delay-independent criteria for exponential admissibility of...

Applied Mathematics and Computation 228 (2014) 432–445

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Delay-independent criteria for exponential admissibility ofswitched descriptor delayed systems q

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.11.107

q This work was supported by the National Natural Science Foundation of China under Grants 11326128.⇑ Corresponding author.

E-mail address: [email protected] (X. Ding).

Xiuyong Ding a,⇑, Xiu Liu a, Shouming Zhong b

a School of Mathematics, Southwest Jiaotong University, Chengdu, Sichuan 610031, PR Chinab School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, PR China

a r t i c l e i n f o a b s t r a c t

Keywords:Switched descriptor systemsRegular-impulsive-freeSwitching-impulsiveness-freeMinimum dwell timeExponential admissibility

This paper deals with the uniform exponential admissibility problem for switched descrip-tor systems with time-varying delays. A criterion for regularity-impulsiveness-free of eachsubsystem is presented. A solution to switching-impulsiveness-free for switched descrip-tor systems is provided. A novel type of piecewise Lyapunov functionals is introduced. Thistype of Lyapunov functionals can efficiently overcome the switching jump of adjacentLyapunov functionals at switching times. By applying this type of Lyapunov functionalsand algebraic manipulations, the delay-independent conditions for uniform exponentialadmissibility is established on the minimum dwell time.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

As is well known, descriptor systems (known as singular, generalized or differential algebraic systems) describe intercon-nections of subsystems, electrical networks, robots and more generally mechanical structures, or can even be seen as approx-imations of singularly perturbed systems. There have been reported many works on stability analysis and control synthesisof descriptor systems, the interested reader can examine [1–6] and some references therein.

In recent years, switched systems have also received growing attention. Switched systems consist of a family of distinctactive subsystems subject to a certain switching rule which chooses one of them being active during a certain time. Suchsystems arise, for example, when different controllers are being placed in the feedback loop with a given process, or whena given process exhibits a switching behavior caused by abrupt changes of the environment. For a discussion of various is-sues related to switched systems, see the survey article [7]. There are three basic problems in stability analysis and design ofswitched systems: (1) find conditions for stability under arbitrary switching (see, e.g., [8–11]). (2) construct an appropriateswitching strategy to stabilize the system (see, for instance, [12,13]). (3) identify the limited but useful class of stabilizingswitching signals (see, for example, [14–16]). To tackle these three basic problems, a considerable number of classical tech-niques have been proposed, such as the common Lyapunov function approach [8,9,11], the multiple Lyapunov function ap-proach [14], the piecewise Lyapunov function approach [16], the switched Lyapunov function approach [10], and the dwelltime or average dwell-time scheme [15,13].

When the singular and switching phenomena are simultaneously encountered, the switched descriptor systems are nat-urally arisen. The previous work for such systems mainly focuses on serval hot topics of Lyapunov stability and stabilizationtheory [17–22,24,23,25–27], controller synthesis [17–19,28–30,20], and reachability [31]. On the other hand, many systems,in practice, arising in disciplines, such as physics, chemistry, biology and engineering, often involve after effects or time lags.

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X. Ding et al. / Applied Mathematics and Computation 228 (2014) 432–445 433

It has been well recognized that time delays not only degrade the performance of a control system, but also can destabilizethe system [32]. Consequently, switched delayed systems have received more and more attention in recent years (see, e.g.,[33,34]). At the same time, the switched descriptor delayed systems (SDDSs) starts to attract the researches’ attention grad-ually [19,21,24,23,26,27]. Basically, the key idea is Lyapunov functional technique and average dwell time approach. Ref.[19], by the switched Lyapunov approach, focused on the robust stability and H1 control problems for discrete-time SDDSsunder arbitrary switching, which focuses on the first basic problem. Ref. [21], for continuous-time SDDSs, presented theexponential admissibility (called exponential stability for general systems) criteria by the piecewise Lyapunov method.Ref. [24,23] extended this result to the nonlinear SDDSs. Ref. [26,27] employed the slow switching idea to stabilizable theSDDSs.

Unfortunately, there are some shortcomings for the existed results. First, all the stability criteria are subjected to theupper bounds of time delays. More precisely, these criteria are only apply the small delay cases. However, when the upperbounds of the time delays tend to infinity, it will be difficult to apply these results in the above-mentioned literature to de-rive the admissibility of SDDSs. Indeed, in practical applications, for many real-world control systems the upper bounds oftime delays may be unknown, and some characteristics of control components may change as the process evolves. For exam-ple, the transmission delays will become large due to environmental change. If these cases occur, it is natural to expect thatthe increasing delays will not destroy the stability of SDDSs. To guarantee the admissibility in the face of the increasing de-lays, the stability conditions should be designed to be independent of the sizes of time delays. However, presenting the de-lay-independent conditions cannot be achieved by the existing results in [19,21,24,23,26]. Second, for switched descriptorsystems, the switching impulse [22] is a unique phenomenon, and may destroy the system stability. For linear time-invariantcase, Zhai and Xu [22] gave a algebraic condition to verify the existence of switching impulse. However, for the delay case,the relevant work [19,21,24,23,26] fails to deal with this problem. That is to say, when a switching impulse occurs, the sta-bility criteria will become invalid.

The observation above inspires our research. In this paper, we are interested in the exponential admissibility of SDDSsunder a class of limited switching signals. We aim to solve the two problems: (i) under what conditions, the switching im-pulse will not occur? (ii) derive an exponential admissibility criterion which is irrespective of the sizes of time delays, i.e.,delay-independent. The layout of the paper is as follows. The notation and preliminaries are stated in Section 2. In Section 3,we first discuss the regularity-impulsiveness-free of each subsystem. Moreover, a algebraic condition will be presented tocheck the existence of switching impulse. Section 4 focuses on the uniform exponential admissibility. As we shall see, dif-ferent from the traditional piecewise Lyapunov functional, a new class of piecewise functional is constructed to establishthe minimum dwell time criterion. This criterion guaranteeing the uniform exponential admissibility of SDDSs can be ap-plied to any bounded time delay. Two numerical examples are presented in Section 5. Finally, Section 6 concludes this paper.

2. Preliminaries

Throughout, R denotes the real number set. Rn stands for the n-dimensional real vector set and Rn�m is the set of n�mmatrices with real entries. C denotes the space of all real-valued continuous functions. For matrix A in Rn�n;A > 0ð< 0Þmeansthat A is a symmetric positive (negative) definite matrix and A P 0ð6 0Þ means that A is a symmetric positive (negative)semi-definite matrix. We use kminðAÞ and kmaxðAÞ to denote the smallest and largest eigenvalue of A, respectively. detðAÞ de-notes the determination of A. N presents the set of all nonnegative integers and k � k denotes the Euclidean norm of vectors.For two sets M;N;M # N implies that M is a subset of N. For a polynomial pðxÞ in x;degðPðxÞÞ denotes the highest degree.

