ℓ T
i g i
h ′
1 (︁ µ i )︁ h 2 (︁ µ i )︁ − h ′
2 (︁ µ i )︁ h 1 (︁ µ i )︁ )︄ for which µ i = σ i }︄
(7.14)
If det( h 2 ( ˜
s ) L σ − h 1 ( ˜
s ) L ) = 0 for all ˜
s ∈ { µ i } n
i = 1 ∪ { σ i } n
i = 1 , then the tr ansfe r function
H ( s ) = G ( h 2 ( s ) L σ − h 1 ( s ) L ) − 1 F T
interpolates the data.
Proof.
S imple calculations yield that, when constr ucting the Loewner pencil with the transformed
interpolation data
(7.14)
, the Loewner matrix and the shifted Loewner matr ix coincide with
− ˜︁
A 1
and ˜︁
A 2 given in (7.10) and (7.12). Theorem 7.5 completes the proof. ■
Cor ollar y 7.7 allows one to t ransfer many r esults of the standar d Loewner framework to the general
framework c onsider ed in this subsection. In particular , this allo ws us to keep the system matri ces
r eal if the set of interpolation data is closed under complex conjugation. The details are formulated
in Lemma 7.8.

118 CHAPTER 7. STR UCTURED INTERPOL A T OR Y REALIZ A TIONS
Lemma 7.8.
Let the set of interpolation data be closed under complex conjugation, i.e., ther e exist
unitar y matrices T F , T G ∈ C n × n with
T ∗
F M T F ∈ R n × n , T ∗
F L T ∈ R n , T ∗
F F T ∈ R n ,
T ∗
G S T G ∈ R n × n , R T G ∈ R n , G T G ∈ R n .
M oreo ver , assume that the set of
θ i
’ s (for the case
µ i = σ i
) is closed under complex conjugation.
Then, the r ealization (
T ∗
F ˜︁
A 1 T G , T ∗
F ˜︁
A 2 T G , T ∗
F F T , G T G
) with (
˜︁
A 1 , ˜︁
A 2 , F T , G
) from Theor em 7.5
consists of r eal-valued matrices and interpolates the data.
Proof.
First we note that if th e set of interpolation data is closed under complex conjugation, so is
the set of transformed data in C orollary 7.7. Based on this obser vation, the proof for the case
µ i = σ j
for all
i , j =
1
, . . . , n
simply follo ws the lines of [11, section 2.4.4.]. This can also be comprehended
after multiplying the S ylvester-like equations
(7.7)
and
(7.8)
from left b y
T ∗
F
and from right b y
T G
.
S imilar reasoning pro ves the claim for the µ i = σ i case . ■
Example 7.9.
A special case of Lemma 7.8 applies when the interpolation data is sorted such that
the r eal values have the highest indices, i.e .,
M = diag( µ 1 , µ 1 , . . . , µ 2 ℓ − 1 , µ 2 ℓ − 1 , µ 2 ℓ + 1 , . . . , µ n ),
L = [︂ ℓ 1 ℓ 1 . . . ℓ 2 ℓ − 1 ℓ 2 ℓ − 1 ℓ 2 ℓ + 1 . . . ℓ n ]︂ ,
F = [︂ f 1 f 1 . . . f 2 ℓ − 1 f 2 ℓ − 1 f 2 ℓ + 1 . . . f n ]︂ ,
S = diag( σ 1 , σ 1 , . . . , σ 2 r − 1 , σ 2 r − 1 , σ 2 r + 1 , . . . , σ n ),
R = [︂ r 1 r 1 . . . r 2 r − 1 r 2 r − 1 r 2 r + 1 . . . r n ]︂ ,
G = [︂ g 1 g 1 . . . g 2 r − 1 g 2 r − 1 g 2 r + 1 . . . g n ]︂ .
