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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 183729, 21 pages
http://dx.doi.org/10.1155/2012/183729
Research Article

Almost Sure Stability and Stabilization for Hybrid Stochastic Systems with Time-Varying Delays

1School of Information Science and Technology, Donghua University, Shanghai 200051, China
2College of Information Science and Engineering, Shanxi Agricultural University, Taigu 030801, China
3Department of Applied Mathematics, Donghua University, Shanghai 200051, China

Received 21 June 2012; Accepted 1 August 2012

Academic Editor: Bo Shen

Copyright © 2012 Hua Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The problems of almost sure (a.s.) stability and a.s. stabilization are investigated for hybrid stochastic systems (HSSs) with time-varying delays. The different time-varying delays in the drift part and in the diffusion part are considered. Based on nonnegative semimartingale convergence theorem, Hölder’s inequality, Doob’s martingale inequality, and Chebyshev’s inequality, some sufficient conditions are proposed to guarantee that the underlying nonlinear hybrid stochastic delay systems (HSDSs) are almost surely (a.s.) stable. With these conditions, a.s. stabilization problem for a class of nonlinear HSDSs is addressed through designing linear state feedback controllers, which are obtained in terms of the solutions to a set of linear matrix inequalities (LMIs). Two numerical simulation examples are given to show the usefulness of the results derived.

1. Introduction

In the past decades, the problems of stability analysis and stabilization synthesis of stochastic systems have received significant attentions, and many results have been reported; see, for example [17] and the references therein. Commonly, the above problems can be solved not only in moment sense [810] but also in a.s. sense [11, 12]. However, in recent years, much interest has been focused on a.s. stability problems for stochastic systems; see, for example [8, 13] and the references therein.

It is well known that a lot of dynamical systems have variable structures subject to abrupt changes in their parameters, which are usually caused by abrupt phenomena such as component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbances. The HSSs, which are regarded as the stochastic systems with Markovian switching in this paper, have been used to model the previous phenomena; see, for example [1418] and the references therein. The HSSs combine a part of the state 𝑥(𝑡) that takes values in 𝑛 continuously and another part of the state 𝑟(𝑡) that is a Markov chain taking discrete values in a finite space 𝑆={1,2,,𝑁}. One of the important issues in the study of HSSs is the analysis of stability. In particular, it is not necessary for the stable HSSs to require every subsystem to be stable; in other words, even all the subsystems are unstable; as the result of Markovian switching, the HSSs may be stable. These reveal that the Markovian jumps play an important role in the stability analysis of HSSs. Therefore, in the past few decades, a great deal of literature has appeared on the topic of stability analysis and stabilization synthesis of HSSs; see, for example [2, 13, 14, 19, 20].

On the other hand, time delays are frequently encountered in a variety of dynamic systems, such as nuclear reactors, chemical engineering systems, biological systems, and population dynamics models. They are often a source of instability and poor performance of systems. So the problems of stability analysis and stabilization synthesis of HSDSs have been of great importance and interest. The classical efforts can be classified into two categories, namely, moment sense criteria, see, for example [2123], and a.s. sense criteria, see, for example [24, 25]. Among the existing results, in [25], based on the techniques proposed in [26] which were developed via the results of [11], a.s. stability and stabilization of HSDSs were studied. In [24], the a.s. stability analysis problem for a general class of HSDSs was derived from extending the results in [25] to HSSs with mode-dependent interval delays. However, to the author’s best knowledge, when the different time-varying delays in the drift part and in the diffusion part are considered, the a.s. stability analysis and stabilization synthesis problems for nonlinear HSDSs have not been adequately addressed and remain an interesting and challenging research topic. This situation motivates the present study.

In this paper, we are concerned with a.s. stability analysis and stabilization synthesis problems for HSDSs. The purpose of stability is to develop conditions such that the underlying systems are a.s. stable. Following the same idea as in dealing with the stability problem, linear state feedback controllers are designed such that the special nonlinear or linear closed-loop systems are a.s. stable. The explicit expressions for the desired state feedback controllers are given by means of the solutions to a set of LMIs. Two numerical simulation examples are exploited to verify the effectiveness of the theoretical results. The main contribution of this paper is mainly twofold: (1) the different time-varying delays in the drift part and in the diffusion part are considered for nonlinear HSDSs; (2) for a class of nonlinear HSDSs, the stabilization synthesis problem is investigated in the a.s. sense.

This paper is organized as follows. In Section 2, we formulate some preliminaries. In Section 3, we investigate the a.s. stability for the hybrid stochastic systems with time-varying delays. In Section 4, the results of Section 3 are then applied to establish a sufficient criterion for the stabilization. In Section 5, two examples are discussed for illustration. Finally, conclusions are drawn in Section 6.

Notation 1. The notation used here is fairly standard unless otherwise specified. 𝑛 and 𝑛×𝑚 denote, respectively, the 𝑛 dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices, and let +=[0,+). (Ω,,{𝑡}𝑡0,) be a complete probability space with a natural filtration {𝑡}𝑡0 satisfying the usual conditions (i.e., it is right continuous, and 0 contains all -null sets). If 𝑥,𝑦 are real numbers, then 𝑥𝑦 stands for the maximum of 𝑥 and 𝑦, and 𝑥𝑦 the minimum of 𝑥 and 𝑦. 𝑀𝑇 represents the transpose of the matrix 𝑀. 𝜆max(𝑀) and 𝜆min(𝑀) denote the largest and smallest eigenvalue of 𝑀, respectively. || denotes the Euclidean norm in 𝑛. 𝔼{} stands for the mathematical expectation. {} means the probability. 𝐶([𝜏,0];𝑛) denotes the family of all continuous 𝑛-valued function 𝜑 on [𝜏,0] with the norm |𝜑|=sup{|𝜑(𝜃)|𝜏𝜃0}. 𝐶𝑏0([𝜏,0);𝑛) being the family of all 0-measurable bounded 𝐶([𝜏,0);𝑛)-value random variables 𝜉={𝜉(𝜃)𝜏𝜃0}. 𝐿1(+;+) denotes the family of functions 𝜆++ such that 0𝜆(𝑡)𝑑𝑡<.

2. Problem Formulation

In this paper, let 𝑟(𝑡),𝑡0 be a right-continuous Markov chain on the probability space taking values in a finite state space 𝑆={1,2,,𝑁} with generator Γ=(𝛾𝑖𝑗)𝑁×𝑁 given by 𝛾{𝑟(𝑡+Δ)=𝑗𝑟(𝑡)=𝑖}=𝑖𝑗Δ+𝑜(Δ)if𝑖𝑗,1+𝛾𝑖𝑖Δ+𝑜(Δ)if𝑖=𝑗,(2.1) where Δ>0 and 𝛾𝑖𝑗0 is the transition rate from mode 𝑖 to mode 𝑗 if 𝑖𝑗 while 𝛾𝑖𝑖=𝑗𝑖𝛾𝑖𝑗. Assume that the Markov chain 𝑟() is independent of the Brownian motion 𝐵(). It is known that almost all sample paths of 𝑟() are right-continuous step functions with a finite number of simple jumps in any finite subinterval of 𝑅+=[0,).