Consider the SDDS given by

E _xðtÞ ¼ ArðtÞxðtÞ þ BrðtÞxðt � hðtÞÞ;xðhÞ ¼ uðhÞ; h 2 ½�h; 0�;

ð1Þ

where xðtÞ 2 Rn is the state vector, rðtÞ : ½0;1Þ !M is the switching signal, rðtÞ ¼ ik 2M for t 2 ½tk; tkþ1Þ,M ¼ f1;2; . . . ;mg;m; k 2 N. Under the control of a switching signal r, system (1) enters from the ik�1th subsystem to theikth subsystem at the point t ¼ tk; tk is switching point and satisfies t0 < t1 < � � � < tk < � � � with limk!1tk ¼ 1 and t0 ¼ 0.The time-varying delay hðtÞ satisfies 0 6 hðtÞ 6 h and _hðtÞ 6 d with given scalars h and d < 1. The matrix E 2 Rn�n may besingular, and we assume that rankðEÞ ¼ r 6 n. For each possible value rðtÞ ¼ ik, the system matrices Aik , Bik 2 Rn�n are knownconstant matrices. Besides, u 2 Cð½�h;0�;RnÞ is the initial function with kukh ¼ sup�h6h�0kuðhÞk.

Assumption 2.1. In SDDS (1), there exists a positive real number sD such that, for any given switching signal rðtÞ,infk2Nftkþ1 � tkgP sD.

In the literature, this is a standard assumption to rule out Zeno behavior for all types of switching [35]. How to identify oravoid Zeno phenomena is a challenging topic which is beyond the scope of this paper. In fact, sD satisfying the above assump-tion is called minimum dwell time of SDDS (1). According to the value of sD, we define two switching signal sets:

SminðsDÞ ¼ rðtÞjrðtÞ ¼ i 2M; infk2Nftkþ1 � tkgP sD

� �

and

434 X. Ding et al. / Applied Mathematics and Computation 228 (2014) 432–445

Sall ¼ rðtÞjrðtÞ ¼ i 2M; infk2Nftkþ1 � tkg > 0

� �:

This paper mainly considers uniform exponential admissibility over switching signal sets SminðsDÞ and Sall. Next, threelemmas that are useful in deriving the principal contribution of this paper are presented.

Lemma 2.1 [36]. Let E; F and H be real matrices of appropriate dimensions with kFk � 1. Then for any scalars e > 0,

EFH þ HT FT ET6 e�1EET þ eHT H:

Lemma 2.2 [34]. Let P; S 2 Rn�n be symmetric positive definite and symmetric matrices, respectively. Then there exists a positiveconstant c P 1 such that for xðtÞ 2 Rn

xTðtÞSxðtÞ 6 cxTðtÞPxðtÞ:

Lemma 2.3 [37]. Given a matrix D, let a positive definite matrix S and a positive real number m 2 ð0;1Þ exist such thatDT SD� mS < 0. Then the matrix D satisfies kDrk 6 xe��r with positive real numbers x P 1 and �.

3. Regularity-impulsiveness-free and switching-impulsiveness-free

For general descriptor systems, the existence and uniqueness of solution relates to the regularity [1]. Let E;A 2 Rn�n, amatrix pair ðE;AÞ is said to be regular if the characteristic polynomial, detðsE� AÞ is not identically zero. Also, to avoidthe state impulse, the impulsiveness-free is involved. Matrix pair ðE;AÞ is said to be impulse-free ifdegðdetðsE� AÞÞ ¼ rankðEÞ (see [1]). Moreover, the descriptor system E _xðtÞ ¼ AxðtÞ is regular-impulsive-free if and only ifðE;AÞ is regular and impulsive-free. By this, we introduce the regularity-impulsiveness-free of descriptor delayed systems.

Definition 3.1. The descriptor delayed system P : E _xðtÞ ¼ AxðtÞ þ Bxðt � hÞ is regular-impulsive-free if the matrix pairs ðE;AÞand ðE;Aþ BÞ are both regular and impulsive-free.

Here we give a well-defined regularity-impulsiveness-free of descriptor delayed system. In the literature (see, e.g.,[38,39]), the researches usually define the regularity-impulsiveness-free of P as the regularity and impulsiveness-free of ma-trix pair ðE;AÞ. This definition may be not strict. Indeed, the regularity and impulsiveness-free of matrix pair ðE;Aþ BÞ is nec-essary. Such a requirement ensures the regularity and impulsiveness-free of P under the case h ¼ 0.

When we focus on the switched descriptor systems, the regularity-impulsiveness-free of each subsystem and the switch-ing-impulsiveness-free have to be involved [22]. In this section we will first give a criterion to check the regularity-impul-siveness-free of subsystems of SDDS (1). Furthermore, we will answer under what conditions the switching would not causestate impulses.

First of all, we discuss the regularity-impulsiveness-free of subsystems. For brevity, for i 2M, write

Ai ¼ ½Ai Bi�; I1 ¼ ½I 0�; I2 ¼ ½0 I�; E ¼ diagfE; Ig:

Proposition 3.1. Consider SDDS (1), for each i 2M, given positive real numbers c; ci with ci P 1, if there exist positive definitematrices Pi;Gi; Lai; Lbi, Hai;Hbi 2 Rn�n, and any matrices Ti 2 Rn�2n; Si 2 Rn�ðn�rÞ such that

Hi ¼ IT1ðE

T Pi þ SiRTi ÞAi þAT

i ðPiEþ RiSTi ÞI1 � ETLi � LiE þ ETHi þHiE þ c þ c

ci

� �I T

1GiI1 þ I T1ET Ti þ TT

i EI1 < 0; ð2Þ

where Li ¼ diagfLai; Lbig;Hi ¼ diagfHai;Hbig, and Ri 2 Rn�ðn�rÞ is any matrix with full column rank and satisfying ET Ri ¼ 0. Thenthe each subsystem of SDDS (1) is regular-impulsive-free.

Proof. For i 2M, we first check the regular impulsive-free properties of ðE;AiÞ. Since rankE ¼ r 6 n, there exist two nonsin-gular matrices U; V 2 Rn�n such that

E ¼ UEV ¼Ir 00 0

� �:

Then, Ri can be parameterized as

Ri ¼ UT 0Zi

� �;

where Zi 2 Rðn�rÞ�ðn�rÞ is any nonsingular matrix. Now, write, for l ¼ a; b,


Ti ¼ Ti11 Ti12½ �; Lli ¼Lli11 Lli12

Lli21 Lli22

� �; Hli ¼

Hli11 Hli12

Hli21 Hli22

� �:

We further define,

Pi ¼ U�T PiU�1 ¼ Pi11 Pi12

Pi21 Pi22

" #; Ai ¼ UAiV ¼

Ai11 Ai12

Ai21 Ai22

" #;

Lai ¼ U�T LaiV ¼Lai11 Lai12

Lai21 Lai22

" #; Ri ¼ U�T Ri ¼

0Zi

� �;