I n this case possible choices for T F and T G are given b y block diagonal unitar y matr ices
T • = blkdiag (︄ 1
⎷ 2 [︄ 1 − ı
1 ı ]︄ , . . . , 1
⎷ 2 [︄ 1 − ı
1 ı ]︄ , 1, . . . , 1 )︄ ,
wher e
• ∈ { F , G }
. One can also obtain the r eal r ealization directly from Theor em 7.1 b y choosing
P T
F = T ∗
F and P G = T G (see discussion after Theor em 7.1). ♠
R emark 7.10.
The r esult from Corollary 7.7 can (formally) be obtained by r ewriting the transfer
function
H ( s ) = ˜︁
C ( h 1 ( s ) ˜︁
A 1 + h 2 ( s ) ˜︁
A 2 ) − 1 ˜︁
B = ˜︁
C (︃ h 1 ( s )
h 2 ( s ) ˜︁
A 1 + ˜︁
A 2 )︃ − 1
˜︁
B 1
h 2 ( s ) .
This corr esponds to a similar strategy as in [71]. ♣

7.3. STR UCTURED REALIZ A TION FOR THE CASE K ≥ 3 119
R emark 7.11.
S imilar as in the Loewner framework, we can paramet erize the realization with a
feedthrough term D . Assuming {︁ µ i }︁ n
i = 1 ∩ {︁ σ i }︁ n
i = 1 = ∅ , simple calculations lead to the realization
H ( s ) = ˜︁
C (︁ h 1 ( s ) ˜︁
A 1 + h 2 ( s ) ˜︁
A 2 )︁ − 1 ˜︁
B + D
with ˜︁
C = G − D R , ˜︁
B = F T − L T D ,
˜︁
C = G − D R , [︁ ˜︁
A 1 ]︁ i , j =
h 2 ( µ i ) (︂ f T
i − ℓ T
i D )︂ r j − ℓ T
i (︂ g j − D r j )︂ h 2 ( σ j )
h 2 ( µ i ) h 1 ( σ j ) − h 1 ( µ i ) h 2 ( σ j ) ,
˜︁
B = F T − L T D , [︁ ˜︁
A 2 ]︁ i , j =
h 1 ( µ i ) (︂ f T
i − ℓ T
i D )︂ r j − ℓ T
i (︂ g j − D r j )︂ h 1 ( σ j )
h 1 ( µ i ) h 2 ( σ j ) − h 2 ( µ i ) h 1 ( σ j ) ,
which interpolates the data
(6.5)
for all matrices
D ∈ R n y × n u
. F or the special case
h 1
(
s
)
= s
and
h 2
(
s
)
≡ −
1, we r eco ver the r esults from [150] given b y
˜︁
C = G − D R
,
˜︁
B = F T − L T D
,
˜︁
A 1 = − L
, and
˜︁
A 2 = − L σ − L T D R . ♣
7.3 S tr uctur ed realization for the c ase K ≥ 3
When
K ≥
3, the conditions in Theorem 7.1 do not pro vide enough conditions for the available
degr ees of freedom (even if
P F
and
P G
ar e fixed). H ence, we have some fr eedom in choosing
the matrices
˜︁
A k
with
k =
1
, . . . , n
. W e can exploit these degrees of fr eedom, for instance, b y fitting
the transfer function to additional data. F or simplicity we assume
{︁ µ i }︁ n
i = 1 ∩ {︁ σ i }︁ n
i = 1 = ∅
for the
r emainder of this section.
7.3.1 I nterpolation at additional points
I n this subsection we focus on fitting the transfer function to additional data or , equivalently ,
matching the given data with a smaller state space dimension. T o this end, we assume that we hav e
(
Q F −
1)
n
additional left interpolation points and (
Q G −
1)
n
additional right interpolation points at
hand, which we group in sets of
n
. M ore pr ecisely , the left interpolation data is grouped into the
matrices
M q : = diag( µ q ;1 , µ q ;2 , . . . , µ q ; n ) ∈ C n × n , L q : = [︂ ℓ q ;1 ℓ q ;2 · · · ℓ q ; n ]︂ ∈ C n y × n ,
F q : = [︂ f q ;1 f q ;2 · · · f q ; n ]︂ ∈ C n u × n , (7.15a)
wher e
q =
1
, . . . , Q F
. H ere , we set
µ 1; i : = µ i
,
f 1; i : = f i
, and
ℓ 1; i : = ℓ i
, such that we have
M 1 = M
,
L 1 = L , and F 1 = F . Similarly , we introduce for q = 1, . . . , Q G the matrices
S q : = diag( σ q ;1 , σ q ;2 , . . . , σ q ; n ) ∈ C n × n , R q : = [︂ r q ;1 r q ;2 · · · r q ; n ]︂ ∈ C n u × n ,
G q : = [︂ g q ;1 g q ;2 · · · g q ; n ]︂ ∈ C n y × n . (7.15b)
T o use the full capacity of the available degr ees of freedom, we assume
K = Q F + Q G
, with
Q F , Q G ≥
1.