Let us consider a class of stochastic systems with time-varying delays: 𝑥𝑑𝑥(𝑡)=𝑓(𝑡),𝑥𝑡𝜏1𝑥(𝑡),𝑡,𝑟(𝑡)𝑑𝑡+𝑔(𝑡),𝑥𝑡𝜏2(𝑡),𝑡,𝑟(𝑡)𝑑𝐵(𝑡)(2.2) with initial data 𝑥0={𝑥(𝜃)𝜏𝜃0}=𝜉𝐶𝑏0([𝜏,0);𝑛) and 𝑟(0)=𝑟0𝑆, where 𝜏max{𝜏1,𝜏2}, 𝜏1 and 𝜏2 are positive constant and 𝜏1(𝑡) and 𝜏2(𝑡) are nonnegative differential functions which denote the time-varying delays and satisfy 0𝜏1(𝑡)𝜏1,̇𝜏1(𝑡)𝑑𝜏1<1,0𝜏2(𝑡)𝜏2,̇𝜏2(𝑡)𝑑𝜏2<1.(2.3) The nonlinear functions 𝑓𝑛×𝑛×+×𝑆𝑛 and 𝑔𝑛×𝑛×+×𝑆𝑛×𝑚 satisfy the local Lipschitz condition in (𝑥,𝑦,𝑧); that is, for any 𝐾>0, there is 𝐿𝐾>0 such that ||𝑓(𝑥,𝑦,𝑡,𝑖)𝑓𝑥,||||𝑔𝑦,𝑡,𝑖(𝑥,𝑧,𝑡,𝑖)𝑔𝑥,||𝑧,𝑡,𝑖𝐿𝐾||𝑥𝑥||+||𝑦𝑦||+||𝑧𝑧||,(2.4) for all |𝑥||𝑦||𝑧||𝑥||𝑦||𝑧|𝐾,𝑡0 and 𝑖𝑆, and moreover, sup𝑡0,𝑖𝑆{|𝑓(0,0,𝑡,𝑖)||𝑔(0,0,𝑡,𝑖)|𝑡0,𝑖𝑆}𝐾0 with some nonnegative number 𝐾0.

Remark 2.1. It should be pointed out that the systems (2.2) can be seen as the specialization of multiple time-varying delays systems which are of the form 𝑥𝑑𝑥(𝑡)=𝑓(𝑡),𝑥𝑡𝜏1(𝑡),𝑥𝑡𝜏2(𝑡),𝑡,𝑟(𝑡)𝑑𝑡+𝑔𝑥(𝑡),𝑥𝑡𝜏1(𝑡),𝑥𝑡𝜏2(𝑡),𝑡,𝑟(𝑡)𝑑𝐵(𝑡).(2.5) But it is easy to see that the results in this paper can be applied to the systems (2.5) by the similar assumption in (2.4).
Let 𝐶2,1(𝑛×+×𝑆;+) denote the family of all nonnegative functions 𝑉(𝑥,𝑡,𝑖) on 𝑛×+×𝑆 that are twice continuously differentiable in 𝑥 and once in 𝑡. If 𝑉𝐶2,1(𝑛×+×𝑆;+), define an operator associated with (2.2) from 𝑛×𝑛×𝑛×+×𝑆 to by 𝑉(𝑥,𝑦,𝑧,𝑡,𝑖)=𝑉𝑡(𝑥,𝑡,𝑖)+𝑉𝑥+1(𝑥,𝑡,𝑖)𝑓(𝑥,𝑦,𝑡,𝑖)2𝑔trace𝑇(𝑥,𝑧,𝑡,)𝑉𝑥𝑥+(𝑥,𝑡,𝑖)𝑔(𝑥,𝑧,𝑡,𝑖)𝑁𝑗=1𝛾𝑖𝑗𝑉(𝑥,𝑡,𝑗).(2.6)

Remark 2.2. 𝑉 is thought as a single notation and is defined on 𝑛×𝑛×𝑛×+×𝑆 while 𝑉 is defined on 𝑛×[𝜏,)×𝑆.

Definition 2.3. The system (2.2) is said to be a.s. stable if for all 𝜉𝐶𝑏0([𝜏,0);𝑛) and 𝑟0𝑆lim𝑡𝑥𝑡;𝜉,𝑟0=0=1.(2.7)

3. Main Results

Theorem 3.1. Assume that there exist nonnegative functions 𝑉𝐶2,1(𝑛×+×𝑆;+), 𝜆𝐿1(+;+), 𝜔1,𝜔2,𝜔3𝐶(𝑛;+) such that 𝑉(𝑥,𝑦,𝑧,𝑡,𝑖)𝜆(𝑡)𝑘1𝜔1(𝑥)+𝑘2𝜔2(𝑦)+𝑘3𝜔3(𝑧),(𝑥,𝑦,𝑧,𝑡,𝑖)𝑛×𝑛×𝑛×+𝜔×𝑆,(3.1)1(𝑥)>𝜔2(𝑥)+𝜔3(𝑥),𝑥0,(3.2)lim|𝑥|inf𝑡0,𝑖𝑆𝑉(𝑥,𝑡,𝑖)=,(3.3) where 𝑘1,𝑘2 and 𝑘3 are positive numbers satisfying 𝑘1max{𝑘2/(1𝑑𝜏1),𝑘3/(1𝑑𝜏2)}. Then system (2.2) is almost surely stable.

To prove this theorem, let us present the following lemmas.

Lemma 3.2 (see [24, 25]). If 𝑉𝐶2,1(𝑛×+×𝑆;+), then for any 𝑡0, the generalized Itô’s formula is given as 𝑥𝑑𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))=𝑉(𝑡),𝑥𝑡𝜏1(𝑡),𝑥𝑡𝜏2(𝑡),𝑡,𝑟(𝑡)𝑑𝑡+𝑉𝑥(𝑥(𝑡),𝑡,𝑟(𝑡))𝑔𝑥(𝑡),𝑥𝑡𝜏2(+𝑡),𝑡,𝑟(𝑡)𝑑𝐵(𝑡)[]𝑉(𝑥(𝑡),𝑡,𝑟(𝑡)+𝑙(𝑟(𝑡),𝛼))𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))×𝜇(𝑑𝑡,𝑑𝛼),(3.4) where function 𝑙(,) and martingale measure 𝜇(,) are defined as, for example, (2.6) and (2.7) in [25].

Lemma 3.3 (see [27]). Let 𝐴1(𝑡) and 𝐴2(𝑡) be two continuous adapted increasing processes on 𝑡0 with 𝐴1(0)=𝐴2(0)=0a.s., let 𝑀(𝑡) be a real-valued continuous local martingale with 𝑀(0)=a.s., and let 𝜁 be a nonnegative 0-measurable random variable such that 𝔼𝜁<. Denote 𝑋(𝑡)=𝜁+𝐴1(𝑡)𝐴2(𝑡)+𝑀(𝑡) for all 𝑡0. If 𝑋(𝑡) is nonnegative, then lim𝑡𝐴1(𝑡)<lim𝑡𝑋(𝑡)<lim𝑡𝐴2(𝑡)<a.s.,(3.5) where 𝐶𝐷a.s. means (𝐶𝐷𝑐=0)=0. In particular, if lim𝑡𝐴1(𝑡)<a.s., then, lim𝑡𝑋(𝑡)<,lim𝑡𝐴2(𝑡)<,<lim𝑡𝑀(𝑡)<a.s..(3.6)
That is, all of the three processes 𝑋(𝑡),𝐴2(𝑡), and 𝑀(𝑡) converge to finite random variables with probability one.

