On the Stabilization through Linear Output Feedback of a Class of Linear Hybrid Time-Varying Systems with Coupled Continuous/Discrete and Delayed Dynamics with Eventually Unbounded Delay

Ci a ion: De la Sen, M. On he
S abiliza ion h ough Linea Ou pu
Feedback o a Class o Linea Hyb id
Time-Va ying Sys ems wi h Coupled
Con inuous/Disc e e and Delayed
Dynamics wi h E en ually
Unbounded Delay. Ma hema ics 2022,
10, 1424. h ps://doi.o g/10.3390/
ma h10091424
Academic Edi o : An ónio M. Lopes
Recei ed: 22 Ma ch 2022
Accep ed: 21 Ap il 2022
Published: 23 Ap il 2022
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ma hema ics
A icle
On he S abiliza ion h ough Linea Ou pu Feedback o a Class
o Linea Hyb id Time-Va ying Sys ems wi h Coupled
Con inuous/Disc e e and Delayed Dynamics wi h E en ually
Unbounded Delay
Manuel De la Sen
Depa men o Elec ici y and Elec onics, Facul y o Science and Technology, Ins i u e o Resea ch and
De elopmen o P ocesses, Uni e si y o he Basque Coun y (UPV/EHU), 48940 Leioa, Bizkaia, Spain;
[email p o ec ed]
Abs ac :
This esea ch s udies a class o linea , hyb id, ime- a ying, con inuous ime-sys ems wi h
ime- a ying delayed dynamics and non-necessa ily bounded, ime- a ying, ime-di e en iable
delay. The conside ed class o sys ems also in ol es a con ibu ion o he whole delayed dynamics
wi h espec o he las p eceding sampled alues o he solu ion acco ding o a p e ixed cons an
sampling pe iod. Such sys ems a e also subjec o linea ou pu - eedback ime- a ying con ol,
which picks-up combined in o ma ion on he ou pu a he cu en ime ins an , he delayed one,
and i s disc e ized alue a he p eceding sampling ins an . Closed-loop asymp o ic s abiliza ion is
add essed h ough he analysis o wo “ad hoc” K aso skii–Lyapuno - ype unc ional candida es,
which in ol e quad a ic o ms o he s a e solu ion a he cu en ime ins an oge he wi h an in eg al-
ype con ibu ion o he s a e solu ion along a ime- a ying p e ious ime in e al associa ed wi h
he ime- a ying delay. An analy ic me hod is p oposed o syn hesize he s abilizing ou pu - eedback
ime- a ying con olle om he solu ion o an associa ed algeb aic sys em, which has he objec i e
o acking p esc ibed sui ed e e ence closed-loop dynamics. I his is no possible—in he e en
ha he men ioned algeb aic sys em is no compa ible— hen a bes app oxima ion o such a ge ed
closed-loop dynamics is made in an e o -no m sense minimiza ion. Su iciency- ype condi ions
o asymp o ic s abili y o he closed-loop sys em a e also de i ed based on he wo men ioned
K aso skii–Lyapuno unc ional candida es, which in ol e e alua ions o he con ibu ions o he
delay- ee and delayed dynamics.
Keywo ds:
hyb id dynamic sys ems; ime- a ying delay; sampled sys ems; asymp o ic s abili y;
asymp o ic s abiliza ion; linea ou pu eedback
MSC: 93C05; 93D20; 93C55
1. In oduc ion
So-called hyb id dynamic sys ems, which essen ially consis o mixed, and in gen-
e al, coupled, con inuous- ime and ei he digi al o disc e e- ime dynamics, a e o an
un-doub able in e es in ce ain enginee ing con ol p oblems. Such in e es a ises om
he ac ha he e a e ce ain eal-wo ld p oblems which e ain combined con inuous- ime
and disc e e- ime in o ma ion, and his ci cums ance is e lec ed in he dynamics. The
con inuous- ime in o ma ion is modelled h ough di e en ial equa ions (such as o dina y,
unc ional o pa ial di e en ial equa ions) while he disc e e- ime dynamics a e modelled
h ough di e ence equa ions. In his way, hyb id sys ems can some imes be e y complex
o analyze, since hey migh in ol e combina ions and couplings o andems o mo e
elemen a y subsys ems. See, o ins ance, [
1
–
4
]. A majo equi emen in he design o
con ol schemes is s abiliza ion ia eedback by syn hesizing a s abilizing con olle . E en
Ma hema ics 2022,10, 1424. h ps://doi.o g/10.3390/ma h10091424 h ps://www.mdpi.com/jou nal/ma hema ics
Ma hema ics 2022,10, 1424 2 o 23
i an open-loop sys em (i.e., ha esul ing in he absence o eedback) is s able, he e is
o en a need o imp o e i s s abili y [
5
–
16
]. A use ul p ocedu e o discuss bo h s abili y and
s abiliza ion conce ns is he use o Lyapuno - ype o Co duneanu- ype unc ionals and
hei gene aliza ions ( o ins ance, Lu ’e, K aso skii, Razumikhin, Popo , e c.). See, o
ins ance, [1–5,8–16] and e e ences he ein.
To ix basic ideas on hyb id sys ems, no e ha a well-known ypical elemen a y ex-
ample o such sys ems is ha consis ing o a con inuous- ime sys em in ope a ion unde
a disc e e- ime con olle . In his way, he con olle does no need o keep in o ma ion
on he con inuous- ime signals o all imes, bu only a sampling ins an s. O he ypical
hyb id sys ems in ol e he combined use o neu al ne s and uzzy logic o ope a e on he
con inuous- ime and/o disc e e- ime dynamics, o elec ical and mechanical d i elines.
On he o he hand, hyb id dynamic sys ems wi h coupled con inuous- ime and digi al
dynamics ha e been desc ibed in [
17
]. Thei p ope ies o con ollabili y, eachabili y
and obse abili y ha e been cha ac e ized in [
18
–
21
] and some o he e e ences he ein.
Adap i e con ol me hods o such sys ems in he case o a pa ial lack o knowledge o
hei pa ame ical alues ha e been add essed in [
22
,
23
], while op imal “ad hoc” designs
ha e been s a ed and discussed in [
24
] and some o he e e ences he ein. In he abo e
opics, i migh be impo an o adap he design o he mul i a e con ex , since some imes
he disc e ized s a es and/o he inpu s can be subjec o di e en sampling a es, ei he
due o accommoda ing he design o he na u e o such signals o imp o ing he con ol
pe o mances. The ini e- ime s abiliza ion o mul i a e ne wo ked con ol sys ems based
on p edic i e con ol is discussed in [
25
]. Ano he mo e gene al p oblem which can be con-
side ed in combina ion wi h di e en mul i a e designs is he e en ual use o ime- a ying
sampling a es, again o be e accommoda e he expec ed pe o mances by adap ing he
sampling a es o he a es o a ia ions in he in ol ed signals [26].
Dynamic sys ems in gene al, and some hyb id dynamic sys ems in pa icula , can also
ypically in ol e linea and non-linea dynamics, and hey can be subjec o he p esence
o in e nal delays (i.e., in he s a e ec o ) and/o ex e nal delays (i.e., in hei inpu s o
ou pu s). See, o ins ance, [
1
,
2
,
6
–
16
]; al hough, i mus be poin ed ou ha he ela ed
backg ound li e a u e is ex ensi e. Typical exis ing eal-li e sys ems in ol ing delays
include a numbe o biological models, such as epidemic models, popula ion g ow h o
di usion models, sun lowe equa ion, wa and peace models, economic models, e c.
This pape o mula es and desc ibes a class o linea ime- a ying, con inuous- ime
sys ems wi h ime- a ying, con inuous- ime delayed dynamics. Such a class o sys ems
is hyb id in he sense ha i can conside an added con ibu ion o delayed dynamics o
i s cu en con inuous- ime dynamics wi h espec o p e iously sampled alues o he
solu ion, o a ce ain de ined sampling pe iod. Such a dynamic con ibu es o he whole
solu ion, oge he wi h bo h he delay- ee, con inuous- ime dynamics and he con inuous
delayed dynamics. The la e is associa ed wi h a ime- a ying, con inuously di e en iable
delay, which is, in gene al, unbounded and o a con inuous- ime de i a i e na u e, being
e e ywhe e less han one. The class o hyb id sys ems unde s udy migh also be subjec o
linea ou pu - eedback ime- a ying con ol unde combined in o ma ion o he ou pu
a he cu en ime ins an , he delayed one and he p e ious disc e e- ime alue in a
closed-loop con igu a ion. The gene al solu ion is calcula ed in a closed explici o m.
Special emphasis is paid o he closed-loop s abiliza ion ia linea ou pu eedback h ough
he app op ia e design o he s abilizing con ol ma ices. The s abiliza ion p ocess is
in es iga ed ia K aso skii–Lyapuno unc ionals.
Nex , he pape deals wi h he de i a ion and analysis o su iciency- ype condi ions
o he closed-loop asymp o ic s abili y, which a e ob ained h ough he de ini ion o wo
K aso skii–Lyapuno unc ional candida es. One o hose unc ional candida es has a
cons an , leading posi i e-de ini e ma ix o de ine he non-in eg al pa as a quad a ic
unc ion o he solu ion alue a each ime ins an , while he second candida e p oposes a
ime- a ying, ime-di e en iable ma ix unc ion o he same pu pose. The e a e also some
ex a assump ions in oked which ocus on he maximum a ia ion o he ime-in eg al o
Ma hema ics 2022,10, 1424 3 o 23
he squa ed no ms o he emaining ma ices o delayed dynamics associa ed wi h bo h
he con inuous- ime delay and wi h he memo y on he sampled pa o he hyb id sys em.
These ex a assump ions essen ially ely on he ac ha hose ime in eg als a y mo e
slowly han linea ly, wi h any conside ed ime in e al leng h, in o de o pe o m he
in eg als o e ime. The subsequen pa o he manusc ip is de o ed o con olle syn hesis
o he e en ual achie emen o closed-loop s abiliza ion ia linea ou pu eedback, in
such a way ha he asymp o ic s abili y esul s o he p e ious sec ion a e ul illed by
he eedback sys em. In he ime-in a ian , delay- ee case, he e a e some backg ound
esul s a ailable on s abiliza ion ia s a ic linea ou pu eedback (see, o ins ance, [
27
–
29
]
and some o he e e ences he ein). The syn hesized con olle possesses se e al gain
ime- a ying ma ix unc ions. One is designed o s abilize he delay- ee dynamics, while
he emaining ones ha e, as hei objec i e, minimiza ion in some app op ia e sense o
he con ibu ion o he na u al and he sampled delayed dynamics o he whole closed-
loop dynamics. To s abilize he delay- ee ma ix o dynamics, he con olle gain ma ix
unc ion is calcula ed ia a K onecke p oduc o ma ices [
29
,
30
], associa ed wi h an
algeb aic sys em. The p oblem is well-posed, p o ided ha such a sys em is compa ible o
some sui able ma ix unc ion desc ibing he delay- ee closed-loop dynamics. In case he
men ioned algeb aic sys em is no compa ible, he con olle gain is syn hesized so as o
app oxima e he esul ing closed-loop ma ix o a sui able dynamic in a bes app oxima ion
con ex o i s no m de ia ion, wi h espec o he p e ixed and sui able closed-loop ma ix o
delay- ee dynamics. This pape also discusses how o syn hesize he emaining ma ices,
which in ol e na u al delays, and he delayed dynamics associa ed wi h he disc e e
in o ma ion, in such a way ha he esul ing ma ix unc ion o delayed dynamics has
small no ms in a sense o he bes app oxima ion o ze o.
I can be poin ed ou ha he p e iously ci ed li e a u e on hyb id sys ems does no ely
on he ou pu - eedback s abiliza ion o sys ems, which include bo h disc e e in o ma ion
on he p e iously sampled solu ion alues and combina ions o bo h delay- ee, con inuous
dynamics and delayed, con inuous, ime- a ying dynamics. This pape also ocuses on
he closed-loop s abiliza ion o he solu ion ia linea ou pu eedback. These conce ns a e
he main no el y o his manusc ip , and also he mo i a ion o he s udy, since he class
o hyb id sys ems unde conside a ion is mo e gene al han hose p e iously s udied in
he li e a u e.
The pape is o ganized as ollows. Sec ion 2s a es and desc ibes he linea hyb id
ime- a ying con inuous ime sys em wi h combined ime- a ying delay- ee and delayed
dynamics, as well as i s solu ion in closed explici o m in bo h un o ced and o ced
cases. The o ced solu ion also conside s a pa icula si ua ion whe e he o cing con ol is
ob ained ia linea eedback o combined in o ma ion on he cu en ou pu , he delayed
ou pu and he p e iously sampled alue o he ou pu . Sec ion 3deals wi h de i a ion
o su iciency- ype condi ions o closed-loop asymp o ic s abili y, which a e ob ained
h ough he de ini ion o wo K aso skii–Lyapuno unc ionals o asymp o ic s abili y
analysis pu poses. One in ol es a cons an posi i e-de ini e ma ix o he de ini ion o he
delay- ee e m, while he o he in ol es a posi i e-de ini e ime- a ying con inuous- ime
di e en iable ma ix. Con olle syn hesis o closed-loop asymp o ic s abiliza ion ia
linea ou pu eedback is also discussed. Finally, conclusions end he pape .
Nomencla u e
The ollowing no a ion is used:
R+={ ∈R: >0}
is a se o posi i e eal numbe s and
R0+=R+∪{0}
is a se o
non-nega i e eal numbe s. Simila ly, he posi i e and non-nega i e in ege numbe s a e
de ined by he espec i e se s Z+={z∈Z:z>0}and Z0+=Z+∪{0}.
Le
M
,
N∈Rn×n
, hen
M
0 deno es ha he ma ix
M
is posi i e-de ini e;
M<
0
deno es ha i is posi i e-semide ini e;
M≺
0 ( espec i ely,
M4
0) deno es ha i is nega i e-
de ini e ( espec i ely, nega i e-semide ini e);
MN⇔M−N0
;
M<N⇔M−N<0
.
Ma hema ics 2022,10, 1424 4 o 23
I
M=Mij=