The next r esult, which is a generalization of the
K =
2 case , gives us the necessar y and sufficient
conditions that the matrices in the realization
H
(
s
) must satisfy to interpolate all pr escr ibed infor -
mation.

120 CHAPTER 7. STR UCTURED INTERPOL A T OR Y REALIZ A TIONS
Theor em 7.12.
Let
H
(
s
)
= ˜︁
C ˜︂
K
(
s
)
− 1 ˜︁
B
with
˜︂
K
(
s
)
= ∑︁ K
k = 1 h k
(
s
)
˜︁
A k
and suppose that
˜︂
K
(
s
) is non-
singular for all ˜
s ∈ { µ q ; i } Q F
q = 1 ∪ { σ q ; i } Q G
q = 1 for all i = 1, . . . , n .
(i)
The left interpolation conditions
ℓ T
q ; i ˜︁
H
(
µ q ; i
)
= f T
q ; i
ar e satisfied for all
i =
1
, . . . , n
and all
q = 1, . . . , Q F if and onl y if ther e exist matrices P F , q with q = 1, . . . , Q F that satisfy
F T
q = P T
F , q ˜︁
B and
K
∑︂
k = 1
h k ( M q ) P T
F , q ˜︁
A k = L T
q ˜︁
C . (7.16)
(ii)
The right interpolation conditions
˜︁
H
(
σ q ; i
)
r q ; i = g q ; i
ar e satisfied for all
i =
1
, . . . , n
and all
q = 1, . . . , Q G if and onl y if ther e exist matrices P G , q with q = 1, . . . , Q G that satisfy
G q = ˜︁
C P G , q and
K
∑︂
k = 1 ˜︁
A k P G , q h k ( S q ) = ˜︁
B R q . (7.17)
Proof.
The r esult follo ws dir ectly from Theorem 7.1. F or the sake of completeness we give the proof
of the first statement again. The second identity in
(7.16)
implies
ℓ T
q ; i ˜︁
C = e T
i P T
F , q ∑︁ K
k = 1 h k
(
µ q ; i
)
˜︁
A k
.
Thus, b y the first identity and the definition of ˜︁
H we conclude
ℓ T
q ; i ˜︁
H ( µ q ; i ) = e T
i P T
F , q ˜︁
B = f T
q ; i
for i = 1, . . . , n and q = 1, . . . , Q F . ■
E vidently , in order to satisfy the interpolation conditions
(6.12a)
it will be sufficient to r equire
that
(7.16)
and
(7.17)
hold simultaneously . This gives us the follo wing strategy to determine the
r ealization matr ices
˜︁
A k , ˜︁
B
, and
˜︁
C
. S uppose we can find matrices
P F , q
and
P G , q
that satisfy the first
identity in
(7.16)
and
(7.17)
, r espectively , i.e., that allo w us to fix
˜︁
B
and
˜︁
C
. Then we can compute the
matrices ˜︁
A k as follo ws. V ectorization [104] of the second identity in (7.16) yields
K
∑︂
k = 1 (︂ I n ⊗ h k ( M q ) P T
F , q )︂ vec( ˜︁
A k ) = (︂ ˜︁
C T ⊗ I n )︂ vec( L T
q ),
wher e
⊗
denotes the Kronecker product and
vec
(
X
) denotes the vector of stacked columns of the
matrix X . Similarly , equation (7.17) implies
K
∑︂
k = 1 (︂ h k ( S q ) P T
G , q ⊗ I n )︂ vec( ˜︁
A k ) = (︁ I n ⊗ ˜︁
B )︁ vec( R q ).