Hai ¼ U�T HaiV ¼Hai11 Hai12

Hai21 Hai22

" #; Si ¼ VT Si ¼

Si11

Si21

" #;

Bi ¼ UBiV ¼Bi11 Bi12

Bi21 Bi22

" #; T i ¼ U�T TidiagfV ; Vg ¼ ½T i11; T i12�:

Due to Gi > 0; c > 0, and ci P 1, we can formulate, from (2), the following inequality easily:

Ki ¼ ATi ðPiEþ RiS

Ti Þ þ ðE

T Pi þ SiRTi ÞAi � ET Lai � LaiEþ ET Hai þ HaiEþ ET Ti11 þ TT

i11E < 0:

Pre- and post-multiplying Ki < 0 by VT and V , respectively, yields

Ki ¼ VTKiV ¼ ATi PiEþ AT

i Ri STi þ ET PiAi þ SiRT

i Ai � ET Lai � LTaiEþ ET Hai þ HT

aiEþ ET T i11 þ TTi11E ¼ Ki11 Ki12

� ATi22ZiST

i21 þ Si21ZTi Ai22

" #< 0:

Obviously, we have

ATi22ZiST

i21 þ Si21ZTi Ai22 < 0

and thus Ai22 is nonsingular. Then it can be shown that ðE;AiÞ is regular and impulsive-free (see [1]).On the other hand, also by (2), pre- and post-multiplying Hi < 0 by ½I I� and ½I I�T , respectively, we get

Pi ¼ ðAi þ BiÞTðPiEþ RiSTi Þ þ ðE

T Pi þ SiRTi ÞðAi þ BiÞ � ET Lai � LaiEþ ET Hai þ HaiEþ ETðTi11 þ Ti12Þ þ ðTi11 þ Ti12ÞT E < 0:

Moreover, pre- and post-multiplying Pi < 0 by VT and V , then by the identical discussion above, the regularity and non-impulsiveness of ðE;Ai þ BiÞ follow immediately.

Finally, by Definition 3.1, we thus conclude that SDDS (1) is regular-impulsive-free. This completes the proof. h

For switched descriptor systems, even if all the descriptor subsystems are regular and impulsive-free (namely, the solu-tion of each subsystem is existence, uniqueness, and no impulse), the variable (state) vector jump caused by switching mayalso occur due to the inconsistent of the algebraic systems (see [28,22]). To see this phenomenon, consider a switcheddescriptor system composed of two subsystems below:

P1 :1 00 0

� �_x1

_x2

� �¼

1 00 1

� �x1

x2

� �;

P2 :1 00 0

� �_x1

_x2

� �¼

1 11 1

� �x1

x2

� �:

It is easy to verify that the two subsystems P1;P2 are both regular and impulsive-free. However, the algebraic system of P1

is 0 ¼ x2, this is obvious inconsistent the algebraic system 0 ¼ x1 þ x2 of P2 when x1 – 0. Under this case, when the switchingoccurs, the states maybe have a ‘‘jump’’, i.e., switching impulsive. Hence, based on this observation, when the switcheddescriptor systems are studied, no switching impulse must be assumed, and call such switching-impulsive-free.

Assumption 3.1. SDDS (1) is switching-impulsive-free.So, in what follows we will steadily make this assumption for SDDS (1). Now the question is, how to check the existence of

the switching impulse. In fact, under Proposition 3.1, SDDS (1) is regular-impulsive-free, then one can choose appropriatenonsingular matrices ~U; ~V 2 Rn�n (see [39]) such that, for one of the system matrices, without loss of generality, assumeA1, we have

~E ¼ ~UE~V ¼Ir 00 0

� �; ~A1 ¼ ~UA1

~V ¼K 00 In�r

� �: ð3Þ

Corresponding to (3), let


~Ai ¼ ~UAi~V ¼

~Ai11~Ai12

~Ai21~Ai22

" #; i 2M=f1g;

~Bi ¼ ~UBi~V ¼

~Bi11~Bi12

~Bi21~Bi22

" #; i 2M;

~xðtÞ ¼ ~V�T xðtÞ ¼ ~xT1ðtÞ ~xT

2ðtÞ� T

;

ð4Þ

with ~x1ðtÞ 2 Rr and ~x2ðtÞ 2 Rn�r . Then, the ith subsystem of (1) is restricted system equivalent to:

_~x1ðtÞ ¼ K~x1ðtÞ þ ~B111~x1ðt � hðtÞÞ þ ~B112~x2ðt � hðtÞÞ; ð5Þ0 ¼ ~x2ðtÞ þ ~B121~x1ðt � hðtÞÞ þ ~B122~x2ðt � hðtÞÞ ð6Þ

and for i 2M=f1g,
_~x1ðtÞ ¼ ~Ai11~x1ðtÞ þ ~Ai12~x2ðtÞ þ ~Bi11~x1ðt � hðtÞÞ þ ~Bi12~x2ðt � hðtÞÞ; ð7Þ0 ¼ ~Ai21~x1ðtÞ þ ~Ai22~x2ðtÞ þ ~Bi21~x1ðt � hðtÞÞ þ ~Bi22~x2ðt � hðtÞÞ: ð8Þ
Due to ~Ai22 is non-singular. Now consider the algebraic equations (6) and (8). It is clear that the impulses occur if the alge-braic equations are not consistent. In contrast, switching in the state space where all the m algebraic equations are satisfiedwill not result in impulses. That is,

~B121~x1ðt � hðtÞÞ þ ~B122~x2ðt � hðtÞÞ ¼ ~A�1i22

~Ai21~x1ðtÞ þ ~A�1i22

~Bi21~x1ðt � hðtÞÞ þ ~A�1i22

~Bi22~x2ðt � hðtÞÞ:

This implies that ~Ai21 is singular, ~B121 ¼ ~A�1i22

~Bi21, and ~B122 ¼ ~A�1i22

~Bi22 for all i 2M=f1g, which provides an easy-to-check condi-tion for switching-impulse-free.

Proposition 3.2. Consider SDDS (1), if each subsystem of SDDS (1) is regular and impulsive-free. Moreover, for all i 2M=f1g; ~Ai21

is singular, ~B121 ¼ ~A�1i22

~Bi21, and ~B122 ¼ ~A�1i22

~Bi22, where ~Ai21;~Ai22; ~B121; ~B122; ~Bi21, and ~Bi22 are defined by (4). Then SDDS (1) is

switching-impulsive-free.This result ensures Assumption 3.1 holds. For non-delayed switched descriptor system E _x ¼ AixðtÞ ði 2MÞ, paper [22]

showed that the singularity of matrix ~Ai21 can guarantee its switching-impulsive-free property. This is a special case of Prop-osition 3.2. Therefore, the above result is a natural generalization of the result presented by [22]. In contrast, also for SDDSs,[19,40,21,24,23,26,27] fail to consider the switching impulse phenomenon or directly assume that the SDDSs do not existswitching impulses. In other word, the existed results did not present some criteria to decide the existence of the switchingimpulses. When the switching impulses occur, the stability criteria in [19,40,21,24,23,26,27] become invalid.