M11 M12 ··· M1m
M21 M22··· M2m
.
.
.
Mn1
.
.
.
Mn2···
.
.
.
Mnm




=




MT
1
MT
2
.
.
.
MT
n




∈Rn×m
and
N=
Nij∈Rp×q
, hen
M⊗N=Mij N∈Rnp×mq
is he K onecke p oduc o he ma ices
Mand N, and ecM =MT
1,MT
2,··· ,MT
nT.
A squa e eal o complex ma ix is a s abili y ma ix i all i s eigen alues ha e nega i e
eal pa s.
A†
deno es he Moo e–Pen ose gene alized in e se, o Moo e–Pen ose pseudo-in e se,
o
A∈Rn×m
. I
ankA =s≤min(n,m)
hen he e exis s
C∈Rn×s
and
D∈Rs×m
such
ha A=CD and
A†=DTDDT−1CTC−1CT
. I sa is ies he condi ions
AA†A=A
,
A†AA†=A†and i coincides wi h he in e se o Ai Ais squa e non-singula .
A closed-loop sys em, in he s anda d e minology, is ha esul ing om a s a e o
ou pu - eedback con ol law. The s abili y is e med o be global i he solu ion is bounded
o all ime and any gi en admissible unc ion o ini ial condi ions. I is o global asymp o ic
ype i , in addi ion, i con e ges asymp o ically o he equilib ium s a e.
We pay special a en ion in his manusc ip o he syn hesis o a s abilizing ou pu
linea eedback con ol. In he con ex o his manusc ip , a hyb id sys em is one which
in ol es mixed con inuous- ime and disc e e- ime dynamics. We conside ha , in gene al,
i also in ol es delayed con inuous- ime dynamics and disc e e- ime dynamics associa ed
wi h a gi en sampling pe iod.
2. The Hyb id Con inuous-Time/Disc e e-Time Di e en ial Sys em Subjec o a
Time-Va ying Delay
Conside he ollowing dynamic con ol sys em subjec o, in gene al, a ime- a ying
delay:
.
x( )=A( )x( )+Ad( )x( −h( )) +Aa( )x( −kT)+B( )u( )+Ba( )u( −kT)(1)
y( )=C( )x( )(2)
∀ ∈R0+
unde a bounded piecewise con inuous unc ion o ini ial condi ions
ϕ:[−h(0), 0]→Rn
, whe e
T>
0 is he sampling pe iod,
k=k( )=(maxz ∈Z0+:zT ≤ )
,
x:[−h(0),∞)→Rn
,
y:[−h(0),∞)→Rp
and
u:[−h(0),∞)→Rm
a e, espec i ely, he
s a e solu ion on
[−h(0),∞)
and he ou pu and inpu ec o unc ions wi h
max(p,m)≤
n
and
x( )=ϕ( )
;
∈[−h(0), 0]
wi h
x0=x(0)=ϕ(0)
and
xk=x(kT)
;
∀k∈
Z0+
. The ma ix unc ions o dynamics
A:[0, ∞)→Rn×n
,
Aa:[−h(0),∞)→Rn×n
and
Ad:[−h(0),∞)→Rn×n
, and he con ol
B:[0, ∞)→Rn×m
and ou pu
C:[0, ∞)→Rp×n
ma ix unc ions, a e piecewise, con inuous and bounded. The con ol ec o is piecewise
and cons an wi h e en ual ini e jumps a he sampling ins an s
k=kT
;
k∈Z0+
( he se
o non-nega i e in ege numbe s) and is he inpu (o con ol) ec o
u( )
; wi h
u(kT)=uk
;
∀k∈Z0+
( he se o non-nega i e eal numbe s), and
h:[0, ∞)→R0+
is he ime- a ying
delay subjec o
h( )≤
;
∀ ∈R+
and
h(0)
ini e. The abo e sys em is con inuous-disc e e
hyb id in he sense ha he s a e e ol es o ced by i s cu en alue a ime
wi h a memo y
e ec on i s las p eceding sampled alue a he sampling ins an
kT
unde a pe iodic
sampling o pe iod
T
and he con ol ope a ing join ly a bo h ins an s
and
−kT
. The
majo in e es o he subsequen in es iga ion is he ou pu - eedback con ols o he o m:
u( )=K( )y( )+Kd( )y( −h( )) +Ka( )y( −kT);∀ ∈R0+(3)
whe e
K:[0, ∞)→Rm×p
,
Kd:[−h(0),∞)→Rm×p
and
Ka:[−h(0),∞)→Rm×p
a e he
con olle gain ma ices o be syn hesized and
k=k( )=(maxz ∈Z0+:zT ≤ )
. The
eplacemen o he ou pu ec o by he s a e ec o in (3) leads o he mos es ic i e
s a e ou pu - eedback con ol ype. Th ough he pape , we will e e o (1) and (2) as he
open-loop sys em, since he con ol ia eedback is no ye selec ed. I s un o ced solu ion is
Ma hema ics 2022,10, 1424 5 o 23
ha co esponding o jus he ini ial condi ions, ha is, when
u≡
0. The o ced solu ions
co espond o nonze o con ols. No e ha he con olled sys em (1) and (2) as well as
he closed-loop con igu a ion (1)–(3) esul ing ia eedback con ol a e pa ame e ized, in
gene al, by ime- a ying ma ices. The closed-loop sys em is he combina ion o (1) o
(3), ha is, ha esul ing a e eplacing he con ol law (3) in (1). The solu ion o (1) is
cha ac e ized in he subsequen heo em.
Theo em 1.
The solu ion o he un o ced sys em (1), o any bounded piecewise con inuous unc ion
o ini ial condi ions ϕ:[−h(0), 0]→Rn, is unique and gi en by:
x( )=Ψ( , 0)x0+Z0
−h(0)
Ψ( ,τ)ϕ(τ)dτ;∀ ∈R0+(4)
whe e he e olu ion ma ix unc ion
Ψ:R0+×(R0+∪[−h(0))) →Rn×n
is subjec o
Ψ( ,τ)=
0 o τ> , Ψ( , )=In( he n- he iden i y ma ix); ∀ ∈R0+, and i sa is ies:
.
Ψ( ,τ)=A( )Ψ( ,τ)+Ad( )Ψ( −h( ),τ)+Aa( )Ψ( −kT,τ)
;∀τ(≤ )∈R0+∪[−h(0), 0),∀ ∈[kT,(k+1)T),∀k∈Z0+
(5)
whe e he do symbol deno es he ime de i a i e wi h espec o he i s a gumen . The whole
solu ion o (1), including he un o ced and he o ced con ibu ions, is:
x( )=Ψ( , 0)x0+Z0
−h(0)
Ψ( ,τ)ϕ(τ)dτ+Z
0
Ψ( ,τ)B(τ)u(τ)dτ+Z
0
Ψ( ,τ)Ba(τ)u(τ−k(τ))dτ;∀ ∈R0+(6)
wi h k( )=(maxz ∈Z0+:zT ≤ ).
P oo .
The uniqueness o he solu ion is ob ious since he ma ix unc ions which pa ame-
e ize (1) a e bounded, piecewise, and con inuous, and he exp ession (4), subjec o (5),
is he solu ion o he un o ced (1), as i can be di ec ly e i ied as ollows. One ob ains
by eplacing (5) in o he ime-de i a i e o (4) wi h he subsequen use o he claimed
solu ion (4):
.
x( )=.
Ψ( , 0)x0+Z0
−h(0)
.
Ψ( ,τ+h(0))ϕ(τ)dτ
=(A( )Ψ( , 0)+Ad( )Ψ( −h( ), 0)+Aa( )Ψ( −kT, 0))x0
+Z0
−h(0)(A( )Ψ( ,τ)+Ad( )Ψ( −h( ),τ)+Aa( )Ψ( −kT,τ))ϕ(τ)dτ
=A( )Ψ( , 0)x0+Z0
−h(0)
Ψ( ,τ)ϕ(τ)dτ+Ad( )Ψ( −h( ), 0)x0+Z0
−h(0)
Ψ( −h( ),τ)ϕ(τ)dτ
+Aa( )Ψ( −h( ), 0)x0+Z0
−h(0)
Ψ( −kT,τ)ϕ(τ)dτ
=A( )x( )+Ad( )x( −h( )) +Aa( )x( −kT);∀ ∈[kT,(k+1)T),∀k∈Z0+
(7)
wi h
x( )=ϕ( )
o
∈[−h(0), 0]
, hus (7) coincides wi h he un o ced di e en ial sys em
(1) so ha he un o ced solu ion is (4) and he e olu ion ma ix unc ion
Ψ:R0+×(R0+∪[−h(0))) →Rn×n
subjec o
Ψ( ,τ)=
0 o
τ>
,
Ψ( , )=In
sa is-
ies (5). As a esul , he whole solu ion o (1) is (6). 
Rema k 1.
I
A( )
commu es wi h
eR
0A(τ)dτ
o all
∈R0+
hen he e olu ion ma ix unc ion o
(1) which is he solu ion o (5) is:
Ψ( ,τ)=eR
τA(σ)dσIn+Z
τe−Rς
0A(σ+τ)dσ(Ad( )Ψ( −h( ),ς)+Aa( )Ψ( −kT,ς))dς