Lemma 3.4 (see [25]). Under the conditions of Theorem 3.1, for any initial data {𝑥(𝜃)𝜏𝜃0}=𝜉𝐶𝑏0([𝜏,0);𝑛) and 𝑟(0)=𝑖0𝑆, (2.2) has a unique global solution.

Proof. Fix any initial data 𝜉, 𝑟0, and let 𝛽 be the bound for 𝜉. For each integer 𝑘𝛽, define 𝑓(𝑘)(𝑥,𝑦,𝑡,𝑖)=𝑓|𝑥|𝑘||𝑦|||𝑥|𝑥,𝑘||𝑦||𝑦,𝑡,𝑖,(3.7) where we set (|𝑥|𝑘/|𝑥|)𝑥=0 when 𝑥=0. Define 𝑔(𝑘)(𝑥,𝑧,𝑡,𝑖) similarly. By (2.4), we can observe that 𝑓(𝑘) and 𝑔(𝑘) satisfy the global Lipschitz condition and the linear growth condition. By the known existence-and-uniqueness theorem, there exists a unique global solution 𝑥𝑘(𝑡) on 𝑡[𝜏,) to the equation 𝑑𝑥𝑘(𝑡)=𝑓(𝑘)𝑥𝑘(𝑡),𝑥𝑘𝑡𝜏1(𝑡),𝑡,𝑟(𝑡)𝑑𝑡+𝑔(𝑘)𝑥𝑘(𝑡),𝑥𝑘𝑡𝜏2(𝑡),𝑡,𝑟(𝑡)𝑑𝐵(𝑡)(3.8) with initial data {𝑥𝑘(𝜃)𝜏𝜃0}=𝜉 and 𝑟(0)=𝑟0.
Define the stopping time 𝜎𝑘||𝑥=inf𝑡0𝑘||(𝑡)𝑘,(3.9) where we set inf= as usual. It is easy to show that 𝑥𝑘(𝑡)=𝑥𝑘+1(𝑡) if 0𝑡𝜎𝑘, which implies that 𝜎𝑘 is increasing in 𝑘. Letting 𝜎=lim𝑘𝜎𝑘, the property above also enables us to define 𝑥(𝑡) for 𝑡[𝜏,𝜎) as 𝑥(𝑡)=𝑥𝑘(𝑡) if 𝜏𝑡𝜎𝑘.
It is clear that 𝑥(𝑡) is a unique solution of (2.2) for 𝑡[𝜏,𝜎). To complete the proof, we only need to show {𝜎=}=1. By Lemma 3.2, we have that for any 𝑡>0, 𝑥𝔼𝑉𝑘𝑡𝜎𝑘,𝑡𝜎𝑘,𝑟𝑡𝜎𝑘𝑥=𝔼𝑉𝑘(0),0,𝑟(0)+𝔼𝑡𝜎𝑘0(𝑘)𝑉𝑥𝑘(𝑠),𝑥𝑘𝑠𝜏1(𝑠),𝑥𝑘𝑠𝜏2(𝑠),𝑠,𝑟(𝑠)𝑑𝑠,(3.10) where operator (𝑘)𝑉 is defined similarly as 𝑉 was defined by (2.6). By the definitions of 𝑓(𝑘) and 𝑔(𝑘), if 0𝑠𝑡𝜎𝑘, we hence observe that (𝑘)𝑉𝑥𝑘(𝑠),𝑥𝑘𝑠𝜏1(𝑠),𝑥𝑘𝑠𝜏2𝑥(𝑠),𝑠,𝑟(𝑠)=𝑉𝑘(𝑠),𝑥𝑘𝑠𝜏1(𝑠),𝑥𝑘𝑠𝜏2.(𝑠),𝑠,𝑟(𝑠)(3.11) By the conditions of (3.1) and (3.2), we derive that 𝑥𝔼𝑉𝑘𝑡𝜎𝑘,𝑡𝜎𝑘,𝑟𝑡𝜎𝑘𝜉𝑉(0),0,𝑟0+𝔼𝑡0𝑘1𝜔1(𝑥(𝑠))+𝑘2𝜔2𝑥𝑠𝜏1(𝑠)+𝑘3𝜔3𝑥𝑠𝜏2+(𝑠)𝑑𝑠𝑡0𝜆(𝑠)𝑑𝑠𝑉𝜉(0),0,𝑟0+𝔼𝑡0𝑘1𝜔1(𝑥(𝑠))𝑑𝑠+𝔼𝑡𝜏1(𝑡)𝜏1𝑘21𝑑𝜏1𝜔2(𝑠)𝑑𝑠+𝔼𝑡𝜏2(𝑡)𝜏2𝑘31𝑑𝜏2𝜔3(𝑠)𝑑𝑠+𝑡0𝜆(𝑠)𝑑𝑠𝑉𝜉(0),0,𝑟0+𝔼0𝜏𝑘1𝜔2(𝜉(𝜃))+𝜔3(𝜉(𝜃))𝑑𝜃𝔼𝑡0𝑘1𝜔1(𝑠)𝜔2(𝑠)𝜔3(𝑠)𝑑𝑠+𝑡0𝜆(𝑠)𝑑𝑠𝑉𝜉(0),0,𝑟0+𝔼0𝜏𝑘1𝜔2(𝜉(𝜃))+𝜔3(𝜉(𝜃))𝑑𝜃+𝑡0𝜆(𝑠)𝑑𝑠.(3.12)
On the other hand, 𝑥𝔼𝑉𝑘𝑡𝜎𝑘,𝑡𝜎𝑘,𝑟𝑡𝜎𝑘{𝜎𝑘𝑡}𝑉𝑥𝑘𝑡𝜎𝑘,𝑡𝜎𝑘,𝑟𝑡𝜎𝑘𝜎𝑑𝑘𝑡inf|𝑥|𝑘,𝑡0,𝑖𝑆𝑉(𝑥,𝑡,𝑖).(3.13)
This yields 𝜎𝑘𝑉𝑡𝜉(0),0,𝑟0+𝔼0𝜏𝑘1𝜔2(𝜉(𝜃))+𝜔3(𝜉(𝜃))𝑑𝜃+𝑡0𝜆(𝑠)𝑑𝑠inf|𝑥|𝑘,𝑡0,𝑖𝑆.𝑉(𝑥,𝑡,𝑖)(3.14)
Letting 𝑘 and using (3.3), we obtain (𝜎𝑡)=0. Since 𝑡 is arbitrary, we must have (𝜎=)=1. The proof is therefore complete.

Let us now begin to prove our main result.