Example 1. As an illustrative example, consider SDDS (1) composed of two subsystems with

E ¼1 00 0

� �; A1 ¼

�9 00 �5

� �; B1 ¼

�2 0�5=3 �5=6

� �;

A2 ¼�3 40 �6

� �; B2 ¼

2 �1�2 �1

� �:

First, choose c ¼ 2 and c1 ¼ c2 ¼ 4, moreover, choose R1 ¼ ½0 1�T and R2 ¼ ½0 2�T associated the structure of E, by solving LMIs(2) in P1 and P2, a feasible solution is

P1 ¼ P2 ¼35:4427 0

0 35:4427

� �:

Each subsystem of SDDS (1), from Proposition 3.1, is thus regular-impulsive-free. Based on this, we choose non-singularmatrices ~U ¼ I and ~V ¼ diagf1;�1=5g such that condition (3) is satisfied. Moreover, by a simply computation we get

~A2 ¼�3 �4=50 6=5

� �; ~B1 ¼

�2 0�5=3 1=6

� �; ~B2 ¼

2 1=5�2 1=5

� �:

This gives that ~A221 ¼ 0, ~B121 ¼ ~A�1222

~B221 ¼ �5=3, and ~B122 ¼ ~A�1222

~B222 ¼ 1=6. Finally, by Proposition 3.2, SDDS is switching-impulsive-free.

4. Uniform exponential admissibility

In this section, we will devote to derive the uniform exponential admissibility of SDDS (1) with the help of the regularity-impulsiveness-free and switching-impulsiveness-free criteria given in above section. Firstly, we introduce the definition ofuniform exponential admissibility.


Definition 4.1. SDDS (1) is said to be uniformly exponentially admissible over SminðsDÞ (or Sall) if the following conditionsare satisfied:

(i) Each subsystem of SDDS (1) is regular-impulsive-free, i.e., Proposition 3.1 holds.(ii) SDDS (1) is switching-impulsive-free, i.e., Assumption 3.1 is true.

(iii) SDDS (1) is exponentially stable, i.e., there exist positive scalars b and d0 such that, for any compatible initial conditionuðhÞ and every switching signal rðtÞ 2 SminðsDÞ (or rðtÞ 2 Sall), the solution xðtÞ of (1) satisfies:

kxðtÞk 6 bkukhe�d0ðt�t0Þ; t P t0: ð9Þ

To accomplish the aim, a new piecewise Lyapunov functional, for each mode, shall be employed as the form:

WðtÞ ¼ edtvðtÞxTðtÞET Pik ExðtÞ þ edtZ t

t�hðtÞxTðsÞQxðsÞds; ð10Þ

where d > 0 is a given sufficiently small constant, Pik ;Q 2 Rn�n are positive definite matrices for all ik 2M. The function vðtÞis formulated by the following manner.

Firstly, consider matrices Pik�1and Pik . By Lemma 2.2 there exists a real number cik

P 1 such that

xTðtÞPik xðtÞ 6 cikxTðtÞPik�1

xðtÞ: ð11Þ

With this cik, we further define a function:

vkðtÞ ¼c

ðtkþ1 � tkÞ21� 1

cik

!ðt � tkÞ2 þ

ccik

; t 2 ½tk; tkþ1Þ; ð12Þ

where c > 0. Now, a piecewise continuously function vðtÞ : ½0;þ1Þ ! ½0;þ1Þ is defined as:

vðtÞ ¼vkðtÞ; t 2 ðtk; tkþ1Þ;vðtþk Þ ¼ vkðtkÞ; t ¼ tk:

�ð13Þ

Remark 4.1. Consider the function vðtÞ given by (13), for each interval ½tk; tkþ1Þ, two facts are checked easily:

(a) vðtkÞ ¼ c=cikand vðt�kþ1Þ ¼ c.

(b) vðtÞ and _vðtÞ are monotone and bounded, respectively, i.e., vðtkÞ 6 vðtÞ 6 vðt�kþ1Þ and 0 6 _vðtÞ 6 ð2c=sDÞð1� 1=cikÞ.

In (10), if vðtÞ ¼ 1;WðtÞ reduces to the general piecewise Lyapunov functional, and thus has a ‘‘jump’’ phenomenon at theswitching instant t ¼ tk. Indeed, in this case, it follows from (10) and (11) that WðtkÞ 6 cik

edtxTðtÞET Pik�1ExðtÞþ

edtR t

t�hðtÞ xTðsÞQxðsÞds 6 cikWðt�k Þ. Therefore, the role of vðtÞ is to overcome this ‘‘jump’’ phenomenon.

Lemma 4.1. The piecewise Lyapunov functional WðtÞ given by (10) is non-increasing at the switching times t ¼ tk, i.e.,WðtkÞ 6Wðt�k Þ.

Proof. At the switching times t ¼ tk, according to Remark 4.1, one can verify that

WðtkÞ ¼ edtk vðtkÞxTðtkÞET Pik ExðtkÞ þ edtk

Z tk

tk�hðtkÞxTðsÞQxðsÞds 6 edtkcik

vðtkÞxTðtkÞET Pik�1ExðtkÞ þ edtk

Z tk

tk�hðtkÞxTðsÞQxðsÞds

¼ edtk cxTðt�k ÞET Pik�1

Exðt�k Þ þ edtk

Z t�k

t�k�hðt�

kÞxTðsÞQxðsÞds ¼ edt�

k vðt�k ÞxTðt�k ÞET Pik�1

Exðt�k Þ þ edt�k

Z t�k

t�k�hðt�

kÞxTðsÞQxðsÞds

¼Wðt�k Þ:

That is, WðtÞ is non-increasing at switching times t ¼ tk. This completes the proof. h

Remark 4.2. This conclusion implies that the delicately constructed piecewise Lyapunov functional (10) can efficiently elim-inate the ‘‘jump’’ phenomena between adjacent Lyapunov functionals. This functional will play a important role in derivingthe delay-independent criterion by minimum dwell time technique. In contrast, if the general Lyapunov functionals areemployed (see, for example [21,24,26]), to overcome the ‘‘jump’’ phenomena, we have to resort to the so-called averagedwell time approach. However, the usage of this approach must be subjected to the delay bounds. That is to say, the stabilitycriteria are delay-dependently by general Lyapunov functional technique.

Based on the discussion above, we now derive the uniform exponential admissibility of SDDS (1).


Theorem 4.1. Consider SDDS (1), suppose that Assumption 2.1 and 3.1 hold. Furthermore, for any i 2M, given positive realnumbers sD; d, and ci with ci P 1 and 1 6 c 6 ci þ 1, if there exist positive definite matrices Pi;Q ;Gi; Lai; Lbi, Hai;Hbi 2 Rn�n, andany matrices Ti 2 Rn�2n; Si 2 Rn�ðn�rÞ such that (2) and the following linear matrix inequalities are satisfied:

Xi ¼�Gi �ET Ti

� X22i

" #< 0; ð14Þ

where Li;Hi;Ri are defined in Proposition 3.1, and

X22i ¼ cETLi þ cLiE �cciETHi �

cciHiE þ I T

1UiI1 � ð1� dÞe�dI T2QI2;

Ui ¼2csD

1� 1ci

� �ET PiE�

c2

ciGi þ Q :

Then SDDS (1) is uniformly exponentially admissible over Smin.