Ma hema ics 2022,10, 1424 6 o 23
o ≥τ≥0. In pa icula , i A( )is cons an , hen
Ψ( ,τ)=eA( −τ)In+Z
τe−Aς(Ad( )Ψ( −h( ),ς)+Aa( )Ψ( −kT,ς))dς(8)
o ≥τ≥0.
An in e es ing p ope y o he e olu ion ma ix h ough ime is gi en in he subsequen
esul , which is use ul o cha ac e ize analy ically and e en ually compu e he solu ion:
P oposi ion 1.
Conside a bi a y ime ins an s
2≥ 1≥
0. Then, he e olu ion ma ix unc ion
sa is ies:
Ψ( 2,τ)=Ψ( 2, 1)Ψ( 1,τ)Z0
−h( 1)
Ψ( 2, 1+σ)Ψ( 1+σ,τ)dσ;∀τ∈[−h( 1), 0](9)
P oo .
x( 2)=Ψ( 2, 0)x0+Z0
−h(0)
Ψ( 2,τ)ϕ(τ)dτ
=Ψ( 2, 1)x( 1)+Z0
−h( 1)
Ψ( 2, 1+τ)x( 1+τ)dτ
=Ψ( 2, 1)Ψ( 1, 0)x0+Z0
−h(0)
Ψ( 1,τ)ϕ(τ)dτ+Z0
−h( 1)
Ψ( 2, 1+τ)Ψ( 1+τ, 0)x0+Z0
−h(0)
Ψ( 1+τ,σ)ϕ(σ)dσdτ
=Ψ( 2, 1)Ψ( 1, 0)+Z0
−h( 1)
Ψ( 2, 1+τ)Ψ( 1+τ, 0)dτx0
+Z0
−h(0)
Ψ( 2, 1)Ψ( 1,τ)ϕ(τ)dτ+Z0
−h( 1)Z0
−h(0)
Ψ( 2, 1+τ)Ψ( 1+τ,σ)ϕ(σ)dσdτ
=Ψ( 2, 1)Ψ( 1, 0)+Z0
−h( 1)
Ψ( 2, 1+τ)Ψ( 1+τ, 0)dτx0
+Z0
−h(0)Ψ( 2, 1)Ψ( 1,τ)+Z0
−h( 1)
Ψ( 2, 1+σ)Ψ( 1+σ,τ)dσϕ(τ)dτ
(10)
The i s and he igh -hand-side exp essions o (10) ha e o be iden ical o any gi en
unc ion o ini ial condi ions ϕ:[−h(0), 0]→Rnso ha (9) holds. 
Le us de ine by
ˆ
x( 1)
he s ip o he solu ion o
x( )
he in e al
[ 1−h( 1), 1]
o he
gi en unc ion o ini ial condi ions
ϕ:[−h(0), 0]→Rn
, wi h
ˆ
x(0)
being
ϕ:[−h(0), 0]→Rn
.
In acco dance wi h (4), de ine he in e al- o-poin e olu ion ope a o
S:R0+→L(X)
as
ollows:
x( )=S( , 0)( ˆ
x( 0)) =Ψ( , 0)x( 0)+Z0
−h( 0)
Ψ( , 0+τ)x( 0+τ)dτ;∀ ∈R0+(11)
o any
≥
0≥
0, whe e
X
is he space o he un o ced solu ions o (1), o any gi en
unc ion o ini ial condi ions
ϕ:[−h(0), 0]→Rn
wi h
x( )=ϕ( )
o
∈[−h(0), 0]
, so
ha , o any 0, 1(≥ 0), 2(≥ 1)∈R0+,
Ma hema ics 2022,10, 1424 7 o 23
x( 2)=S( 2, 1)( ˆ
x( 1)) =Ψ( 2, 1)x( 1)+Z0
−h( 1)
Ψ( 2, 1+τ)x( 1+τ)dτ
=Ψ( 2, 1)x( 1)+Z0
−h( 1)
Ψ( 2, 1+τ)Ψ( 1+τ, 0)dτx( 0)+Z0
−h( 1)Z0
−h( 0)
Ψ( 2, 1+τ)Ψ( 1+τ, 0+σ)x( 0+σ)dσdτ
=Ψ( 2, 1)Ψ( 1, 0)x( 0)+Ψ( 2, 1)Z0
−h( 0)
Ψ( 1, 0+τ)x( 0+τ)dτ
+Z0
−h( 1)
Ψ( 2, 1+τ)Ψ( 1+τ, 0)dτx( 0)+Z0
−h( 1)Z0
−h( 0)
Ψ( 2, 1+τ)Ψ( 1+τ, 0+σ)x( 0+σ)dσdτ
=Ψ( 2, 1)S( 1, 0)( ˆ
x( 0))
+Z0
−h( 1)
Ψ( 2, 1+τ)Ψ( 1+τ, 0)x( 0)+Z0
−h( 0)
Ψ( 1+τ, 0+σ)x( 0+σ)dσdτ
=Ψ( 2, 1)S( 1, 0)( ˆ
x( 0))+Z0
−h( 1)
Ψ( 2, 1+τ)(S( 1+τ, 0)ˆ
x( 0))dτ
=S( 2, 0)( ˆ
x( 0))
(12)
so ha he e olu ion ope a o sa is ies o 0, 1(≥ 0), 2(≥ 1)∈R0+:
S( 2, 0)( ˆ
x( 0)) =Ψ( 2, 1)S( 1, 0)( ˆ
x( 0))+Z0
−h( 1)
Ψ( 2, 1+τ)(S( 1+τ, 0)ˆ
x( 0))dτ
I can be no iced ha he in e al- o-poin e olu ion ope a o is ela ed o he e olu ion
ma ix unc ion ia he iden i ies (12), and, unde he addi ional assump ion ha he delay
unc ion is non-inc easing discussed in he subsequen esul , i is also ela ed o an in e al-
o-in e al e olu ion ope a o .
P oposi ion 2. I h :[0,∞)is non-inc easing, hen he ollowing p ope ies hold:
i h(0)=sup
∈R0
h( )and 1−h( 1)≤ 2−h( 2) o any 1, 2(≥ 1)∈R0+.
ii
De ine he in e al- o-in e al e olu ion ope a o
ˆ
S:R0+→L(X)
as ollows o any
1, 2(≥ 1)∈R0+:
ˆ
S( 2, 1)( ˆ
x( 1)) =S( 2, 1)( ˆ
x( 1)) ∪{x( ): ∈[ 2−h( 2), 2)}
=S( 2, 0)( ˆ
x( 0)) ∪{x( ): ∈[ 2−h( 2), 2)}(13)
so ha o any 0, 1(≥ 0), 2(≥ 1)∈R0+, one has:
ˆ
x( 2)=ˆ
S( 2, 1)( ˆ
x( 1)) =ˆ
S( 2, 1)ˆ
S( 1, 0)(ˆ
x( 0))
=ˆ
S( 2, 0)( ˆ
x( 0))
=S( 2, 0)( ˆ
x( 0)) ∪{x( ): ∈[ 2−h( 2), 2)}
(14)
and ˆ
S:R0+∪[−h(0), 0)→L(X)is a s ongly con inuous one-pa ame e semig oup.
P oo : h(0)=sup
∈R0
h( )
ollows di ec ly since
h
:
[0,∞)
is non-inc easing. Now assume, on
he con a y o he second p ope y, ha
1−h( 1)> 2−h( 2)
o some
1
,
2(≥ 1)∈
R0+
. Then,
h( 2)> 2− 1+h( 1)>h( 1)
which con adic s ha
h
:
[0,∞)
is non-
inc easing. Thus,
1−h( 1)≤ 2−h( 2)
o any
1
,
2(≥ 1)∈R0+
so ha P ope y (i)
is p o ed. No e ha , since
h
:
[0,∞)
is non-inc easing, hen (13) is well-posed, since
ˆ
x( 2)={x( ): ∈[ 2−h( 2), 2]}=ˆ
S( 2, 1)( ˆ
x( 1))
o any
1
,
2(≥ 1)∈R0+
and (14)
ollows om (12), (13). Now, no e ha
ˆ
S( 0, 0)
is he iden i y ope a o on
X
o any
0∈R0+
,
ˆ
S( 2, 0)=ˆ
S( 2, 1)ˆ
S( 1, 0)
(see (14)), and
lim
0→0+kˆ
S( 0.0)ˆ
x( 0)−ˆ
x( 0)k=
0 so
ha he in e al- o-in e al e olu ion ope a o is con inuous in he s ong ope a o opol-
ogy. P ope y (ii) has been p o ed. 
Ma hema ics 2022,10, 1424 8 o 23
No e ha P oposi ion 2 also holds in pa icula i he delay is cons an .
The ollowing esul is closely ela ed o Theo em 1, excep o ha he hyb id sys em
conside s he con ibu ion o he dynamics o he las p eceding sampling ins an o he
cu en con inuous one ins ead o he delay be ween hem bo h.
Co olla y 1. Conside he di e en ial sys em:
.
x( )=A( )x( )+Ad( )x( −h( )) +Aa( )x(kT)+B( )u( )+Ba( )u(kT);
∀ ∈[kT,(k+1)T),∀k∈Z0+
(15)
The un o ced solu ion o any bounded, piecewise, con inuous unc ion o ini ial condi ions
ϕ:[−h(0), 0]→Rnis unique, and gi en by
x( )=Ψ( , 0)x0+Z0
−h(0)
Ψ( ,τ)ϕ(τ)dτ;∀ ∈R0+(16)
whe e he e olu ion ma ix unc ion
Ψ:R0+×(R0+∪[−h(0))) →Rn×n
is subjec o
Ψ( ,τ)=
0 o τ> , Ψ( , )=In;∀ ∈R0+, and i sa is ies:
.
Ψ( ,τ)=A( )Ψ( ,τ)+Ad( )Ψ( −h( ),τ)+Aa( )Ψ(kT,τ);∀τ(≤ )∈R0+
,∀ ∈[kT,(k+1)T)(17)
and he whole solu ion o (1) is:
x( )=Ψ( , 0)x0+R0
−h(0)Ψ( ,τ)ϕ(τ)dτ+R
0Ψ( ,τ)B(τ)u(τ)dτ+∑k( )
j=0R
jT Ψ( ,τ)Ba(τ)dτuj;
∀ ∈R0+
(18)
wi h k( )=(maxz ∈Z0+:zT ≤ )and uj=u(jT);∀j∈Z0+.
The p oo o Co olla y 1 is simila o ha o Theo em 1 by no ing ha an auxilia y
delay
( )= −kT
o
∈[kT,(k+1)T)
allows us o w i e