All equations together yield the linear algebraic system
A α = β
with
A ∈ C K n 2 × K n 2
,
α , β ∈ C K n 2
given

7.3. STR UCTURED REALIZ A TION FOR THE CASE K ≥ 3 121
b y
A : =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
I n ⊗ h 1 (︁ M 1 )︁ P T
F ,1 · · · I n ⊗ h K (︁ M 1 )︁ P T
F ,1
.
.
. .
.
.
I n ⊗ h 1 (︂ M Q F )︂ P T
F , Q F · · · I n ⊗ h K (︂ M Q F )︂ P T
F , Q F
h 1 (︁ S 1 )︁ P T
G ,1 ⊗ I n · · · h K (︁ S 1 )︁ P T
G ,1 ⊗ I n
.
.
. .
.
.
h 1 (︂ S Q G )︂ P T
G , Q G ⊗ I n · · · h K (︂ S Q G )︂ P T
G , Q G ⊗ I n
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
,
α : = ⎡
⎢
⎢
⎣
vec( ˜︁
A 1 )
.
.
.
vec( ˜︁
A K )
⎤
⎥
⎥
⎦ , and β : =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
(︂ ˜︁
C T ⊗ I n )︂ vec (︂ L T
1 )︂
.
.
.
(︂ ˜︁
C T ⊗ I n )︂ vec (︂ L T
Q F )︂
(︁ I n ⊗ ˜︁
B )︁ vec (︁ R 1 )︁
.
.
.
(︁ I n ⊗ ˜︁
B )︁ vec (︂ R Q G )︂
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
.
(7.18)
N ote that the solution of the linear equation system
A α = β
depends on
P F , q
and
P G , q
and ther e is
some fr eedom in choosing these matr ices . A simple possibility is given b y
P T
F , q : = [︂ F T
q ⋆ ]︂ , P G , q : = [︄ G q
⋆ ]︄ , ˜︁
B : = [︄ I n u
0 ]︄ , and ˜︁
C : = [︂ I n y 0 ]︂ , (7.19)
which satisfies the first identity in
(7.16)
and
(7.17)
for any choice of
⋆
. Ho wever , the trivial choice
of setting these blocks to zer o makes the system matrix
A
singular , see also Remar k 7.3. In stead, we
propose to fill the
⋆
part of the matrices
P F , q
and
P G , q
such that
P F , q
and
P G , q
ar e nonsingular
assuming that
F q
and
G q
have full ro w rank. A more specific choice of
⋆
may even lead to r eal-valued
r ealizations as stated in the follo wi ng lemma.
Lemma 7.13.
Let each of the interpolation data sets be closed under complex conjugation, i.e.,
ther e exist unitar y matrices T F , q , T G , q ∈ C n × n with
T ∗
F , q M q T F , q ∈ R n × n , T ∗
F , q L T
q ∈ R n , T ∗
F , q F T
q ∈ R n , for q = 1, . . . , Q F ,
T ∗
G , q S q T G , q ∈ R n × n , R q T G , q ∈ R n , G q T G , q ∈ R n , for q = 1, . . . , Q G .
M oreo ver , let the matrices
P F , q
and
P G , q
be as in
(7.19)
with fr ee entries
⋆
chosen such that
T ∗
F , q P T
F , q ∈ R n
and
P G , q T G , q ∈ R n
hold. Then, the matrices
˜︁
A 1 , . . . , ˜︁
A K
,
˜︁
B
, and
˜︁
C
from Theo-
r em 7.12 ar e real matrices (if they e xist).
Proof.