Proof. The proof process is too long and thus is moved in A.

Remark 4.3. By a direct expansion, it easily checks that the conditions (2) and (14) can be replaced by the following twoinequalities:

!i ðET Pi þ SiRTi ÞBi þ ET Ti2

� �2Lbi þ 2Hbi

" #< 0

and

�Gi �ET Ti1 �ET Ti2

� Ci 0� � 2cLbi � 2c

ciHbi � ð1� dÞe�dQ

264

375 < 0;

where !i ¼ ðET Pi þ SiRTi ÞAi þ AT

i ðPiEþ RiSTi Þ � ET Lai � LaiEþ ET Hai þ HaiEþ ðc þ c

ciÞGi þ ET Ti1 þ TT

i1E and Ci ¼ cET Lai þ cLaiE�cciðET Hai þ HaiEÞ þ 2c

sDð1� 1

ciÞET PiE� c2

ciGi þ Q .

The above result establishes on the minimum dwell time set Smin. If choosing ci ¼ 1, the set Smin can be extended to Sall.This maybe greatly improve the design of switching laws.

Corollary 4.1. Consider SDDS (1), under Assumption 2.1 and 3.1, for any i 2M, given a positive real number d, if there existpositive definite matrices Pi, Q ;Gi;2 Rn�n, and any matrices Ti1; Ti2 2 Rn�n; Si 2 Rn�ðn�rÞ such that

�!i ðET Pi þ SiRTi ÞBi þ ET Ti2

� �2Lbi þ 2Hbi

" #< 0 ð15Þ

and

�Gi �ET Ti1 �ET Ti2

� �Ci 0� � 2Lbi � 2Hbi � ð1� dÞe�dQ

264

375 < 0; ð16Þ

where Ri is defined in Proposition 3.1, �!i ¼ ðET Pi þ SiRTi ÞAi þ AT

i ðPiEþ RiSTi Þ � ET Lai � LaiEþ ET Hai þ HaiEþ 2cGi þ ET Ti1 þ TT

i1E, and�Ci ¼ cET Lai þ cLaiE� cET Hai � cHaiE� c2Gi þ Q. Then the SDDS (1) is uniformly exponentially admissible over Sall.

Remark 4.4. Notice that, the method presented in this paper is not invalid for the general case of the singular matrix E hav-ing switching mode, i.e., E is changed to Ei; i 2 M. Since there does not exist one common state space coordinate basis forevery different subsystems. However, it is easy to check that our method is applicable for some special cases, such as, fori 2M

(i) All matrices Ei are not singular, i.e., rankðEiÞ ¼ n, especially, Ei � I. In this case, SDDS (1) reduces to the generalswitched delayed systems.

(ii) There exist nonsingular matrices Ui;V 2 Rn�n such that UiEiV ¼ diagfIr ;0g; r 6 n.(iii) NðE1Þ ¼ � � � ¼ N ðEmÞ, where NðEiÞ denotes the right zero subspace of Ei.

In this end of this section, we give a illustrative example to show the effectiveness of the proposed results.

Table 1Table 1 Comparison of allowable bounds of dwell time and delays in Example 2.

Methods c1 ¼ c2 h Dwell times

Theorem 1 [21] 2 6.72 6.9315Theorem 1 [23] 2 119:9 89.0216Theorem 4.1 2 h P 0 0.6330Theorem 1 [21] 1.1 6.72 0.9531Theorem 1 [23] 1.1 128.0 84.4180Theorem 4.1 1.1 h P 0 0.6330Theorem 1 [21] 1 Infeasible InfeasibleTheorem 1 [23] 1 Infeasible InfeasibleCorollary 4.1 1 h P 0 sD > 0

Table 2Table 2 Comparison of allowable bounds of dwell time and delay in Example 3.

Methods c1 c2 h Dwell times

Theorem 1 [42] 80.15 1.23 20.25 14.79Theorem 1 [41] 80.15 1.23 h P 0 14.79Theorem 4.1 80.15 1.23 h P 0 4.845


Example 2. Consider SDDS (1) given in Example 1. According to the structure of E, we can choose R1 ¼ ½0 1�T and R2 ¼ ½0 2�T .For d ¼ 0:1; d ¼ 0:1; c ¼ 2, by employing our Theorem 4.1, the obtained lower bounds of the minimum dwell time and theupper bounds of the delay are showed in Table 1 for different ci ði ¼ 1;2Þ. Also, by the results of [21,23] (here the nonlinearfunctions and the lower bound of delays in [23] are chosen to 0 and 0:1, respectively), we found the corresponding lowerbounds of the so-called average dwell time and the delay bounds. From Table 1, one can see that, when ci > 1 ði ¼ 1;2Þ, ourminimum dwell time bounds are less than the results of [21,23]. In addition, our results hold for any bounded delays. Incontrast, in [21,23], the dwell time bounds are restricted by the delay sizes. This implies that, for example, if c1 ¼ c2 ¼ 1:1and the dwell time 0:9531 in [21] are fixed, the result in [21] cannot guarantee the uniform exponential admissibility ofSDDS (1) with time delay hðtÞ > 6:72. Therefore, it is clear that our Theorem 4.1 provides a less conservative criterion forstability. Now for a special case when ci ¼ 1 ði ¼ 1;2Þ, Obviously, vðtÞ � 2, the Lyapunov functional (10) reduces to thetraditional functional. From Corollary 4.1, SDDS (1) is uniformly exponentially admissible over Sall. See the last two lines inTable 1. However, in this case, Theorem 1 in [21] and Theorem 1 in [23] do not work.

Example 3. Consider SDDS (1) composed two subsystems with E ¼ I and the following parameters which are borrowed from[41]

A1 ¼0 1�10 �1

� �; B1 ¼

0:1 0�0:01 0:05

� �; A2 ¼

0 1�0:1 �0:5

� �; B2 ¼

0:02 0�0:01 0:02

� �:

For d ¼ 0:1; c ¼ 1:09; d ¼ 0:1, Table 2 shows if we choose the same dwell time, the result of [41] is less conservative thanthe result of [42] according to the delay bounds. In addition, in the case of any bounded delay, the lower bound of minimumdwell time is much less than it in [41], and Theorem 4.1 thus gives a better criterion for uniform exponential stability ofswitched delayed systems.