Proof. Let 𝜔(𝑥)=𝜔1(𝑥)𝜔2(𝑥)𝜔3(𝑥) for all 𝑥𝑛. Inequality (3.2) implies 𝜔(𝑥)>0 whenever 𝑥0. Fix any initial value 𝜉 and any initial state 𝑟0, and for simplicity write 𝑥(𝑡;𝜉,𝑟0)=𝑥(𝑡).
By Lemma 3.2 and condition (3.1), we have 𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))=𝑉𝜉(0),0,𝑟0+𝑡0𝑉𝑥(𝑠),𝑥𝑠𝜏1(𝑠),𝑥𝑠𝜏2(+𝑠),𝑠,𝑟(𝑠)𝑑𝑠𝑡0𝑉𝑥(𝑥(𝑠),𝑠,𝑟(𝑠))𝑔𝑥(𝑠),𝑥𝑠𝜏2+(𝑠),𝑠,𝑟(𝑠)𝑑𝐵(𝑠)𝑡0𝑉𝑥(𝑠),𝑠,𝑟0+𝑙(𝑟(𝑠),𝛼)𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))𝜇(𝑑𝑠,𝑑𝛼)𝑉𝜉(0),0,𝑟0+𝑡0𝜆(𝑠)𝑑𝑠𝑡0𝑘1𝜔1(+𝑥(𝑠))𝑡0𝑘2𝜔2𝑥𝑠𝜏1(𝑠)+𝑘3𝜔3𝑥𝑠𝜏2+(𝑠)𝑑𝑠𝑡0𝑉𝑥(𝑥(𝑠),𝑠,𝑟(𝑠))𝑔𝑥(𝑠),𝑥𝑠𝜏2+(𝑠),𝑠,𝑟(𝑠)𝑑𝐵(𝑠)𝑡0𝑉𝑥(𝑠),𝑠,𝑟0+𝑙(𝑟(𝑠),𝛼)𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))𝜇(𝑑𝑠,𝑑𝛼)𝑉𝜉(0),0,𝑟0+𝑡0𝜆(𝑠)𝑑𝑠+𝑘10𝜏𝜔2(𝑥(𝑠))+𝜔3(𝑥(𝑠))𝑑𝑠𝑘1𝑡0𝜔(𝑥(𝑠))𝑑𝑠+𝑡0𝑉𝑥𝑥(𝑥(𝑠),𝑠,𝑟(𝑠))𝑔(𝑠),𝑥𝑠𝜏2+(𝑠),𝑠,𝑟(𝑠)𝑑𝐵(𝑠)𝑡0𝑉𝑥(𝑠),𝑠,𝑟0+𝑙(𝑟(𝑠),𝛼)𝑉(𝑥(𝑠),𝑠,𝑟(𝑠))𝜇(𝑑𝑠,𝑑𝛼).(3.15) Since 0𝜆(𝑠)𝑑𝑠<, applying Lemma 3.3 we obtain that lim𝑡𝑡0𝜔(𝑥(𝑠))𝑑𝑠=0𝜔(𝑥(𝑠))𝑑𝑠<a.s.,(3.16)lim𝑡sup𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))<a.s..(3.17) Define 𝛽𝑅+𝑅+ as 𝛽(𝑟)=inf|𝑥|𝑟,0𝑡<,𝑖𝑆𝑉(𝑥,𝑡,𝑖). Then, it is obvious to see from (3.17) that sup0𝑡<𝛽||𝑥||(𝑡)sup0𝑡<𝑉(𝑥(𝑡),𝑡,𝑟(𝑡))<a.s..(3.18)
On the other hand, by (3.3) we have sup0𝑡<|𝑥(𝑡)|< a.s.. It is easy to find an integer 𝑘0 such that |𝜉|<𝑘0 a.s. because of 𝜉𝐶𝑏0([𝜏,0);𝑛). Furthermore, for any integer 𝑘>𝑘0, we can define the stopping time 𝜌𝑘||𝑥||=inf𝑡0(𝑡)𝑘,(3.19) where inf= as usual. Clearly, 𝜌𝑘 a.s. as 𝑘. Moreover, for any given 𝜀>0, there is 𝑘𝜀𝑘0 such that {𝜌𝑘<}𝜀 for any 𝑘𝑘𝜀.
It is straightforward to see from (3.16) that lim𝑡inf𝜔(𝑥(𝑡))=0 a.s.; then we claim that lim𝑡𝜔(𝑥(𝑡))=0a.s..(3.20)
The rest of the proof is carried out by contradiction. That is, assuming that (3.20) is false, we have lim𝑡sup𝜔(𝑥(𝑡))>0>0.(3.21)
Furthermore, there exist 𝜀0>0 and 𝜀>𝜀1>0 such that 𝜎2𝑗<𝑗𝑍𝜀0,(3.22) where 𝑍 is a set of natural numbers and {𝜎𝑗}𝑗1 are a sequence of stopping times defined by 𝜎1=inf𝑡0𝜔(𝑥(𝑡))2𝜀1,𝜎2𝑗=inf𝑡𝜎2𝑗1𝜔(𝑥(𝑡))𝜀1𝜎,𝑗=1,2,,2𝑗+1=inf𝑡𝜎2𝑗𝜔(𝑥(𝑡))2𝜀1,𝑗=1,2,.(3.23)
By the local Lipschitz condition (2.4), for any given 𝑘>0, there exists 𝐿𝑘>0 such that ||||||||𝑓(𝑥,𝑦,𝑡,𝑖)𝑔(𝑥,𝑧,𝑡,𝑖)𝐿𝑘,(3.24) for all |𝑥||𝑦||𝑧|𝑘,𝑡0 and 𝑖𝑆.
For any 𝑗𝑍, let 𝑇<𝜎2𝑗𝜎2𝑗1; by Hölder’s inequality and Doob’s martingale inequality, we compute 𝔼𝕀{𝜎2𝑗<𝜌𝑘}sup0𝑡𝑇||𝑥𝜎2𝑗1𝜎+𝑡𝑥2𝑗1||2𝕀=𝔼{𝜎2𝑗<𝜌𝑘}sup0𝑡𝑇||||𝜎2𝑗1𝜎+𝑡2𝑗1𝑓𝑥(𝑠),𝑥𝑠𝜏1+(𝑠),𝑠,𝑟(𝑠)𝑑𝑠𝜎2𝑗1𝜎+𝑡2𝑗1𝑔𝑥(𝑠),𝑥𝑠𝜏2||||(𝑠),𝑠,𝑟(𝑠)𝑑𝐵(𝑠)2𝕀2𝔼{𝜎2𝑗<𝜌𝑘}sup0𝑡𝑇||||𝜎2𝑗1𝜎+𝑡2𝑗1𝑓𝑥(𝑠),𝑥𝑠𝜏1||||(𝑠),𝑠,𝑟(𝑠)𝑑𝑠2𝕀+8𝔼{𝜎2𝑗<𝜌𝑘}sup0𝑡𝑇𝜎2𝑗1𝜎+𝑡2𝑗1||𝑔𝑥(𝑠),𝑥𝑠𝜏2||(𝑠),𝑠,𝑟(𝑠)2𝑑𝑠2𝐿2𝑘𝑇(𝑇+4),(3.25) where 𝕀𝐴 is the indicator of set 𝐴.
Since 𝜔(𝑥) is continuous in 𝑛, it must be uniformly continuous in the closed ball 𝑆𝑘={𝑥𝑛|𝑥|𝑘}. For any given 𝑏>0, we can choose 𝑐𝑏>0 such that |𝜔(𝑥)𝜔(𝑦)|<𝑏 whenever 𝑥,𝑦𝑆𝑘 and |𝑥𝑦|<𝑐𝑏. Furthermore, let us choose 𝜀𝜀=03,𝑘𝑘𝜀,𝑏=𝜀1.(3.26)
By inequality (3.25) and Chebyshev’s inequality, we have 𝜌𝑘𝜎2𝑗𝜎+2𝑗<𝜌𝑘sup0𝑡𝑇||𝜔𝑥𝜎2𝑗1𝑥𝜎+𝑡𝜔2𝑗1||𝜀1𝜌𝑘𝜎2𝑗𝜎2𝑗𝜌=+𝑘𝜎2𝑗𝜎2𝑗𝜎<+2𝑗<𝜌𝑘sup0𝑡𝑇||𝑥𝜎2𝑗1𝜎+𝑡𝑥2𝑗1||𝑐𝜀12𝐿2𝑘𝑇(𝑇+4)𝑐2𝜀1+12𝜀.(3.27)
Meanwhile, we can also choose 𝑇=𝑇(𝜀,𝜀1,𝑘) sufficiently small for 2𝐿2𝑘𝑇(𝑇+4)𝑐2𝜀1𝜀.(3.28)
And then, (3.27) and (3.28) yield 𝜎2𝑗<𝜌𝑘Ω𝑗𝜀,(3.29) where Ω𝑗={sup0𝑡𝑇|𝜔(𝑥(𝜎2𝑗1+𝑡))𝜔(𝑥(𝜎2𝑗1))|<𝜀1}.
In the following, we can obtain from (3.16) and (3.29) that >𝔼0𝜔(𝑥(𝑡))𝑑𝑡𝑗=1𝔼𝕀{𝜎2𝑗<𝜌𝑘}𝜎2𝑗𝜎2𝑗1𝜔(𝑥(𝑡))𝑑𝑡𝑗=1𝜀1𝔼𝕀{𝜎2𝑗<𝜌𝑘}𝜎2𝑗𝜎2𝑗1𝑗=1𝑇𝜀1𝜎2𝑗<𝜌𝑘Ω𝑗𝑗=1𝑇𝜀11𝜀=3𝑗=1𝑇𝜀0𝜀1=.(3.30)
This is a contradiction. So there is an ΩΩ with (Ω)=1 such that lim𝑡𝜔(𝑥(𝑡,𝜔))=0,sup0𝑡<||||𝑥(𝑡,𝜔)<,𝜔Ω.(3.31)
Finally, any fixed 𝜔Ω, {𝑥(𝑡,𝜔)}𝑡0 is bounded in 𝑛. By Bolzano-Weierstrass theorem, there is an increasing sequence{𝑡𝑖}𝑖1 such that {𝑥(𝑡,𝜔)}𝑖1 converges to some 𝑧𝑛 with |𝑧|<. Since 𝜔(𝑥)>0 whenever 𝑥0, we must have 𝜔(𝑥)=0 if and only if 𝑥=0. This implies that the solution of (2.2) is a.s. stable, and the proof is therefore completed.