x(kT)=x( − ( ))
and
u(kT)=x( − ( ))
, which leads o (17) being iden ical o (5) o such a delay. No e ha
he hyb id con inuous/disc e e di e en ial sys em (15) has a ini e memo y con ibu ion
o he s a e and con ol a he sampling ins an s on each nex in e -sample ime in e al,
which is inco po a ed in o he con inuous- ime dynamics.
Rema k 2.
The un o ced and he o al solu ions (16) and (18) o (1) can also be w i en equi alen ly
as ollows, by aking ini ial condi ions on he in e al [kT −h(kT),kT]:
x( )=Ψ( ,kT)xk+Z0
−h(kT)
Ψ( ,kT +τ)x(kT +τ)dτ;∀ ∈[kT,(k+1)T);∀k∈Z0+(19)
x( )=Ψ( ,kT)xk+Z0
−h(kT)
Ψ( ,kT +τ)x(kT +τ)dτ+Z
kT
Ψ( ,τ)B(τ)u(τ)dτ
+Z
kT
Ψ( ,τ)(B(τ)u(τ)+Ba(τ)u(τ−kT))dτ;∀ ∈[kT,(k+1)T);∀k∈Z0+
(20)
The closed-loop di e en ial sys em (1) is ob ained by eplacing he eedback con ol
(3) in o (1), aking in o accoun (2), o yield:
.
x( )=Acl( )x( )+Adcl( )x( −h( )) +A0
acl( )x( −kT)
+Ba( )[K( )C( )x( −kT)+Kd( )C( )x( −kT −h( )) +Ka( )C( )x( −kT)]
=Acl( )x( )+Adcl(x( −h( ))) +Aaclx( −kT)
+Badx( −kT −h( )) +Baax( −2kT);∀ ∈[kT,(k+1)T);∀k∈Z0+
(21)
Ma hema ics 2022,10, 1424 9 o 23
whe e
Acl( )=A( )+B( )K( )C( )
Adcl( )=Ad( )+B( )Kd( )C( )
A0
acl( )=Aa( )+B( )Ka( )C( )
Aacl( )=A0
acl( )+Ba( )K( )C( )=Aa( )+(B( )Ka( )+Ba( )K( ))C( )
Bad( )=Ba( )Kd( )C( )
Baa( )=Ba( )Ka( )C( )
(22)
The solu ion o (21) and (22) is ound di ec ly by eplacing he e olu ion ma ix
unc ion o Theo em 1 by ha associa ed wi h (21), subjec o (22), which leads o he
subsequen esul :
Theo em 2.
The solu ion o he closed-loop di e en ial sys em (21) and (22) o any gi en bounded,
piecewise, con inuous unc ion o ini ial condi ions
ϕ:[−h(0), 0]→Rn
, is unique, and gi en by:
x( )=Ψcl( , 0)x0+Z0
−h(0)
Ψcl( ,τ)ϕ(τ)dτ;∀τ(≤ )∈R0+∪[−h(0), 0),
∀ ∈[kT,(k+1)T),∀k∈Z0+
(23)
wi h
k( )=(maxz ∈Z0+:zT ≤ )
;
∀ ∈R0+
, whe e he e olu ion ma ix unc ion
Ψcl :R0+×(R0+∪[−h(0))) →Rn×n
is subjec o
Ψcl( ,τ)=
0 o
τ>
,
Ψcl( , )=In
;
∀ ∈R0+, and i sa is ies:
.
Ψcl( ,τ)=Acl( )Ψ( ,τ)+Adcl( )Ψcl( −h( ),τ)+Aacl( )Ψcl( −kT,τ)
+Ba( )[Kd( )C( )Ψcl( −kT −h( ),τ)+Ka( )C( )Ψcl( −2kT,τ)];
∀τ(≤ )∈R0+;∀ ∈[kT,(k+1)T),∀k∈Z0+
(24)
Rema k 3.
A pa allel conclusion o ha o Rema k 1 o he closed-loop sys em is ha , i A( )
commu es wi h eR
0A(τ)dτ o all ∈R0+, hen he e olu ion ma ix unc ion o (23), and solu ion
o (21) subjec o (22), is
Ψcl ( ,τ)=eR
τAcl (σ)dσIn+Z
τe−Rς
0Acl (σ+τ)dσ(Adcl( )Ψcl ( −h( ),ς)+Aacl( )Ψcl( −kT,ς))dς
+Z
τe−Rς
0Acl (σ+τ)dσBa( )(Kd( )C( )Ψcl ( −kT −h( ),τ)+Ka( )C( )Ψcl ( −2kT,τ))dς(25)
o ≥τ≥0.
The ollowing esul add esses he ac ha he global Lyapuno s abili y and asymp-
o ic s abili y o any bounded unc ion o ini ial condi ions o he un o ced di e en ial
sys ems (1) and (15), and ha o he closed-loop hyb id sys em (21) and (22), ob ained ia
he eedback con ol law (3), depend di ec ly on he boundedness and anishing condi ions
o hei espec i e e olu ion ma ix unc ions.
Theo em 3. The ollowing p ope ies hold:
i
The un o ced sys em (1) is globally s able in he Lyapuno
´
s sense, i , and only i , he e olu ion
ma ix unc ion
Ψ:R0+×(R0+∪[−h(0))) →Rn×n
, being he solu ion o (5), and i s
gi en cons ain s, is bounded o any
∈R0+
and
τ∈R0+∪[−h(0))
, wi h
,
τ(≤ )
, and
any gi en bounded unc ions o ini ial condi ions
ϕ:[−h(0), 0]→Rn
. The un o ced sys em
(1) ollows Lyapuno
´
s global asymp o ic s abli y, i and only i , in addi ion,
Ψ( ,τ)→0
as
| −τ|→∞.
ii
The un o ced sys em (15) ollows Lyapuno
´
s global s abili y, i and only i he e olu ion
ma ix unc ion
Ψ:R0+×(R0+∪[−h(0))) →Rn×n
, being he solu ion o (17), and i s
Ma hema ics 2022,10, 1424 16 o 23
and hen P ope y (ii) ollows di ec ly. By modi ying (29) wi h a ime- a ying con inuously
ime-di e en iable P( ) as:
V( ,x )=4xT( )P( )x( )+Z( ,x );∀ ∈[kT,(k+1)T),∀k∈Z0+(50)
wi h
Z( ,x )
de ined in (30), one ob ains by ollowing he same s eps as in he p oo o
Theo em 4 ha (33) is modi ied as ollows:
.
V( ,x )≤ −q−k.
P( )k2kx( )k2−ˆ
q( ,x )≤ − q−sup
∈R0+k.
P( )k2!kx( )k2
∀ ∈[kT ,(k+1)T)∩R0+,∀k∈Z0+
(51)
and
.
V( ,x )
is nega i e o any nonze o x( ) i
sup
∈R0+k.
P( )k2<q
, which combined wi h (44),
leads o:
sup
∈R0+k.
P( )k ≤ min(q,k2
2 ρ−4k2sup
∈R0+kP( )k!
× 2sup
∈R0+kP( )ksup
∈R0+k.
Acl ( )k+sup
∈R0+k.
Ph( )k+sup
∈R0+k.
PT( )k+sup
∈R0+k.
P2T( )k+sup
∈R0+k.
PhT ( )k+sup
∈R0+k.
Ω( )k!! (52)
which comple es he p oo o P ope y (i). 
Rema k 5.
No e ha
kPk ≤ k2d
2(ρ−2k2kPk)
is he simpli ied e sion o he no m cons ain (46) in
he p oo o Theo em 6 being adap ed ad hoc, as associa ed wi h (26) in Theo em 5, by aking in o
accoun ha P is cons an .
Following he ela ions p e ious o (39) in he p oo o Theo em 6 o he pa allel cons ain (26)
in Theo em 5, by aking in o accoun ha P is cons an unde he cons ain
kPk ≤ k2d
2(ρ−2k2kPk)
,
which is a simpli ied e sion o (46) o his case, whe e he cons ain
kP( )k ∈ 0, ρ
4k2
is
weakened o
kPk ∈ 0, ρ
2k2
since he s onge cons ain
kPk ∈ 0, ρ
4k2
o Theo em 6 is emo ed
since
P
is cons an . Thus, (47) becomes simpli ied o
p(kPk)=
4
k2kPk2−
2
ρkPk+k2d≥
0,
which, combined wi h
kPk ∈ 0, ρ
4k2
, esul s in Theo em 5 in he subsequen pa allel cons ain
o (49) ob ained o Theo em 4, and which is a necessa y condi ion o he exis ence o
P
, sa is ying
(26):
kPk