5. Conclusions

The uniform exponential admissibility problem for switched descriptor systems with bounded time-varying delays isstudied in this paper. To ensure the existence and uniqueness of each subsystem of SDDSs, a sufficient condition for regu-larity-impulsiveness-free was presented. Also, to avoid the state impulse caused by switching, we answer the question howthe switching should be done so that no impulse for SDDSs occurs. Under the regularity-impulsiveness-free and switching-impulsiveness-free, the uniform exponential admissibility criteria are presented via the piecewise Lyapunov functionalmethods. This criteria only depend on the minimum dwell time, irrespective the sizes of time-varying delays.

Appendix A. Proof of Theorem 4.1

By Proposition 3.1, SDDS (1) is regular and impulsive-free. Now define


U ¼ U;V ¼ VIr 0

�A�1i22Ai21 A�1

i22

� �;

where U; V ; Ai 2 Rn�n are defined in Proposition 3.1. Obviously, U;V are non-singular since Ai22 (i 2M) is non-singular. More-over, it easily checks that for i 2M

�E ¼ UEV ¼Ir 00 0

� �; �Ai ¼ UAiV ¼

�Ai11�Ai12

0 In�r

" #;

where �Ai11 ¼ Ai11 � Ai12A�1i22Ai21; �Ai12 ¼ Ai12A�1

i22. Correspondingly, for i 2M; l 2 fa; bg, denote

�Bi ¼ UBiV ¼�Bi11

�Bi12

�Bi21�Bi22

" #; �Pi ¼ U�T PiU

�1 ¼�Pi11

�Pi12

�Pi21�Pi22

" #; �Q ¼ VT QV ¼

�Q 11�Q 12

�Q 21�Q 22

" #;

�Lli ¼ U�T LliV ¼�Lli11

�Lli12

�Lli21�Lli22

" #; �Ri ¼ U�T Ri ¼

0�Zi

� �; �Si ¼ VT Si ¼

�Si11

�Si21

" #�Hli ¼ U�T HliV ¼

�Hli11�Hli12

�Hli21�Hli22

" #;

�Gi ¼ VT GiV ¼�Gi11

�Gi12

�Gi21�Gi22

" #; �Ti ¼ U�T TidiagfV ; Vg ¼ ½�Ti11; �Ti12�:

Furthermore, set

�xðtÞ ¼ �V�T xðtÞ ¼ �xT1ðtÞ �xT

2ðtÞ� T

;

with �x1ðtÞ 2 Rr and �x2ðtÞ 2 Rn�r . Then the SDDS (1) can be transformed into an equivalent system

�E _�xðtÞ ¼ �ArðtÞ�xðtÞ þ �BrðtÞ�xðt � hðtÞÞ;�xðhÞ ¼ �V�TuðhÞ ¼ �uðhÞ; h 2 ½�h; 0�:

ðA:1Þ

The ith subsystem can further be rewritten as

_�x1ðtÞ ¼ �Ai11�x1ðtÞ þ �Ai12�x2ðtÞ þ �Bi11�x1ðt � hðtÞÞ þ �Bi12�x2ðt � hðtÞÞ; ðA:2Þ0 ¼ �x2ðtÞ þ �Bi21�x1ðt � hðtÞÞ þ �Bi22�x2ðt � hðtÞÞ:: ðA:3Þ

Next, we shall present the uniform exponential admissibility of (1) through showing the exponential stability of the dif-ferential system with (A.2) and the algebraic system with (A.3), respectively. The rest of the proof is broken into two steps.

Step 1: We shall show that the differential system with (A.2) is exponentially stable.Choose the piecewise Lyapunov functional WðtÞ defined in (10). Firstly, we wish to show that the upper right-hand deriv-

ative of WðtÞ satisfies:

DþWðtÞ � dWðtÞ < 0; t 2 ½tk; tkþ1Þ: ðA:4Þ

To this end, consider the case t 2 ½tk; tkþ1Þ, by applying Remark 4.1, along SDDS (1) we obtain

DþWðtÞ � dWðtÞ 6 edt ½xTðtÞQxðtÞ � ð1� dÞxTðt � hðtÞÞe�dQxðt � hðtÞÞ�

þ 2edtvðtÞðExðtÞÞT Pik Aik xðtÞ þ Bik xðt � hðtÞÞ �

þ 2edt csD

1� 1cik

!xTðtÞET Pik ExðtÞ: ðA:5Þ

Due to ET Rik ¼ 0, for any matrix Sik 2 Rn�n�r , one can know that

0 ¼ 2vðtÞ _xTðtÞET Rik STik

xðtÞ: ðA:6Þ

Substituting (A.6) into (A.5) yields that

DþWðtÞ � dWðtÞ 6 edtvðtÞxTðtÞ ðET Pik þ Sik RTikÞAik þ AT

ikðPik Eþ Rik ST

ikÞ

h i� xðtÞ þ 2edtvðtÞxTðtÞðET Pik þ Sik RT

ikÞBik xðt � hðtÞÞ

� edtð1� dÞxTðt � hðtÞÞe�dQxðt � hðtÞÞ þ edtxTðtÞ 2csD

1� 1cik

!ET Pik Eþ Q

" #xðtÞ:

Now, define nTðtÞ ¼ ½xTðtÞ xTðt � hðtÞÞ�; fTðtÞ ¼ ½vðtÞxðtÞT nTðtÞ�. Then, the above expression can be rewritten as:

DþWðtÞ � dWðtÞ 6 edtxTðtÞ 2csD

1� 1cik

!ET Pik Eþ Q

" #xðtÞ � edtð1� dÞxTðt � hðtÞÞe�dQxðt � hðtÞÞ þ edtvðtÞnTðtÞ½I T

1

� ðET Pik þ Sik RTikÞAik þ A

TikðPik Eþ Rik ST

ikÞI1�nðtÞ: ðA:7Þ

By Remark 4.1, for any positive definite matrices Llik ;Hlik ;Gik 2 Rn�n, ðl ¼ a; bÞ, the following two inequalities obviously holds:


ðc � vðtÞÞnTðtÞðETLik þ LikEÞnðtÞ þ vðtÞ � ccik

!nTðtÞðETHik þ HikEÞnðtÞP 0; ðA:8Þ

ðc � vðtÞÞ vðtÞ � ccik

!xTðtÞGik xðtÞP 0: ðA:9Þ

Furthermore, together with (A.7), (A.8), and (A.9) yields that:

DþWðtÞ � dWðtÞ 6 edtvðtÞxTðtÞð�Gik ÞvðtÞxðtÞ þ edtðd� 1ÞxTðt � hðtÞÞe�dQxðt � hðtÞÞ þ edtnTðtÞ cðETLik þ Lik EÞ �ccik

ðETHik þ Hik EÞ" #

nðtÞ

þ edtxT ðtÞ 2csD

1� 1cik

!ET Pik Eþ Q � c2

cik

Gik

" #xðtÞ

þ edtvðtÞnT ðtÞ I T1ðE

T Pik þ Sik RTikÞAik þ A

TikðPik Eþ Rik ST

ikÞI1 � ETLik � Lik E þ E

THik þ Hik E þ IT1ET Tik þ TT

ikEI1

h inðtÞ

þ edtvðtÞxTðtÞ c þ ccik

!Gik xðtÞ � 2edtvðtÞxTðtÞET Tik nðtÞ

¼ edtfTðtÞXik fðtÞ þ edtvðtÞnTðtÞHik nðtÞ:

Applying (2) and (14), it follows that, for t 2 ½tk; tkþ1Þ,

DþWðtÞ � dWðtÞ 6 edtfTðtÞXik fðtÞ þ edtvðtÞnTðtÞHik nðtÞ < 0: ðA:10Þ

We thus conclude that (A.4) holds.Now set

k1 ¼mini2Mfkminð�XiÞg > 0; k2 ¼ min

i2Mfkminð�HiÞg > 0; c ¼max

i2Mfcig:

By Remark 4.1, we can induce from (A.10) that

DþWðtÞ � dWðtÞ 6 �k1edtkfðtÞk2 � ck2

cedtknðtÞk2

6 � k1 þc2k1

c2 þck2

c

� �edtkxðtÞk2 � k1 þ

ck2

c

� �edtkxðt � hðtÞÞk2

6 � k1 þc2k1

c2 þck2

c

� �edtkxðtÞk2

; t 2 ½tk; tkþ1Þ:

Note that �xðtÞ ¼ V�1xðtÞ, we further have

DþWðtÞ � dWðtÞ 6 � k1 þc2k1

c2 þck2

c

� �edtkxðtÞk2

6 � k1 þc2k1

c2 þck2

c

� �edtkVkk�xðtÞk2

; t 2 ½tk; tkþ1Þ:

Set �k ¼ ðk1 þ c2k1c2 þ ck2

c ÞkVk, the above expression can be rewritten as

DþWðtÞ � dWðtÞ 6 ��kedtk�xðtÞk2; t 2 ½tk; tkþ1Þ: ðA:11Þ

On the other hand, from (10), we have

WðtÞ ¼ edtvðtÞxTðtÞET Pik ExðtÞ þ edtZ t

t�hðtÞxTðsÞQxðsÞds;¼ edtvðtÞ�xTðtÞVT ET Pik EV�xðtÞ þ edt

Z t

t�hðtÞ�xTðsÞVT QV�xðsÞds;

¼ edtvðtÞ�xTðtÞ�ET �Pik�E�xðtÞ þ edt

Z t

t�hðtÞ�xTðsÞ�Q�xðsÞds;¼ edtvðtÞ�xT

1ðtÞ�Pik11�x1ðtÞ þ edtZ t

t�hðtÞ�xTðsÞ�Q�xðsÞds:

Moreover, by Remark 4.1, there exist positive real numbers jj ðj ¼ 1;2;3Þ such that, for t 2 ½0;þ1Þ,

j0edtk�x1ðtÞk26WðtÞ 6 j1edtk�xðtÞk2 þ j2edt

Z t

t�hk�xðsÞk2ds: ðA:12Þ

Choose d sufficiently small such that

�k P dðj1 þ j2hedhÞ: ðA:13Þ

Then, combining (A.11), (A.12), and (A.13), it follows that, for t 2 ½tk; tkþ1Þ,

DþWðtÞ 6 edt dj1 � �k �

k�xðtÞk2 þ dj2

Z t

t�hk�xðsÞk2ds

� �: ðA:14Þ

Now integrating both sides of (A.14) from tk to t gives


WðtÞ 6WðtkÞ þZ t

tk

eds dj1 � �k �

k�xðsÞk2 þ dj2

Z s

s�hk�xðhÞk2dh

� �ds:

Applying Lemma 4.1 and (A.12), by directly deducing we further get, for t 2 ½0;1Þ,

j0edtk�x1ðtÞk26WðtÞ 6Wð0Þ þ

Z t

0eds dj1 � �k

�k�xðsÞk2 þ dj2

Z s

s�hk�xðhÞk2dh

� �ds: ðA:15Þ

Observe that
Z t
0edsds

Z s

s�hk�xðhÞk2dh 6 hedh

Z t

0edsk�xðsÞk2dsþ hedh

Z 0

�hedsk�xðsÞk2ds:

Therefore, it follows from (A.13) and (A.15) that

j0edtk�x1ðtÞk26Wð0Þ þ dj2hedh

Z 0

�hedsk�xðsÞk2ds:

It turns out that, for t 2 ½0;1Þ,

k�x1ðtÞk 6 b1k �ukhe�d1ðt�t0Þ: ðA:16Þ

with

d1 ¼d2;b1 ¼

j1 þ j2hþ dj2h2edh

j0

!12

:

Therefore, the differential system with (A.2) is exponentially stable.Step 2: Now it remains to show that the algebraic system with (A.3) is exponentially stable. For i 2M, it follows from (2)

and (14) that Hi þXi22 < 0. Pre- and post-multiplying by diag fVT ;VTg and its transpose, respectively, then one can obtain

D ¼ D11�Si21

�ZTi�Bi22

� D22

" #< 0;

with D11 ¼ �Q22 þ �Si21�ZT

i þ �Zi�ST

i21 þ ðc þ cci� c2

ciÞ�Gi22;D22 ¼ ðd� 1Þe�d �Q22 þ 2ðc � 1Þ�Lbi22 þ 2ð1� c

ciÞ�Hbi22. Also, pre- and post-mul-

tiplying the above inequality by ½��BTi22 I� and its transpose, respectively, and note that �Gi22 > 0; �Q i22 > 0; d > 1; ci P 1

and 1 6 c 6 ci, it follows that

�BTi22

�Q 22�Bi22 � e�d �Q 22 < 0:

By Lemma 2.3, there exist positive real numbers xi P 1 and �i such that

kðe13d�Bi22Þ

rk 6 xie��ir ; r > 0: ðA:17Þ

Now set

t0 ¼ t; tl ¼ tl�1 � hðtl�1Þ; l ¼ 1;2; . . .

Obviously, tl�1 P tl for any l, there exists a finite positive integer Ki0 such that tKi0 2 ½�h; t0�. Furthermore, setting t 2 ½tk; tkþ1Þand K ¼ f0;1; . . . ;Ki0g, we now, according to the time interval, classify the time points tlðl ¼ 0;1;2; . . .Þ as follows:

ftKikþ1 ; tKikþ1þ1; . . . ; tKik

�1g# ½tk; tkþ1Þ;ftKik ; tKik

þ1; . . . ; tKik�1�1g# ½tk�1; tkÞ;

..

.

ftKi1 ; tKi1þ1; . . . ; tKi0

�1g# ½t0; t1Þ;

where tKikþ1 ¼ t0 and Kis 2 K for s ¼ 1; . . . ; kþ 1. (Notice that, when the size of time delays are larger than the length of timeintervals, some time intervals maybe not include any tl; l 2 K. However, in this case, the following discussion is alsoapplicable.)