Remark 3.5. The techniques proposed in Theorem 3.1 can be used to deal with the a.s. stability problem for other HSDSs, such as the ones in [25]. In a very special case when 𝜏1(𝑡)=𝜏2(𝑡)=𝜏 for all 𝑡0 and 𝑖𝑆, it is easy to see that ̇𝜏1(𝑡)=̇𝜏2(𝑡)=0, and Theorem 3.1 is exactly Theorem  2.1 in [25]. Similarly, Theorem  2.2 in [25] can be generalized to system (2.2) as a LaSalle-type theorem (see [24, 26]) for HSSs with multiple time-varying delays.

4. Almost Sure Stabilization of Nonlinear HSDSs

Consider the following nonlinear HSDSs: 𝐴𝑑𝑥(𝑡)=(𝑟(𝑡))𝑥(𝑡)+𝐴𝑑(𝑟(𝑡))𝑥𝑡𝜏1𝑥(𝑡)+𝑓(𝑡),𝑥𝑡𝜏1(𝑡),𝑡,𝑟(𝑡)+𝐵𝑢(𝑟(𝑡))𝑢(𝑡)𝑑𝑡+𝑔𝑥(𝑡),𝑥𝑡𝜏2(𝑡),𝑡,𝑟(𝑡)𝑑𝐵(𝑡),(4.1) where 𝐵𝑢(𝑟(𝑡)) are known constant matrices with appropriate dimensions and 𝐵(𝑡) represents a scalar Brownian motion (Wiener process) on (Ω,,{𝑡}𝑡0,) that is independent of Markov chain 𝑟(𝑡) and satisfies: 𝐸{𝑑𝐵(𝑡)}=0,𝐸𝑑𝐵(𝑡)2=𝑑𝑡,(4.2)𝑓 and 𝑔 are both functions from 𝑛×𝑛×+×𝑆 to 𝑛 which satisfy local Lipschitz condition and the following assumptions: ||𝑓𝑥(𝑡),𝑥𝑡𝜏1||(𝑡),𝑡,𝑟(𝑡)2𝑥𝑇(𝑡)𝐹1(𝑟(𝑡))𝑥(𝑡)+𝑥𝑇𝑡𝜏1𝐹(𝑡)2(𝑟(𝑡))𝑥𝑡𝜏1,||𝑔(𝑡)𝑥(𝑡),𝑥𝑡𝜏2||(𝑡),𝑡,𝑟(𝑡)2𝑥𝑇(𝑡)𝐺1(𝑟(𝑡))𝑥(𝑡)+𝑥𝑇𝑡𝜏2𝐺(𝑡)2(𝑟(𝑡))𝑥𝑡𝜏2,(𝑡)(4.3) where, for each 𝑟(𝑡)=𝑗𝑆, 𝐴(𝑟(𝑡)), 𝐴𝑑(𝑟(𝑡)) are known constant matrices with appropriate dimensions, and 𝐹𝑖(𝑟(𝑡))𝑛×𝑛, 𝐺𝑖(𝑟(𝑡))𝑛×𝑛(𝑖=1,2) are positive definite matrices.

In the sequel, we denote the matrix associated with the 𝑖th mode by Γ𝑖Γ(𝑟(𝑡)=𝑖),(4.4) where the matrix Γ could be 𝐴,𝐴𝑑,𝐵𝑢,𝐹1,𝐹2,𝐺1,𝐺2,𝐺, or 𝐺𝑑.

As the given HSDSs (4.1) is nonlinear, we here consider the resulting systems can be stabilized only by linear state feedback controller which is of the form 𝑢(𝑡)=𝐾(𝑟(𝑡))𝑥(𝑡),(4.5) where 𝐾(𝑟(𝑡)) are controller parameters to be designed.

Under control law (4.5), the closed-loop system can be given as follow: 𝐴𝑑𝑥(𝑡)=(𝑟(𝑡))𝑥(𝑡)+𝐴𝑑(𝑟(𝑡))𝑥𝑡𝜏1𝑥(𝑡)+𝑓(𝑡),𝑥𝑡𝜏1(𝑡),𝑡,𝑟(𝑡)+𝐵𝑢(𝑥𝑟(𝑡))𝐾(𝑟(𝑡))𝑥(𝑡)𝑑𝑡+𝑔(𝑡),𝑥𝑡𝜏2(𝑡),𝑡,𝑟(𝑡)𝑑𝐵(𝑡).(4.6) The stabilization problem is therefore to design matrices 𝐾(𝑟(𝑡)) for the closed-loop system (4.6) to be a.s. stable. In order to guarantee the solvability of 𝐾(𝑟(𝑡)), the following theorem is given.