∈[(0, kPk1]∪[kPk2, 0)] ∩0, ρ
2k2=0, ρ−√ρ2−4k4d
4k2∪ρ−√ρ2−4k4d
4k2,ρ
2k2i ρ>2k2√d
∈0, ρ
2k2i ρ≤2k2√d
3.2. Closed-Loop Asymp o ic S abiliza ion
No e ha he second condi ions o Theo ems 3 and 4, isualized by he Lyapuno
ma ix inequali y (26) and he Lyapuno ma ix Equa ion (39), espec i ely, ely on he ac
ha ma ix o delay- ee closed-loop dynamics
Acl( )
is a s abili y ma ix o all ime. In
iew o he i s iden i y o (22), he open-loop delay- ee dynamics can be s abilized ia
linea ou pu eedback i , and only i , he e exis s some ma ix unc ion
K:R0+→Rm×p
,
such ha
Acl( )
equalizes some s abili y ma ix
Am( )
o all
∈R0+
. The subsequen
esul cha ac e izes he linea ou pu - eedback s abilizing gain ma ix o he delay- ee,
closed-loop dynamics. I also discusses how o add ess he hi d s ipula ion o Theo ems
4 and 5 by he choice o he o he wo con olle gain ma ix unc ions
Kd( )
and
Ka( )
in (22) o he delayed dynamics. Each o hose con ol gain ma ices is in ended o be
calcula ed o cancel, i possible, he co esponding delayed closed-loop dynamics i he

