Frequency Domain Methods and Decoupling of Linear Constant Coefficient Infinite Dimensional Differential Algebraic Systems [original]

Technische Universit¨at Berlin

Institut f¨ur Mathematik

Frequency Domain Methods and

Decoupling of Linear Constant Coefficient

Infinite Dimensional Differential

Algebraic Systems

Timo Reis and Caren Tischendorf

Preprint 1-2005

Preprint-Reihe des Instituts f¨ur Mathematik

Technische Universit¨at Berlin

http://www.math.tu-berlin.de/preprints

Report 1-2005 2005

FREQUENCY DOMAIN METHODS AND DECOUPLING OF

LINEAR CONSTANT COEFFICIENT INFINITE DIMENSIONAL

DIFFERENTIAL ALGEBRAIC SYSTEMS∗

TIMO REIS†AND CAREN TISCHENDORF‡

Abstract. We discuss the analysis of constant coefficient linear differential algebraic equations

E˙x(t) = Ax(t) + q(t) on infinite dimensional Hilbert spaces. We give solvability criteria of these

systems which are mainly based on Laplace transformation. Furthermore, we investigate decoupling

of these systems, motivated by the decoupling of finite dimensional differential algebraic systems by

the Kronecker normal form. Applications are given by the analysis of mixed systems of ordinary

differential, partial differential and differential algebraic equations.

Key words. partial differential-algebraic equations, index, infinite dimensional linear system

theory

AMS subject classifications. 34A09, 34A30, 93A10, 34G10, 35E20

1. Introduction. In today’s engineering applications, there an increasing in-

terest in partial differential algebraic equations (PDAE’s), which are mainly coupled

systems of partial differential equations (PDE’s) and differential-algebraic equations

(DAE’s). More concrete, they appear e.g. in modelling and simulation of electrical

circuits with further effects which are modelled by PDE’s. These effects can be para-

sitic like transmission lines or heat conduction [2, 14, 23] as well as they could be the

result of a more reliable modelling of complex components like semiconductor devices

[3, 29, 33]. Moreover, PDAE’s are the outcome of mathematical models of several

mechanical systems like elastic multibody systems [10] or biomechanical systems like

blood flow networks. In order to study these problems in a mathematically systematic

way, we are led to differential algebraic systems

F( ˙x(t), x(t), t) = 0 (1.1)

in an abstract setting, the so called abstract DAE’s (ADAE’s). The unknown function

x(·) is now a path in an appropriate (mostly infinite dimensional) Hilbert space, and

the Frech´et derivative d

d ˙xF( ˙x, x, t). has a nontrivial nullspace, in general. In this

work, we focus on the linear constant coefficient case

E˙x(t) = Ax(t) + q(t) (1.2)

and making use of that gaining additional structure. E:X→Zis now a bounded

linear operator and X, Z are some Hilbert spaces. In many practical cases, Ais often

acting on some product spaces and it is a block operator containing differential and

evaluation operators, for example. Hence, it is natural to assume that it is unbounded

in general and is defined on some proper subspace D(A)⊂X.

The aim of this work is a step-by-step generalization of the known theory for the finite

dimensional version of (1.2), in which case we have square matrices E, A ∈Rn×n.

The finite dimensional systems are well-studied and subject of various textbooks like

e.g. [4], [5] and [15]. The matrix pair (E, A) is said to be regular, if det(sE −A) does

∗This work was supported by the DFG Graduiertenkolleg Mathematik und Praxis in Kaiser-

slautern and by the DFG Research Center ”Mathematics for key technologies” in Berlin.

†Fachbereich Mathematik, TU Kaiserslautern, (reis@mathematik.uni-kl.de)

‡Institut f¨ur Mathematik, TU Berlin, (tischend@math.tu-berlin.de)

2T. REIS AND C. TISCHENDORF

not vanish identically, i.e. det(sE −A)6≡ 0. For regular matrix pairs, it is known that

there exist invertible matrices T, W ∈Rn×n, such that

(W ET, W AT ) = N

I,I¯

A,(1.3)

where N∈Rn∞×n∞is nilpotent and ¯

A∈R(n−n∞)×(n−n∞)is an arbitrary square

matrix. The representation (1.3) is called Kronecker normal form of (E, A). Further,

the nilpotency index ν∈Nof N, i.e. the number with Nν−16= 0, Nν= 0, is well-

defined by (E, A) and called the Kronecker index. Multiplying a finite dimensional

DAE of the form (1.2) from the left side with Wand insert the identity I=T T −1,

we get

W ET (T−1˙x(t)) = W ET (T−1x(t)) + W q. (1.4)

If we introduce (x1x2) = T−1xand (x1x2) = W q, the equivalent DAE in Kronecker

normal form is obtained, namely the following decoupled differential equations

N˙x1(t) = x1(t) + q1(t) (1.5a)

˙x2(t) = ¯

Ax2(t) + q2(t).(1.5b)

(1.5a) contains algebraic equations and some further hidden relations, being algebraic,

when (1.5a) is differentiated, and thus it is called the (hidden) algebraic constraints.