The matrices
˜︁
B
and
˜︁
C
from
(7.20)
ar e real-valued. In addition, the second equalities in
(7.16)

122 CHAPTER 7. STR UCTURED INTERPOL A T OR Y REALIZ A TIONS
and (7.17) ar e equivalent to
K
∑︂
k = 1
T ∗
F , q h k ( M q ) T F , q T ∗
F , q P T
F , q ˜︁
A k = T ∗
F , q L T
q ˜︁
C and
K
∑︂
k = 1 ˜︁
A k P G , q T G , q T ∗
G , q h k ( S q ) T G , q = ˜︁
B R q T G , q .
S ince the matr ices
˜︁
A k
ar e the solutions of these linear matr ix equations and since their coefficient
matrices as well as the right hand sides are r eal-valued, we conclude that the matrices
˜︁
A k
ar e also
r eal-valued. ■
T o complete the discussion, we analyze the r egularity of
A
in the SISO case , that is
p = m =
1. H ere ,
we set
P F , q : = diag( F q ), P G , q : = diag( G q ), ˜︁
B : = ⎡
⎢
⎢
⎣
1
.
.
.
1
⎤
⎥
⎥
⎦ , and ˜︁
C : = [︂ 1 . . . 1 ]︂ . (7.20)
With these settings , the (
i , j
) components of the second matrix equations in
(7.16)
and
(7.17)
r ead as
f q ; i ∑︁ K
k = 1 h k
(
µ q ; i
)[
˜︁
A k
]
i , j =
1 and
g q ; j ∑︁ K
k = 1 h k
(
σ q ; j
)[
˜︁
A k
]
i , j =
1, r espectively . Putting this into matrix
notation yields the linear system
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
f 1; i . . .
f Q F ; i
g 1; j . . .
g Q G ; j
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
h 1 ( µ 1; i ) . . . h K ( µ 1; i )
.
.
. .
.
.
h 1 ( µ Q F ; i ) . . . h K ( µ Q F ; i )
h 1 ( σ 1; j ) . . . h K ( σ 1; j )
.
.
. .
.
.
h 1 ( σ Q G ; j ) . . . h K ( σ Q G ; j )
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎡
⎢
⎢
⎢
⎢
⎣
[ ˜︁
A 1 ] i , j
[ ˜︁
A 2 ] i , j
.
.
.
[ ˜︁
A K ] i , j
⎤
⎥
⎥
⎥
⎥
⎦ = ⎡
⎢
⎢
⎢
⎢
⎣
1
1
.
.
.
1
⎤
⎥
⎥
⎥
⎥
⎦ , (7.21)
wher e the system matr ix is the product of a diagonal matrix and a generalized V ander monde matrix.
W e notice that reor dering the entr ies of
α
in
(7.18)
yields an orthogonal similarity transformation
that decouples
(7.18)
in smaller systems of the form
(7.21)
. This generalized V ander monde matrix is
also called a Haar matrix [60] and is nonsingular if the dr iving fr equencies
µ q ; i
and
σ q ; j
ar e distinct
and the functions
h k
satisfy the Haar condition [60]. I n par ticular , the H aar condition is satisfied
for monomials, and thus r elevant for second-or der systems (cf. T able 6.1 ). The diagonal matrix is
nonsingular if the driving frequencies
µ q ; i
and
σ q ; j
ar e distinct from the roots of the original transfer
function. In this case , the system in
(7.21)
has a unique solution for each (
i , j
) combination and
hence , via transformations, we can infer that A is nonsingular .
W e illustrate the construction of the r ealization with additional data with the follo wing example .