Ma hema ics 2022,10, 1424 17 o 23
esul ing algeb aic sys em is sol able, o o ob ain he bes app oxima ion o ze oing such
co esponding dynamics i he co esponding algeb aic sys em is incompa ible.
Theo em 6. The ollowing p ope ies hold:
(i) The algeb aic sys em:
B( )K( )C( )=Am( )−A( );∀ ∈R0+(53)
is sol able in
K( )
, o some s abili y ma ix A
m
( );
∀ ∈R0+
, equi alen ly, he se o algeb aic
linea sys em o equa ions:
B( )⊗CT( ) ec K( )= ec(Am( )−A( ));∀ ∈R0+(54)
is sol able in ecK( );∀ ∈R0+, i and only i
B( )B( )†(Am( )−A( ))C( )†C( )=Am( )−A( );∀ ∈R0+(55)
equi alen ly, i and only i
ankB( )⊗CT( )= ankB( ) ank C( )= ank B( )⊗CT( ), ec(Am( )−A( ));
∀ ∈R0+
(56)
so ha he ma ix o delay- ee, closed-loop dynamics
Acl( )
is s able since i is ixed o
Am( )
;
∀ ∈R0+.
(ii) I (53) is sol able by a s abilizing ma ix unc ion o he closed-loop, delay- ee dynamics
gained by linea ou pu eedback, hen he se o solu ions o such a gain is gi en by
K( )=B( )†(Am( )−A( ))C( )†+K0( )−B( )†B( )K0( )C( )C( )†;∀ ∈R0+(57)
and equi alen ly, by,
ecK( )=B( )⊗CT( )† ec(Am( )−A( )) +Ipm −B( )⊗CT( )†B( )⊗CT( ) ecK0( )
:∀ ∈R0+
(58)
whe e K0( )∈Rm×p;∀ ∈R0+is a bi a y.
(iii) Assume ha o a gi en s abili y ma ix
Am( )
, (53), and equi alen ly (54), is algeb aically
incompa ible ( ha is, (55), equi alen ly (56), does no hold) o some
∈R0+
. Then, he bes
app oxima e solu ion o (54) is ob ained by aking K0( )≡0in (58).
(i ) The subsequen choices o
Kd( )
and
Ka( )
minimize
kAdcl( )k
and
kAacl( )k
, espec-
i ely:
Kd( )=−B( )†Ad( )C( )†;∀ ∈R0+(59)
equi alen ly,
ecKd( )=−B( )⊗CT( )† ecAd( ):∀ ∈R0+(60)
and
Ka( )=−B( )†(Aa( )+Ba( )K( )C( ))C( )†;∀ ∈R0+(61)
equi alen ly
ecKa( )=−B( )⊗CT( )† ec(Aa( )+Ba( )K( )C( )):∀ ∈R0+(62)
P oo .
No e ha (53) is he i s iden i y o (22) o A
cl
( ) = A
m
( );
∀ ∈R0+
, which is
sol able in
K( )
;
∀ ∈R0+
, i and only i (56) holds om Rouché- Capelli heo em, and
Ma hema ics 2022,10, 1424 18 o 23
equi alen ly, i and only i (55) holds, which is he necessa y and su icien condi ion o
sol abili y o (53) ia he Moo e–Pen ose pseudo-in e ses [29,30].
No e ha (55), and equi alen ly (56), is a necessa y condi ion o he second s ipu-
la ions o Theo em 4 and Theo em 5 o hold, since
Acl( )
has o be a s abili y ma ix o
sa is y he espec i e Lyapuno ma ix inequali y and equa ion in such heo ems. No e
also ha he solu ion o delay- ee con olle gain
K( )
is, in gene al, non-unique, wi h
he algeb aic linea sys em (54) being a compa ible inde e mina e. This p o es P ope y
(i). P ope y (ii) ollows di ec ly om P ope y (i) by making he solu ion explici in he
equi alen o ms (57) and (58) unde he necessa y and su icien condi ion o i s exis ence.
P ope y (iii) ollows, since i no solu ions exis , hen (58), and equi alen ly, (57), unde he
choice
K0( )≡
0, minimizes he e o no m
kB( )⊗CT( ) ecK( )− ec(Am( )−A( ))k
wi h espec o all he choices o he a bi a y ma ix K0( ), [29,30].
To p o e P ope y (i ), no e ha in (28), he ollowing ela ion can be w i en
Z −1
1( 2)
−1
1( 1)kAdcl(τ)k21−.
h(τ)dτ=Z 2
1kAdcl −1
1(τ)k2dτ≤µ1( 2− 1)(63)
o
2> 1≥
0, and close equi alences apply o he emaining h ee condi ions gi en in
(28). Now, he alues o
µ1
and
µ2
become as small as possible by educing as much as
possible
kAacl( )k
and
kBad( )k
h ough he choices o
Kd( )
and
Ka( )
, espec i ely. Thus,
i he equa ions Ad( )+B( )Kd( )C( )=Adcl( )=0
Aa( )+(B( )Ka( )+Ba( )K( ))C( )=Aacl( )=0
om (22) a e ei he sol able,
Kd( )
and
Ka( )
o algeb aically incompa ible, hen he espec-
i e minimiza ions o
kAacl( )k
and
kBad( )k
a ise by he choices (59) and (61), espec i ely.