The second expression (1.5b) is nothing but an ordinary differential equation extracted

from the DAE (1.2) and is therefore called the inherent ODE. Altogether, solutions

of these equations are given by

x1(t) = −

ν−1

k=0

Nkq(k)

1(t), x2(t) = e¯

Atx2(0) + Zt

e¯

A(t−τ)q2(τ)dτ. (1.6)

In [4] and [13], for example, algorithms for the computations of Tand Ware presented.

Due to (1.6), it can be seen that the Kronecker index νof (E, A) is the minimal integer

which satisfies an inequality

kx(T)k ≤ cT·

k=0

kq(k)(·)kL2([0,T ],Rn)(1.7)

for some positive constant c.L2([0, T ],Rn) denotes the Lebesque space of square

integrable functions with values in Rn.

In this work, we will perform an analysis as in (1.7) as well as we generalize the

decoupling framework to infinite dimensional descriptor systems. We will obtain a

form as follows





0 0

˙x1(t)

˙x2(t)=



I K

0R

x1(t)

x2(t)+



q1(t)

q2(t)

q3(t)

.(1.8)

The second two lines, i.e. the system

˙x2(t) = Ux2+q2(t)

0 = x2(t) + q3(t),(1.9)

INFINITE DIMENSIONAL DAE’s 3

play the role of the inherent ODE for the finite dimensional case. This type of equa-

tions is called an abstract boundary control system, since, in an abstract setting, bound-

ary controlled systems can be written in this way. After solving (1.9) for x2, we obtain

for the first component of the state vector

x1(t) = −

ν−1

k=0

Nk(q(k)

1+Kx2(t)).

An extraordinary role is taken by the coupling term K. Due to the existence of the

Kronecker normal form, in the finite dimensional case there can be always found a

representation with K= 0. However, this is not true for infinite dimensional DAE’s.

It will turn out that Khas to satisfy a certain boundedness condition in order to

guarantee that it can be eliminated. The proof of the existence of the form (1.8) is

constructive and requires some projector chain to be existent and stagnant. There, we

lean against the results of [17]. Besides that Eis bounded, we will make the assump-

tion that the generalized resolvent (sE −A)−1is bounded and analytic for sin some

complex half-plane and has there, it has at most polynomial growth in s. With the

Laplace transformation of the ADAE, we will shift the problem into some frequency

domain spaces, namely the Hardy spaces H2and H∞. From that, we will derive

some criteria for the solvability of ADAE’s. Many examples of practical relevance,

especially coupled systems of DAE’s and PDE’s, fulfill the requirements, we make on

Eand A.

We briefly resume the actual state of affairs concerning ADAE’s. [12] considers sys-

tems (1.2), where he assumes that Eis indeed injective but not boundedly invertible.

Although these assumptions are almost disjoint to those, we make, the mathematical

methods for the analysis of the solvability are based on Laplace transform and hence,

they are somehow similar to our guesses. The application of that work mainly focuses

on PDE’s with spacial singularities. In the papers of Thaller et. al. (e.g. [31, 32]),

besides the boundedness E:X→Z, they additionally assume that

{x∈D(A) : Ax ∈im E} ∩ ker E={0}.

It will turn out that this assumption is equivalent to the index of the ADAE being

less than 2, in our formulation. Moreover, both assume that the spaces, we denoted

by Zand X, coincide. As we will see with the given examples, this is not reasonable

for the consideration of coupled systems. More related to this work is [17]. There,

abstract differential algebraic systems of the form

A(t)d

dt(D(t)x(t)) + B(t)x(t) = q(t) (1.10)

are considered and an extraction of the algebraic relations is performed. Especially,

this approach is applied and in [3, 29, 33] in modelling and simulation of analog cir-

cuits. Indeed, this approach is more general than ours since time-varying operators

are considered. On the other hand, the presented theory is close to the given practi-

cal examples and for the applicability of those results, some inspired homogenizations

have to be performed. Moreover, there is no uniform theory for the solvability of this

type of equations. Since we assume time-invariance, i.e. our operators Eand Ado not

depend on time, a much bigger mathematical framework is disclosed, of which we can

make use of. As an example, we can apply methods based on Laplace transformation.

Possible applications of this paper are given by the structure analysis of coupled sys-

tems. Especially, by the transformation into (1.8), we get inside into the system

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