Example 7.14. G iven scalars a 1 , a 2 , a 3 , b , c ∈ R wi th b c = 0, we consider the system
a 1 ˙
x ( t ) = a 2 x ( t ) + a 3 x ( t − 1) + b u ( t ),
y ( t ) = c x ( t )

7.3. STR UCTURED REALIZ A TION FOR THE CASE K ≥ 3 123
with transfer function
H
(
s
)
= c b
s a 1 − a 2 − e − s a 3
. Setting
Q F =
1 and
Q G =
2, we pick distinct interpolation
points
µ 1;1 = µ
,
σ 1;1 = σ
, and
σ 2;1 = λ
. W e choose
˜︁
B =
1 and
˜︁
C =
1 with
P F ,1 = H
(
µ
)
, P G ,1 = H
(
σ
),
and P G ,2 = H ( λ ). Then the system in (7.21) reads as
⎡
⎢
⎣
H ( µ )
H ( σ )
H ( λ ) ⎤
⎥
⎦ ⎡
⎢
⎣
µ − 1 − e − µ
σ − 1 − e − σ
λ − 1 − e − λ ⎤
⎥
⎦ ⎡
⎢
⎣ ˜︁
A 1
˜︁
A 2
˜︁
A 3
⎤
⎥
⎦ = ⎡
⎢
⎣
1
1
1 ⎤
⎥
⎦ . (7.22)
The inverse of the H aar matrix is given b y
1
µ e µ (e σ − e λ ) + σ e σ (e λ − e µ ) + λ e λ (e µ − e σ ) ⎡
⎢
⎣
e µ (e σ − e λ ) − e σ (e µ − e λ ) e λ (e µ − e σ )
e µ ( σ e σ − λ e λ ) − e σ ( µ e µ − λ e λ ) e λ ( µ e µ − σ e σ )
− e µ e σ e λ ( σ − λ ) e µ e σ e λ ( µ − λ ) − e µ e σ e λ ( µ − σ )
⎤
⎥
⎦
such that the solution of
(7.22)
is
[︂ ˜︁
A 1 ˜︁
A 2 ˜︁
A 3 ]︂ = 1
c b [︂ a 1 a 2 a 3 ]︂
. I n fact, we reco ver the original
transfer function. ♠
Clearly , the realization is real-valued if all quantities in
(7.21)
ar e real. If we pick the driving fr equen-
cies on the imaginary axis, th en in general the H aar matrix will be complex-valued. The following
lemma sho ws ho w to obtain real-valued r ealizations based on complex interpolation data if the
matrices P F , q and P G , q are chosen as in (7.20).
Lemma 7.15.
Let the interpolation data be closed under complex conjugation and sorted as in
Example 7.9 such that the unitar y matrices T F , T G ∈ C n × n from Example 7.9 satisfy
T ∗
F M q T F ∈ R n × n , T ∗
F L T
q ∈ R n , T ∗
F F T
q ∈ R n , for q = 1, . . . , Q F ,
T ∗
G S q T G ∈ R n × n , R q T G ∈ R n , G q T G ∈ R n , for q = 1, . . . , Q G .
M or eo ver , let the matrices P F , q and P G , q be as in (7.20) . Then, the r eali zation
( T ∗
F ˜︁
A 1 T G , . . . , T ∗
F ˜︁
A K T G , T ∗
F ˜︁
B , ˜︁
C T G ),
with (
˜︁
A 1 , . . . , ˜︁
A K , ˜︁
B , ˜︁
C
) from Theor em 7.12, consists of real-v alued matri ces and interpolates the
data.
Proof.
First note that the state space tr ansfor mation b y the unitar y matrices
T ∗
F
and
T G
does not
change the transfer function and thus the interpolation given b y Theorem 7.12 is also valid her e. It
r emains to show that t he r ealization consists of real-valued matrices . Since
˜︁
B
and
˜︁
C
ar e given in
(7.20)
, it is straightforward to see that
T ∗
F ˜︁
B
and
˜︁
C T G
ar e real-valued. As in the proof of Lemma 7.13,
we deduce the r ealness of
T ∗
F ˜︁
A k T G
b y obser ving that the second equalities in
(7.16)
and
(7.17)
ar e
equivalent to
K
∑︂
k = 1
T ∗
F h k ( M q ) T F T ∗
F P T
F , q T F T ∗
F ˜︁
A k T G = T ∗
F L T
q ˜︁
C T G and
K
∑︂
k = 1
T ∗
F ˜︁
A k T G T ∗
G P G , q T G T ∗
G h k ( S q ) T G = T ∗
F ˜︁
B R q T G .