Rema k 6.
No e ha , in gene al, a less es ic i e condi ion han ha gi en in Theo em 6 o he
sol abili y o (53) is he s abiliza ion by linea s a e- eedback, since he s a e space dimension
n
is
usually highe han ha o he ou pu space p. In ha case, he con olle gain ma ices a e o o de s
m×n
ins ead o
m×p
. This educes, o ake
C( )=In
in (53) and (54) so ha he sol abili y
condi ion (55) becomes weakened o:
ank(B( )⊗In)=n× ankB( )= ank(B( ), ec(Am( )−A( )));∀ ∈R0+(64)
On he o he hand, in he pa icula case wi h
m=p=n
, he dimensions o he s a e, inpu
and ou pu a e iden ical, and i can also be discussed as a pa icula case o linea s a e eedback o
he same numbe o inpu s as he numbe o ou pu s, bo h o hem equalizing he s a e dimension.
Howe e , his heo e ical case is no e y use ul in mos applica ions whe e he numbe s o inpu s
and ou pu s a e less han he s a e dimension.
In addi ion, no e ha in he case whe e he algeb aic sys em is incompa ible, he simples solu-
ion (
K0( )≡
0), co esponding o he inde e mina e compa ible case, gi es he bes app oxima ing
solu ion in he sense ha he e o no m be ween bo h sides o (54) is he minimum possible e o
no m o any selec ion o K( ).
I can be poin ed ou ha he e a e o he gene alized in e ses, such as he gene alized Bo –
Du in in e se, which is cons ained by he use o a p ojec ion on a subspace o he solu ion, o he
D azin in e se. I does no sa is y he condi ion AA†A=A, in gene al, [29]. 
Rema k 7.
No e om (21) and (22) ha Theo em 6 (i ) p o ides a way o minimize
kAacl( )k
and
kBad( )k
, bu we s ill need o deal wi h he delayed dynamics associa ed wi h he ma ices
Bad( )
and
Baa( )
. Howe e , he con ol law (3) has no ex a gains o deal wi h hose esul ing
Ma hema ics 2022,10, 1424 19 o 23
con ibu ions o he close-loop dynamics. A modi ica ion o he con ol o ce in (1) can assis wi h
ha ask. Conside he di e en ial sys em:
.
x( )=A( )x( )+Ad( )x( −h( )) +Aa( )x( −kT)+B( )u( )+Ba( )u0( −kT);
∀ ∈R0+
(65)
wi h
u( )
s ill being gene a ed by (3) and
u0( )=K0( )x( −kT)
;
∀ ∈R0+
being ano he
supplemen a y con ol o deal wi h he abo e-men ioned d awback. Then, he o me closed-loop
di e en ial sys em (21) and (22) becomes modi ied as ollows:
.
x( )=Acl( )x( )+Adcl( )x( −h( )) +Abc`( )x( −kT);∀ ∈[kT,(k+1)T);
∀k∈Z0+
(66)
whe e
Abcl( )=Aa( )+B( )Ka( )+Ba( )K0( )C( );∀ ∈R0+(67)
Now,
K( )
and
Kd( )
a e designed as in Theo em 6 o deal wi h
Acl( )
and
Adcl( )
, while
Ka( )and K0( )a e designed o deal wi h Abcl( ) ia he ollowing possibili ies:
(a)
Ka( )=K0( )=−(B( )+Ba( ))†Aa( )C( )†(68)
and equi alen ly,
ecKa( )= ecK0( )=−(B( )+Ba( )) ⊗CT( )† ecAa( )(69)
leading o
Abcl( )=Aa( )−(B( )+Ba( ))(B( )+Ba( ))†Aa( )C( )†C( )(70)
is he bes app oxima ion o Abcl( )=Aa( )+(B( )+Ba( ))Ka( )C( ) o Abcl( )=0.
(b) I Ka( )=0 hen
K0( )=−Ba( )†Aa( )C( )†(71)
and equi alen ly,
ecK0( )=−Ba( )⊗CT( )† ecAa( )(72)
leading o
Abcl( )=Aa( )−Ba( )Ba( )†Aa( )C( )†C( )(73)
is he bes app oxima ion o Abcl( )=Aa( )+Ba( )K0( )C( ) o Abcl( )=0.
(c) I K0( )=0 hen
Ka( )=−B( )†Aa( )C( )†(74)
and equi alen ly,
ecKa( )=−B( )⊗CT( )† ecAa( )(75)
leading o
Abcl( )=Aa( )−B( )B( )†Aa( )C( )†C( )(76)
is he bes app oxima ion o Abcl( )=Aa( )+B( )Ka( )C( ) o Abcl( )=0.
3.3. Example
Conside he ollowing ime- a ying, hi d-o de linea sys em wi h wo inpu s and
wo ou pu s, de ined by:
A( )=Aij( )∈R3×3
Ma hema ics 2022,10, 1424 20 o 23
B( )=