Now, recalling the algebraic system with (A.3), we first consider the interval ½tk; tkþ1Þ. In this case, the ikth subsystem isactivated. It follows from (A.3) that

�x2ðtÞ ¼ �x2ðt0Þ ¼ ��Bik21�x1ðt1Þ � �Bik22�x2ðt1Þ:

That is,

�x2ðtKikþ1 Þ ¼ ��Bik21�x1ðtKikþ1þ1Þ � �Bik22�x2ðtKikþ1

þ1Þ: ðA:18Þ


Similarly, we have

�x2ðtKikþ1þ1Þ ¼ ��Bik21�x1ðtKikþ1

þ2Þ � �Bik22�x2ðtKikþ1þ2Þ;

..

.

�x2ðtKik�1Þ ¼ ��Bik21�x1ðtKik Þ � �Bik22�x2ðtKik Þ:

ðA:19Þ

Combining (A.18) and (A.19) and by a simple induction, one can obtain that

�x2ðtÞ ¼ �x2ðtKikþ1 Þ ¼ ð��Bik22ÞKik�Kikþ1 �x2ðtKik Þ �

XKik�Kikþ1

l¼1

ð��Bik22Þl�1�Bik21�x1ðtKikþ1

þlÞ:

Moreover, for ½tk�1; tkÞ, the ik�1th subsystem is activated. By an identical discussion above, we can derive that

�x2ðtKik Þ ¼ ð��Bik�122ÞKik�1

�Kik �x2ðtKik�1 Þ �XKik�1�Kik

l¼1

ð��Bik�122Þl�1�Bik�121�x1ðtKik

þlÞ:

Repeating this process until ½t0; t1Þ, the i0th subsystem is activated. Then we get

�x2ðtKi1 Þ ¼ ð��Bi022ÞKi0�Ki1 �x2ðtKi0 Þ �

XKi0�Ki1

l¼1

ð��Bi022Þl�1�Bi021�x1ðtKi1

þlÞ:

Based on the discussion above, for the whole time interval ½�h; t�, applying a simple induction, it turns out that

�x2ðtÞ ¼ �x2ðt0Þ ¼ �x2ðtKikþ1 Þ ¼ g1 þ g2 þ g3; ðA:20Þ

where

g1 ¼Yk

s¼0

ð��Bis22ÞKis�Kisþ1 �x2 tKi0

� ;

g2 ¼ �XKik

l¼1

ð��Bik22Þl�1�Bik21�x1ðtlÞ;

g3 ¼ �Xk

p¼1

Yk

s¼p

ð��Bis22ÞKis�Kisþ1

XKip�1�Kip

l¼1

ð��Bip�122Þl�1�Bip�121�x1ðtKipþlÞ

24

35:

Finally, we shall estimate the exponential decay of �x2ðtÞ given by (A.20). To begin with, define

k�B21k ¼max8i2Mk�Bi21k:

We first consider the expression g1. Since t0 P tKi0 ¼ t �PKi0

s¼1hðts�1ÞP t � Ki0 h, this implies that Ki0 P ðt � t0Þ=h. By this,from (A.17) there exist positive real numbers xis and �is ðs 2 f0; . . . ; kgÞ such that

kg1k 6Yk

s¼0

kð�Bis22ÞKis�Kisþ1 kk�x2 tKi0

� k 6

Yk

s¼0

kðe13d�Bis22Þ

Kis�Kisþ1 kk�ukhe�13dKi0 6

Yk

s¼0

xis e��is ðKis�Kisþ1

Þk�ukhe�1

3hdðt�t0Þ: ðA:21Þ

Note that l P ðt � tlÞ=h results from tl ¼ t �Pl

s¼1hðts�1ÞP t � lh. Furthermore, for the term g2, applying (A.16) and (A.17)gives that

kg2k 6 k�B21kXKik

l¼1

kð�Bik22Þl�1kk�x1ðtlÞk ¼ k�B21k

XKik

l¼1

e�13dðl�1Þkðe1

3d�Bik22Þl�1kk�x1ðtlÞk 6 xik b1k�B21k

XKik

l¼1

eð13dþ�ik

Þð1�lÞk�ukhe�d1ðtl�t0Þ

6 xikb1k�B21kXKik

l¼1

eð13dþ�ik

Þe�1hð

13dþ�ik

Þðt�tlÞk�ukhe�d1ðtl�t0Þ;

where xik and �ik are any positive real numbers. Setting d0 ¼minf1h ð13 dþ �ik Þ; d1g, we further obtain

kg2k 6 xikb1k�B21kKik eð13dþ�ik

Þk �ukhe�d0 ðt�t0Þ: ðA:22Þ

Now, consider g3, due to tKipþl ¼ t �PKipþl

s¼1 hðts�1ÞP t � ðKip þ lÞh, then it follows that l P 1h ðt � tKipþlÞ � Kip . Moreover, by the

similar argument above, one can make the following estimation:


kg3k 6 k�B21kXk

p¼1

Yk

s¼p

kð�Bis22ÞKis�Kisþ1 k

XKip�1�Kip

l¼1

kð�Bip�122Þl�1kk�x1ðtKipþlÞk

6 k�B21kXk

p¼1

Yk

s¼p

e�13dðKis�Kisþ1

Þkðe13d�Bis22Þ

Kis�Kisþ1 k �XKip�1�Kip

l¼1

e�13dðl�1Þkðe1

3d�Bip�122Þl�1kk�x1ðtKipþlÞk

6 b1k�B21kXk

p¼1

Yk

s¼p

xis e�ð13dþ�is ÞðKis�Kisþ1

Þ �XKip�1�Kip

l¼1

xip�1 e�ð13dþ�ip�1

Þðl�1Þk �ukhe�d1ðtKipþl�t0Þ

6 b1k�B21kXk

p¼1

xip�1 eð13dþ�ip�1

Þð1þKip ÞYk

s¼p


Þ �XKip�1�Kip

l¼1

e�1hð

13dþ�ip�1

Þðt�tKipþlÞk �ukhe�d1ðt

Kipþl�t0Þ:

By setting d00 ¼ minf1h ð13 dþ �ik Þ; d1g, one can get

kg3k 6 b1k�B21kXk

p¼1


Þð1þKip Þ �Yk

s¼p


Þk �ukhe�d00 ðt�t0Þ: ðA:23Þ

Note that 2d1 ¼ d is a constant which does not depend on the t and k, it turns out from (A.21), (A.22) and (A.23) that

k�x2k 6 b2k �ukhe�d2ðt�t0Þ; ðA:24Þ

with d2 ¼ maxf 13h d; d0; d00g and

b2 ¼Yk

s¼0

xis e��is ðKis�Kisþ1

Þ þxik b1k�B21kKik eð13dþ�ik

Þ þ b1k�B21kXk

p¼1


Þð1þKip ÞYk

s¼p


Þ:

That is, the algebraic system with (A.3) is exponentially stable.Finally, taking Assumption 2.1, Assumption 3.1 and Proposition 3.1 into account, we thus conclude from (A.16) and (A.24)

that SDDS (1) is uniformly exponentially admissible over SminðsDÞ. This completes the proof.

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