124 CHAPTER 7. STR UCTURED INTERPOL A T OR Y REALIZ A TIONS
S traightfor war d computations yield that
T ∗
F P T
F , q T F
and
T ∗
G P G , q T G
ar e real-valued. F rom these
linear matrix equations we can determine the matrices
˜︁
A k
or equivalently the transf ormed analogues
T ∗
F ˜︁
A k T G
. In the latter ca se , we obser ve that the coefficient matrices as well as the right hand sides ar e
r eal-valued and thus we deduce that the matrices
T ∗
F ˜︁
A k T G
ar e also real-valued for
k =
1
, . . . , K
.
■
7.3.2 M atching der iv ative data
H ermite interpolation pro vides a well kno wn and robust approach for polynomial appro ximation
that involves the matching of derivativ e data. When we seek reduced models th at ar e str ucturally
equivalent to standar d first order r ealizations (that is, when in
(6.10)
we have
K =
2,
h 1
(
s
)
= s
, and
h 2
(
s
)
≡ −
1) then first or der necessar y conditions for optimality of the reduced or der appro ximant
with r espect to the
H 2
norm are kno wn and they requir e that the reduced tr ansfer function
˜︁
H
(
s
)
must be a H ermite interpolant of the or iginal
H
(
s
) [89]. E ven though these necessar y conditions do
not extend immediately to mor e general structured systems as in
(6.10)
, it is kno wn for some special
cases such as second or der systems wi th modal damping and port-H amiltonian systems [22], and
for systems with simple delay structures [70, 71], that Hermite interpolation (in a differ ent form
than for the rational case) still plays a fundament al role in the necessary optimality conditions.
Ther efore , if derivative information for the transfer function
H
is accessible then this motivates
finding a structurally equivalent r ealization
H
(
s
) that matches both the evaluation data and the
derivative data. W e ther efore assume that
( f ′
i ) T : = ℓ T
i H ′ ( µ i ) and g ′
i : = H ′ ( σ i ) r i for i = 1, . . . , n (7.23)
ar e available, collected in the matrices
F ′ = [︂ f ′
1 . . . f ′
n ]︂ and G ′ = [︂ g ′
1 . . . g ′
n ]︂ .
I n this section, we derive conditions such that the transfer function
˜︁
H
interpolates the data
(6.5)
with {︁ µ i }︁ n
i = 1 ∩ {︁ σ i }︁ n
i = 1 = ∅ and in addition satisfies the H er mite interpolation condition (7.23).
Theor em 7.16.
Let
H
(
s
)
= ˜︁
C
(
∑︁ K
k = 1 h k
(
s
)
˜︁
A k
)
− 1 ˜︁
B
and suppose that
∑︁ K
k = 1 h k
(
˜
s
)
˜︁
A k
is nonsingular
for all ˜
s ∈ {︁ µ i }︁ n
i = 1 ∪ {︁ σ i }︁ n
i = 1 .
(i)
The left interpolation conditions
ℓ T
i ˜︁
H
(
µ i
)
= f T
i
and the left Hermite interpolation conditions
ℓ T
i ˜︁
H ′
(
µ i
)
= (︁ f ′
i )︁ T
ar e satisfied for
i =
1
, . . . , n
if and only if ther e exist matrices
P F
and
P F ′
that satisfy
F T = P T
F ˜︁
B ,
K
∑︂
k = 1
h k ( M ) P T
F ˜︁
A k = L T ˜︁
C , (7.24)
(︁ F ′ )︁ T = (︁ P F ′ )︁ T ˜︁
B ,
K
∑︂
k = 1
h k ( M ) (︁ P F ′ )︁ T ˜︁
A k = −
K
∑︂
k = 1
h ′
k ( M ) P T
F ˜︁
A k . (7.25)
(ii)
The right interpolation conditions
˜︁
H
(
σ i
)
r i = g i
and the right Hermite interpolation condi-
tions
˜︁
H ′
(
σ i
)
r i = g ′
i
for
i =
1
, . . . , n
ar e satisfied if and only if there e xist matrices
P G
and
P G ′

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