0 1 +sin
1 1
0 1 

C=1
23
21
−1
23
21
whe e
A11( )=−5(1+sin )−1
2 (1+sin )
A12( )=1+3
2( (1+ )−6(1+sin ))
A13( )= (1+ )−6(1+sin )
A21( )=1
2 (1+ )−sin −2
A22( )=3−5
2( (1+ )+sin )
A23( )=−( (1+ )+sin )
A31( )=2+1
2
A32( )=−3
2
A33( )=3−
The s abiliza ion objec i e is he achie emen o dynamics gi en by he s abili y ma ix:
Am=

010
−2−3 0
−3−9−3

whose eigen alues a e
−
1,
−
2 and
−
3. The algeb aic sys em o equa ions o be sol ed
sol e o his pu pose is
B( )K( )C=Am−A( )
which is sol able in he con olle gain
K( )
, since (56) is ul illed, [
29
–
31
]. The s abilizing
con olle gain which sa is ies he abo e equa ion is:
K( )=6+sin −4+ 2
−8 2 + (77)
The i s condi ion o Theo em 4 is ul illed wi h P=I3, since
λmaxAT
m+Am+q+1+1
4sup
∈R0+
λmax(Ph( )+PT( )+P2T( )+PhT( ))
≤ −2.05 +q+1+1
4sup
∈R0+
λmax(Ph( )+PT( )+P2T( )+PhT( )) ≤0
(78)
is ul illed acco ding o (27). I o some
q∈(0, 1.05)
, any disc e e dynamics and con inuous-
ime dynamics sa is y he ollowing cons ain o
k=max(z∈Z0+:zT ≤ )
, since his
cons ain gua an ees ha , in addi ion, (28) holds, so ha Theo em 4 is ul illed i he e
a e e en ual con ibu ions o ex a disc e e and con inuous- ime delayed dynamics which
sa is y:
sup
∈R0+ kAdcl1
−h( )k
2
2
+kAacl1
−kT k
2
2
+kBad1
−kT −h( )k
2
2
+kBaa1
−2kT k
2
2!≤4(1.05 −q)(79)
Ma hema ics 2022,10, 1424 21 o 23
Now, assume, o ins ance, ha he abo e delay- ee dynamics also inco po a es
disc e e dynamics, de ined by he ma ix unc ion:
Aa( )=

(3 /8)(1+sin ) (3 /8)(1+sin )0.25 (1+sin )
3 /8 −0.75 3 /8 −0.75 0.25 −0.5
3 /8 3 /8 0.25 
(80)
The co esponding gain con olle ma ix in he con olle (3) gi en by
Ka( )=1−0.5
−0.5 0.25 (81)
cancels he con ibu ion o such disc e e dynamics in he closed-loop dynamics wi h
Aacl( )=A0
acl( )=
0 and
Ba( )=Baa( )=Bad( )=
0 in (22). Thus, he whole closed-loop
sys em wi h delay- ee and disc e e dynamics is s abilized by he con olle :
u( )=K( )y( )+Ka( )y( −kT);∀ ∈R0+(82)
wi h he con olle gains gi en by (77) and (81). I hen su ices o he con inuous- ime
delayed con ibu ion, i any (i.e., i
Ad( )
is no iden ical o ze o in (1)) o he closed-loop
dynamics o sa is y (79). Fo ins ance, i is su icien o he whole con olle (3) o ha e he
gains
K( )
, Equa ion (77) and
Ka( )
, Equa ion (81), wi h an ex a gain
Kd( )
which sa is ies:
kAdcl1
−h( )k2
=kAd1
−h( )+B1
−h( )Kd1
−h( )Ck2
<2
in o de o s abilize he con inuous- ime delayed dynamics subjec o a ime- a ying
di e en iable delay h( )o a ime-de i a i e less han uni y.
In u u e wo ks, i is planned o ex end he esul s o his pape o he hype s abili y
and passi i y heo ies, [
32
–
36
] by designing he con olle gains so ha “ad hoc” Popo
´
s-
ype inequali ies be sa is ied by a eedback con ol loop unde gene ic nonlinea ime-
a ying con ol laws.
4. Conclusions
This pape has s udied a solu ion in closed o m as well as he asymp o ic s abili y and
asymp o ic s abiliza ion o a linea , ime- a ying, hyb id con inuous- ime/disc e e- ime
dynamic sys em subjec also o delayed dynamics, whose dynamics depend no only on
ime bu on p e iously sampled s a e alues as well. The delay unc ion is no necessa ily
bounded, and i is ime-di e en iable wi h bounded ime-de i a i es wi h a bound is less
han one o all ime. The asymp o ic s abili y a e injec ing e en ual eedback e o s is
s udied h ough wo K aso skii–Lyapuno unc ionals, one o hem ha ing a cons an
leading posi i e-de ini e ma ix o de ine he non-in eg al pa as a quad a ic unc ion
o he solu ion, while he o he akes a ime- a ying, ime-di e en iable ma ix unc ion
o he same pu pose. Those K aso skii–Lyapuno unc ionals es ablish su iciency- ype
condi ions o he asymp o ic s abili y o he closed-loop sys em. The sys em is assumed
unde a con ol law based on ime- a ying linea ou pu eedback, which akes combined
in o ma ion o he cu en ou pu alue, he delayed one and i s las p e ious sampled
alue, which a ises om he combined con inuous- ime/disc e e- ime hyb id na u e o
he di e en ial sys em. The associa ed Lyapuno ma ix inequali y, o equali y associa ed
wi h he abo e-men ioned K aso skii–Lyapuno unc ionals, assumes ha he delay- ee
ma ix o he closed-loop sys em dynamics is a s abili y ma ix o all ime, achie ed, unde
ce ain condi ions, by one o he con ol gain ma ix unc ions o he con ol law. The e a e
also ex a assump ions on he maximum a ia ion o he ime-in eg al o squa ed no ms
o he emaining ma ices o delayed dynamics in he sense ha hose ime in eg als a y
mo e slowly han linea ly wi h any conside ed ime in e al leng h.

Ma hema ics 2022,10, 1424 22 o 23
Funding:
This esea ch was unded by he Spanish Go e nmen and he Eu opean Commission
h ough G an RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE).
Ins i u ional Re iew Boa d S a emen : No applicable.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : No applicable.
Acknowledgmen s:
The au ho is g a e ul o he Spanish Go e nmen and he Eu opean Commission
o i s suppo h ough g an RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and o he Basque
Go e nmen o i s suppo h ough g an IT1207-19. He is also g a e ul o he Edi o and he e e ees
by hei use ul sugges ions.
Con lic s o In e es : The au ho decla es he has no compe ing in e es s.
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