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International Journal of Differential Equations
Volume 2011, Article ID 613695, 13 pages
http://dx.doi.org/10.1155/2011/613695
Research Article

Mean Square Stability of Impulsive Stochastic Differential Systems

Institute of System Science and Mathematics, Naval Aeronautical and Astronautical University, Yantai, Shandong 264001, China

Received 4 December 2010; Revised 17 May 2011; Accepted 26 May 2011

Academic Editor: Xingfu Zou

Copyright © 2011 Shujie 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

Based on Lyapunov-Krasovskii functional method and stochastic analysis theory, we obtain some new delay-dependent criteria ensuring mean square stability of a class of impulsive stochastic equations. Numerical examples are given to illustrate the effectiveness of the theoretical results.

1. Introduction

It is recognized that the theory of impulsive systems provides a natural framework for the mathematical modeling of many real world phenomena, and impulsive dynamical systems have attracted considerable interest in science and engineering during the past decades. Two classical monographs are Lakshmikantham et al. [1] and Bainov and Simeonov [2]. In general, an impulsive dynamical system can be viewed as a hybrid one comprised of three components: a continuous-time differential equation, which governs the motion of the dynamical systems between impulsive or resetting events; a difference equation, which governs the way the system states are instantaneously changed when a resetting event occurs and a criterion for determining when the states of the systems are to be reset, see Chen and Zheng [3]. Stability properties of impulsive systems have been extensively studied in the literatures. We refer to Li et al. [4, 5], Li et al. [6], Yang [7], Autonio, and Alfonso [8] and the references therein.

Besides impulsive effects, a practical system is usually affected by external stochastic perturbations. Stochastic perturbation is also a factor that makes systems unstable. Recently, stochastic modeling has come to play an important role in many branches of science and industry. An area of particular interest has been stability analysis of impulsive systems with stochastic perturbation. In Yang et al. [9] and Chen et al. [10], the stability properties of nonlinear impulsive stochastic systems are studied using Lyapunov function methods. In Mao et al. [11], a linear matrix inequality approach is proposed for stability analysis of linear uncertain impulsive stochastic systems. However, to the best of our knowledge, there are only few results about this problem.

This paper is inspired by Yang et al. [9], in which the authors considered the problems of stability or robust stabilization for impulsive time delay systems. Unfortunately, they need all the impulsive time sequences to satisfy some strict conditions, that is, the length of the intervals between two jumping time instants must have upper bound or lower bound. But in practical systems, it is always impossible or difficult to obtain it. In this article, by using Lyapunov function methods, together with stochastic analysis, we focus on the mean square stability of trival solution of a class of nonlinear impulsive stochastic time-delay differential systems. We obtain some new conditions ensuring mean square stability of trival solution of the impulsive stochastic differential systems with time-delay. This paper improved some related results.

2. Preliminaries

Throughout this paper, unless explicitly given, for symmetric matrices 𝐴 and 𝐵, the notion 𝐴𝐵 (𝐴>𝐵, 𝐴𝐵, 𝐴<𝐵) means 𝐴𝐵 is positive semidefinite (positive definite, negative semidefinite, negative definite) matrix. 𝜆max() (𝜆min()) represents the maximum (minimum) eigenvalue of the corresponding matrix, respectively. denotes Euclidean norm for vectors or the spectral norm of matrices. Moreover, let (Ω,,{𝑡}𝑡0,𝑃) be a complete probability space with a filtration {𝑡}𝑡0 satisfying the usual conditions, that is, the filtration contains all P-null sets and is right continuous. Let PC([𝜏,0],𝑛) denote the set of piecewise right continuous function 𝜙[𝜏,0]𝑛 with the norm defined by 𝜙𝜏=sup𝜏𝜃0𝜙(𝜃), where 𝜏 is a known positive constant, PC(𝛿)={𝜑𝜑PC([𝜏,0],𝑛),𝜑𝜏𝛿}, PC𝑏0([𝜏,0],𝑛) denote the family of all 0-measurable PC([𝜏,0],𝑛)-valued stochastic process 𝜑={𝜑(𝑠)𝜏𝑠0} with sup𝜏𝑠0𝔼{𝜑(𝑠)2}<, where 𝔼{} represents the mathematical expectation operator with respect to the probability measure 𝑃, PC𝑏0(𝛿)={𝜑𝜑PC𝑏0([𝜏,0],𝑛),sup𝜏𝑠0𝔼{𝜑(𝑠)2}𝛿, denote the well-known -operator given by the Itô's formula.