Theorem 4.1. If there exist sequences of scalars 𝜀1𝑖>0,𝜀2𝑖>0,𝛿𝑖>0, positive definite matrices 𝑋𝑖>0 and matrices 𝑌𝑖 such that the following LMIs 𝑀𝑖1𝑀𝑖2𝑀𝑖4𝑀𝑖30𝑀𝑖5𝑋<0𝑖,𝑗𝑆,(4.7)𝑖𝛿𝑖𝐼(4.8) hold, where 𝑀𝑖1=𝐴𝑖𝑋𝑖+𝑋𝑖𝐴𝑇𝑖+𝐵𝑢𝑖𝑌𝑖+𝑌𝑇𝑖𝐵𝑇𝑢𝑖+𝜀1𝑖𝐴𝑑𝑖𝐴𝑇𝑑𝑖+𝜀2𝑖𝐼+𝛾𝑖𝑖𝑋𝑖,𝑀𝑖2=𝑋𝑖,𝑋𝑖,𝑋𝑖,𝑋𝑖,𝑋𝑖,𝑀𝑖3𝜀=diag2𝑖𝐹11𝑖,𝑐1𝜀2𝑗𝐹12𝑗,𝛿𝑖𝐺11𝑖,𝑐2𝛿𝑖𝐺12𝑗,𝑐1𝜀1𝑗𝐼,𝑀𝑖4=𝛾𝑖1𝑋𝑖,,𝛾𝑖(𝑖1)𝑋𝑖,𝛾𝑖(𝑖+1)𝑋𝑖,,𝛾𝑖𝑁𝑋𝑖,𝑀𝑖5𝑋=diag1,,𝑋𝑖1,𝑋𝑖+1,,𝑋𝑁,𝑐1=1𝑑𝜏1,𝑐2=1𝑑𝜏2,(4.9) then the controlled system (4.6) is a.s. stable and the state feedback controller determined by 𝑢(𝑡)=𝐾𝑖𝑥(𝑡),𝐾𝑖=𝑌𝑖𝑋𝑖1,𝑖𝑆.(4.10)

Proof. Let 𝑃𝑖=𝑋𝑖1 and 𝑉(𝑥,𝑖)=𝑥𝑇𝑃𝑖𝑥+𝑡𝑡𝜏1(𝑡)𝑥𝑇(𝑠)𝑄1𝑥(𝑠)𝑑𝑠+𝑡𝑡𝜏2(𝑡)𝑥𝑇(𝑠)𝑄2𝑥(𝑠)𝑑𝑠.
The operator 𝑉𝑛×𝑛×𝑛×𝑆 has the form 𝑉(𝑥,𝑦,𝑧,𝑖)=𝑥𝑇𝑄1𝑥1̇𝜏1𝑦(𝑡)𝑇𝑄1𝑦+𝑥𝑇𝑄2𝑥1̇𝜏2𝑧(𝑡)𝑇𝑄2𝑧+2𝑥𝑇𝑃𝑖𝐴𝑖𝑥+𝐴𝑑𝑖𝑦+𝑓(𝑥,𝑦,𝑖)+𝐵𝑢𝑖𝐾𝑖𝑥+𝑔𝑇(𝑥,𝑧,𝑖)𝑃𝑖𝑔(𝑥,𝑧,𝑖)+𝑁𝑗=1𝛾𝑖𝑗𝑥𝑇𝑃𝑗𝑥𝑥𝑇𝑄1+𝑄2+𝑃𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐵𝑢𝑖𝐾𝑖+𝐵𝑢𝑖𝐾𝑖𝑇𝑃𝑖+𝜀1𝑖𝑃𝑖𝐴𝑑𝑖𝐴𝑇𝑑𝑖𝑃𝑖+𝜀2𝑖𝑃2𝑖+𝑁𝑗=1𝛾𝑖𝑗𝑃𝑗+𝜀12𝑖𝐹1𝑖+𝛿𝑖1𝐺1𝑖𝑥+𝑦𝑇𝜀11𝑖𝐼+𝜀12𝑖𝐹2𝑖1𝑑𝜏1𝑄1𝑦+𝑧𝑇𝛿𝑖1𝐺2𝑖1𝑑𝜏2𝑄2𝑧.(4.11) So 𝑉(𝑥,𝑦,𝑧,𝑖)𝜔1𝑖(𝑥)+1𝑑𝜏1𝜔2𝑖(𝑦)+1𝑑𝜏2𝜔3𝑖(𝑧),(4.12) where 𝜔1𝑖(𝑥)=𝑥𝑇𝑄1𝑄2𝑃𝑖𝐴𝑖𝐴𝑇𝑖𝑃𝑖𝑃𝑖𝐵𝑢𝑖𝐾𝑖𝐵𝑢𝑖𝐾𝑖𝑇𝑃𝑖𝜀1𝑖𝑃𝑖𝐴𝑑𝑖𝐴𝑇𝑑𝑖𝑃𝑖𝜀2𝑖𝑃2𝑖𝜀12𝑖𝐹1𝑖𝛿𝑖1𝐺1𝑖𝑁𝑘=1𝛾𝑖𝑘𝑃𝑘𝜔𝑥,2𝑖(𝑥)=𝑥𝑇𝑐11𝜀11𝑖𝐼+𝑐11𝜀12𝑖𝐹2𝑖𝑄1𝜔𝑥,3𝑖(𝑥)=𝑥𝑇𝑐21𝛿𝑖1𝐺2𝑖𝑄2𝑥.(4.13)
By assumption 1, it is easy to see that we can choose 𝑄1 and 𝑄2 such that 𝜔2𝑖(𝑥)0,𝜔3𝑖(𝑥)0 for all 𝑥𝑛,𝑖𝑆.
Noting that 𝑃𝑖=𝑋𝑖1 and 𝑌𝑖=𝐾𝑖𝑋𝑖, we can pre- and postmultiply (4.7) by diag(𝑃𝑖,,𝑃𝑖), and using Schur complements, we can obtain Φ𝑖𝑗<0,(4.14) where Φ𝑖𝑗=𝑃𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐵𝑢𝑖𝐾𝑖+𝐵𝑢𝑖𝐾𝑖𝑇𝑃𝑖+𝜀1𝑖𝑃𝑖𝐴𝑑𝑖𝐴𝑇𝑑𝑖𝑃𝑖+𝜀2𝑖𝑃2𝑖+𝛿𝑖1𝐺1𝑖+𝜀12𝑖𝐹1𝑖+𝑁𝑘=1𝛾𝑖𝑘𝑃𝑘+𝑐11𝜀11𝑗𝐼+𝑐11𝜀12𝑗𝐹2𝑗+𝑐21𝛿𝑗𝐺2𝑗.(4.15) This implies 𝜔1𝑖(𝑥)>𝜔2𝑗(𝑥)+𝜔3𝑗(𝑥)0,𝑥0.(4.16)
Let 𝜔1(𝑥)=min𝑖𝑆𝜔1𝑖(𝑥),𝜔2(𝑥)=max𝑖𝑆𝜔2𝑖(𝑥), and 𝜔3(𝑥)=max𝑖𝑆𝜔3𝑖(𝑥).
Clearly 𝜔1(𝑥)>𝜔2(𝑥)+𝜔3(𝑥)0,𝑥0.(4.17) Moreover, by (4.24) we further obtain 𝑉(𝑥,𝑦,𝑧,𝑖)𝜔1(𝑥)+1𝑑𝜏1𝜔2(𝑦)+1𝑑𝜏2𝜔3(𝑧).(4.18)
The required assertion now follows from Theorem 3.1.

If the systems (4.6) reduces to linear HSDSs of the form 𝐴𝑑𝑥(𝑡)=(𝑟(𝑡))𝑥(𝑡)+𝐴𝑑(𝑟(𝑡))𝑥𝑡𝜏1(𝑡)+𝐵𝑢(𝑟(𝑡))𝐾(𝑟(𝑡))𝑥(𝑡)𝑑𝑡+𝐺(𝑟(𝑡))𝑥(𝑡)+𝐺𝑑(𝑟(𝑡))𝑥𝑡𝜏2(𝑡)𝑑𝐵(𝑡),(4.19) where 𝐴(𝑟(𝑡)),𝐴𝑑(𝑟(𝑡)),𝐵𝑢(𝑟(𝑡)),𝐺(𝑟(𝑡)), and 𝐺𝑑(𝑟(𝑡)) are known constant matrices with appropriate dimensions.

Then, the following corollary follows directly from Theorem 4.1.

Corollary 4.2. If there exist sequences of scalars 𝜀1𝑖>0,𝜀2𝑖>0, positive definite matrices 𝑋𝑖>0 and matrices 𝑌𝑖 such that the following LMIs 𝑀𝑖1𝑀𝑖2𝑀𝑖4𝑀𝑖30𝑀𝑖5<0𝑖,𝑗𝑆(4.20) hold, where 𝑀𝑖1=𝐴𝑖𝑋𝑖+𝑋𝑖𝐴𝑇𝑖+𝐵𝑢𝑖𝑌𝑖+𝑌𝑇𝑖𝐵𝑇𝑢𝑖+𝜀1𝑖𝐴𝑑𝑖𝐴𝑇𝑑𝑖+𝛾𝑖𝑖𝑋𝑖,𝑀𝑖2=2𝑋𝑖𝐺𝑇𝑖,𝑋𝑗,𝑋𝑗,2𝑋𝑇𝑗𝐺𝑑𝑗,𝑀𝑖3𝑋=diag𝑖,𝑐1𝜀1𝑗𝐼,𝜀2𝑗𝐼,𝑋𝑗,𝑀𝑖4=𝛾𝑖1𝑋𝑖,,𝛾𝑖(𝑖1)𝑋𝑖,𝛾𝑖(𝑖+1)𝑋𝑖,,𝛾𝑖𝑁𝑋𝑖,𝑀𝑖5𝑋=diag1,,𝑋𝑖1,𝑋𝑖+1,,𝑋𝑁,𝑐1=1𝑑𝜏1,𝑐2=1𝑑𝜏2,(4.21) then the controlled system (4.19) is a.s. stable and the state feedback controller determined by 𝑢(𝑡)=𝐾𝑖𝑥(𝑡),𝐾𝑖=𝑌𝑖𝑋𝑖1,𝑖𝑆.(4.22)

Proof. Let 𝑃𝑖=𝑋𝑖1 and 𝑉(𝑥,𝑖)=𝑥𝑇𝑃𝑖𝑥+𝑡𝑡𝜏1(𝑡)𝑥𝑇(𝑠)𝑄1𝑥(𝑠)𝑑𝑠+𝑡𝑡𝜏2(𝑡)𝑥𝑇(𝑠)𝑄2𝑥(𝑠)𝑑𝑠.
The operator 𝑉𝑛×𝑛×𝑛×𝑆 has the form 𝑉(𝑥,𝑦,𝑧,𝑖)=𝑥𝑇𝑄1𝑥1̇𝜏1𝑦(𝑡)𝑇𝑄1𝑦+𝑥𝑇𝑄2𝑥1̇𝜏2𝑧(𝑡)𝑇𝑄2𝑧+2𝑥𝑇𝑃𝑖𝐴𝑖𝑥+𝐴𝑑𝑖𝑦+𝐵𝑢𝑖𝐾𝑖𝑥+𝐺𝑖𝑥+𝐺𝑑𝑖𝑧𝑇𝑃𝑖𝐺𝑖𝑥+𝐺𝑑𝑖𝑧+𝑁𝑘=1𝛾𝑖𝑘𝑥𝑇𝑃𝑘𝑥𝑥𝑇𝑄1+𝑄2+𝑃𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐵𝑢𝑖𝐾𝑖+𝐵𝑢𝑖𝐾𝑖𝑇𝑃𝑖+𝜀1𝑖𝑃𝑖𝐴𝑑𝑖𝐴𝑇𝑑𝑖𝑃𝑖+𝑁𝑘=1𝛾𝑖𝑘𝑃𝑘+2𝐺𝑇𝑖𝑃𝑖𝐺𝑖𝑥+𝑦𝑇𝜀11𝑖𝐼1𝑑𝜏1𝑄1𝑦+𝑧𝑇𝜀12𝑖𝐼+2𝐺𝑇𝑑𝑖𝑃𝑖𝐺𝑑𝑖1𝑑𝜏2𝑄2𝑧.(4.23) So 𝑉(𝑥,𝑦,𝑧,𝑖)𝜔1𝑖(𝑥)+1𝑑𝜏1𝜔2𝑖(𝑦)+1𝑑𝜏2𝜔3𝑖(𝑧),(4.24) where 𝜔1𝑖(𝑥)=𝑥𝑇𝑄1𝑄2𝑃𝑖𝐴𝑖𝐴𝑇𝑖𝑃𝑖𝑃𝑖𝐵𝑢𝑖𝐾𝑖𝐵𝑢𝑖𝐾𝑖𝑇𝑃𝑖𝜀1𝑖𝑃𝑖𝐴𝑑𝑖𝐴𝑇𝑑𝑖𝑃𝑖𝑁𝑘=1𝛾𝑖𝑘𝑃𝑘2𝐺𝑇𝑖𝑃𝑖𝐺𝑖𝜔𝑥,2𝑖(𝑥)=𝑥𝑇𝑐11𝜀11𝑖𝐼𝑄1𝜔𝑥,3𝑖(𝑥)=𝑥𝑇𝜀12𝑖𝐼+2𝑐21𝐺𝑇𝑑𝑖𝑃𝑖𝐺𝑑𝑖𝑄2𝑥.(4.25)
It is easy to see that we can choose 𝑄1 and 𝑄2 such that 𝜔2𝑖(𝑥)0,𝜔3𝑖(𝑥)0 for all 𝑥𝑛,𝑖𝑆.
Noting that 𝑃𝑖=𝑋𝑖1 and 𝑌𝑖=𝐾𝑖𝑋𝑖, we can pre- and postmultiply (4.7) by diag(𝑃𝑖,,𝑃𝑖), and using Schur complements, we can obtain Φ𝑖𝑗<0,(4.26) where Φ𝑖𝑗=𝑃𝑖𝐴𝑖+𝐴𝑇𝑖𝑃𝑖+𝑃𝑖𝐵𝑢𝑖𝐾𝑖+𝐵𝑢𝑖𝐾𝑖𝑇𝑃𝑖+𝜀1𝑖𝑃𝑖𝐴𝑑𝑖𝐴𝑇𝑑𝑖𝑃𝑖+𝑁𝑘=1𝛾𝑖𝑘𝑃𝑘+2𝐺𝑇𝑖𝑃𝑖𝐺𝑖+𝑐11𝜀11𝑗𝐼+𝜀12𝑗𝐼+2𝑐21𝐺𝑇𝑑𝑗𝑃𝑗𝐺𝑑𝑗.(4.27)
This implies 𝜔1𝑖(𝑥)>𝜔2𝑗(𝑥)+𝜔3𝑗(𝑥)0,𝑥0.(4.28)
Let 𝜔1(𝑥)=min𝑖𝑆𝜔1𝑖(𝑥),𝜔2(𝑥)=max𝑖𝑆𝜔2𝑖(𝑥), and 𝜔3(𝑥)=max𝑖𝑆𝜔3𝑖(𝑥).
Clearly 𝜔1(𝑥)>𝜔2(𝑥)+𝜔3(𝑥)0,𝑥0.(4.29) Moreover, by (4.24) we further obtain 𝑉(𝑥,𝑦,𝑧,𝑖)𝜔1(𝑥)+1𝑑𝜏1𝜔2(𝑦)+1𝑑𝜏2𝜔3(𝑧).(4.30)
The required assertion now follows from Theorem 3.1.

5. Examples

In this section we will provide two examples to illustrate our results. In the following examples we assume that 𝐵(𝑡) is a scalar Brownian motion, 𝛾(𝑡) is a right-continuous Markov chain independent of 𝐵(𝑡) and taking values in 𝑆={1,2}, and the step size Δ=0.0001. By using the YALMIP toolbox, simulations results are shown in Figures 13. Figure 1 gives a portion of state 𝛾(𝑡) of Example 5.1 for clear display. Figure 2 simulates the numerical results for Example 5.1. The simulation results have illustrated our theoretical analysis. Following from Theorem 4.1, the simulation results for Example 5.2 can be founded in Figure 3, which verify our desired results.

183729.fig.001
Figure 1: The state 𝛾(𝑡) of Example 5.1.
183729.fig.002
Figure 2: The state evolution of Example 5.1.
183729.fig.003
Figure 3: The state evolution of Example 5.2.

Example 5.1. Let 𝛾Γ=𝑖𝑗2×2=0.80.80.30.3.(5.1)
Consider scalar nonlinear HSDSs: 𝑥𝑑𝑥(𝑡)=𝑓(𝑥(𝑡),𝑡,𝑟(𝑡))𝑑𝑡+𝑔(𝑡),𝑥𝑡𝜏2(𝑡),𝑡,𝑟(𝑡)𝑑𝐵(𝑡),(5.2) where 𝑓(𝑥,𝑡,1)=65𝑥,𝑔(𝑥,𝑧,𝑡,1)=5𝑥3+25𝑧3,𝑓3(𝑥,𝑡,2)=231+𝑡45𝑥,𝑔(𝑥,𝑧,𝑡,2)=5𝑥35cos(𝑡)+45𝑧3sin(𝑡),(5.3)𝜏2(𝑡)=0.3+0.3sin(𝑡).
To examine the stability of system (5.2), we consider a Lyapunov function candidate 𝑉×𝑆+ as 𝑉(𝑥,𝑖)=𝑥2 for 𝑖=1,2. Then we have 𝑉(𝑥,𝑧,𝑡,1)10𝑥6/5+4𝑧6/5,𝑉(𝑥,𝑧,𝑡,2)3𝑥31+𝑡6𝑥6/5+258𝑧6/5.(5.4)
By the elementary inequality 𝛼𝑐𝛽1𝑐𝑐𝛼+(1𝑐)𝛽 for all 𝛼0,𝛽0, and 0𝑐1, we see that inequality 3𝑥3=61+𝑡5𝜅𝑥6/56/56𝜅55(1+𝑡)21/6𝜅𝑥6/5+𝜅1(1+𝑡)2(5.5) holds for any 𝜅>0, where 𝜅1=(𝜅/5)5.
From inequalities (5.4)–(5.5), we have 𝜅𝑉(𝑥,𝑧,𝑡,𝑖)1(1+𝑡)2(6𝜅)𝑥6/5+4𝑧6/5,(5.6) for all 𝑡0 and 𝑖𝑆. By 𝜏2(𝑡)=0.3+0.3sin(𝑡), it is easy to see that 𝑑𝜏2(𝑡)<1/3; then, we choose constant 𝜅 such that 0<𝜅<(26𝑑𝜏2)/(1𝑑𝜏2), and hence conditions of Theorem 3.1 are satisfied.

Example 5.2. Let 𝛾Γ=𝑖𝑗2×2=0.60.60.50.5.(5.7)
Consider scalar nonlinear closed-loop HSDSs: 𝑓𝑥𝑑𝑥(𝑡)=(𝑡),𝑥𝑡𝜏1(𝑡),𝑡,𝑟(𝑡)+𝐵(𝑟(𝑡))𝐾(𝑟(𝑡))𝑥(𝑡)𝑑𝑡+𝑔𝑥(𝑡),𝑥𝑡𝜏2(𝑡),𝑡,𝑟(𝑡)𝑑𝐵(𝑡)(5.8) with 1𝑓(𝑥,𝑦,𝑡,1)=𝑥+2𝑦+2𝑥3(|𝑥|+1)2𝑧+𝑦sin(𝑡),𝑔(𝑥,𝑧,𝑡,1)=𝑥cos(𝑡)+3(|𝑧|+1)2,𝑥(5.9)𝑓(𝑥,𝑦,𝑡,2)=2𝑥+𝑦+3(|𝑥|+2)2+𝑦3(|𝑦|+1)2,𝑥𝑔(𝑥,𝑧,𝑡,2)=2𝑥sin(𝑡)+32(|𝑥|+1)2+𝑧3(|𝑧|+2)2,(5.10)𝜏1(𝑡)=0.1+0.1sin(𝑡), 𝜏2(𝑡)=0.2+0.2sin(2𝑡), 𝐵1=2, 𝐵2=3, 𝐴1=1, 𝐴2=2, 𝐴𝑑1=1/2, 𝐴𝑑2=1, 𝐹11=8, 𝐹12=𝐺11=2, 𝐺12=𝐹21=𝐹22=2, 𝐺21=1/2,𝐺22=2.
By Theorem 4.1 we can find the feasible solution 𝐾1=3,𝐾2=2 for the a.s. stability.

6. Conclusions

In this paper, we have investigated the a.s. stability analysis and stabilization synthesis problems for nonlinear HSDSs. Some sufficient conditions are given to guarantee the resulting systems to be a.s. stable. Under these conditions, a.s. stabilization problem for a class of nonlinear HSDSs is solved in terms of the solutions to a set of LMIs. Finally, the results of this paper have been demonstrated by two numerical simulation examples.

Acknowledgment

This work is supported in part by the National Natural Science Foundation of P.R. China (no. 60974030).

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