In this paper, we consider a class of Itô impulsive stochastic differential systems with time delay d𝑥(𝑡)=𝑓𝑡,𝑥(𝑡),𝑥𝑡d𝑡+𝑔𝑡,𝑥(𝑡),𝑥𝑡d𝜔(𝑡),𝑡𝑡0,𝑡𝑡𝑘,𝑥𝑡𝑘=𝐻𝑘𝑥𝑡𝑘𝑥𝑡,𝑘=1,2,,0[],+𝜃=𝜑(𝜃),𝜃𝜏,0(2.1) where the initial value 𝜑PC𝑏0(𝛿), the fixed impulsive time moments 𝑡𝑘 satisfy 0𝑡0<𝑡1<𝑡2<<𝑡𝑘<(𝑡𝑘 as 𝑘). 𝑥(𝑡)𝑛 is the system state, 𝑓𝐶(×𝑛×𝑛,𝑛), 𝑔𝐶(×𝑛×𝑛,𝑛×𝑚). 𝜔(𝑡)𝑚 is an standard Brownian motion defined on the complete probability space (Ω,,{𝑡}𝑡0,𝑃). Besides, we assume that 𝐻𝑘(0)=0,(𝑘=1,2,), 𝑓(𝑡,0,0)=0, 𝑔(𝑡,0,0)=0 and 𝐻𝑘𝑥𝑡𝑘𝛾𝑘𝑥𝑡𝑘,𝛾𝑘0,𝑘=1,2,.(2.2)

In the following, we will divide three cases to consider the mean square stability of system (2.1). We denote by 𝒩inf(𝛽) and 𝒩sup(𝛽) the class of impulsive time sequences that satisfy inf𝑘{𝑡𝑘𝑡𝑘1}𝛽 and sup𝑘{𝑡𝑘𝑡𝑘1}𝛽, respectively.

We need the following lemma and definitions.

Lemma 2.1 (Chaplygin Comparison Theorem, see Shi et al. [12]). Assume that 𝑓,𝐹𝐶(𝐺), 𝑔2 and 𝑓(𝑡,𝑥)<𝐹(𝑡,𝑥),(𝑡,𝑥)𝐺.(2.3) If 𝜙(𝑡) (𝑡𝑈1) and Φ(𝑡) (𝑡𝑈2) are the solutions of Cauchy problems 𝑥̇𝑥=𝑓(𝑡,𝑥),(𝑡,𝑥)𝐺,(𝜏)=𝜉,̇𝑥=𝐹(𝑡,𝑥),(𝑡,𝑥)𝐺,𝑥(𝜏)=𝜉,(2.4) respectively, then for 𝑡(𝜏,)𝑈1𝑈2, 𝜙(𝑡)<Φ(𝑡)(2.5) and for 𝑡(,𝜏)𝑈1𝑈2,   𝜙(𝑡)>Φ(𝑡).(2.6)

Definition 2.2. For a given class 𝒩 of admissible impulsive time sequence, the solution of (2.1) is called mean squarely stable if for any 𝜀>0, there exists a scalar 𝛿>0, such that the initial function 𝜑PC𝑏0(𝛿) implies 𝔼{𝑥(𝑡)2}<𝜀, 𝑡𝑡0 for all admissible time sequence in 𝒩.

Definition 2.3 (see Yang et al. [9]). The function 𝑉[𝑡0𝜏,)×𝑛+ belongs to class 𝒱(1,2) if
(1)the function 𝑉(𝑡,𝑥) is continuously differentiable in 𝑡 and twice continuously differentiable in 𝑥 on each of the sets [𝑡𝑘1,𝑡𝑘)×𝑛,(𝑘=1,2,) and for all 𝑡𝑡0, 𝑉(𝑡,0)0, (2)𝑉(𝑡,𝑥) is locally Lipschitaian in 𝑥, (3)for each 𝑘=1,2,, there exist finite limits lim𝑡(𝑡,𝑦)𝑘,𝑥𝑉𝑡(𝑡,𝑦)=𝑉𝑘,𝑥,lim𝑡(𝑡,𝑦)+𝑘,𝑥𝑉𝑡(𝑡,𝑦)=𝑉+𝑘,𝑉𝑡,𝑥+𝑘𝑡,𝑥=𝑉𝑘.,𝑥(2.7)

3. Main Results

Theorem 3.1. Assume that there exist scalars 𝜆2>𝜆1>0, 𝜆𝜏>0, 𝛽>0, 𝜆0, 𝜌>0 matrix 𝑃>0 and Lyapunov-Krasovskii functional 𝑉(𝑡,𝑥(𝑡))𝒱(1,2), such that
(C1)𝜆1𝑥(𝑡)2𝑉(𝑡,𝑥(𝑡))𝜆2𝑥𝑡2𝜏, (C2)𝔼𝑉(𝑡,𝑥(𝑡))𝜆𝔼𝑉(𝑡,𝑥(𝑡))+𝜆𝜏𝔼𝑉(𝑡,𝑥𝑡),𝑡[𝑡𝑘1,𝑡𝑘),𝑘=1,2,, whenever 𝔼𝑉(𝑡,𝑥𝑡)(𝜇+𝜌)𝔼𝑉(𝑡,𝑥(𝑡)),(C3)𝜇=sup𝑘{𝜆𝑘=(𝜆2/𝜆1)𝛾2𝑘}>1 and 𝜆+(𝜇+𝜌)𝜆𝜏((ln(𝜇+𝜌))/𝛽),
then the trivial solution of system (2.1) is mean squarely stable over 𝒩inf(𝜏+𝛽).

Proof. For any given 𝜀>0, choose 0<𝛿𝜆1/(𝜇+𝜌)𝜆2𝜀. We assume that the initial function 𝜑PC𝑏0(𝛿) and denote the solution 𝑥(𝑡,𝑡0,𝜑) of system (2.1) through (𝑡0,𝜑) by 𝑥(𝑡). In the following, we will prove that 𝑥(𝑡) is mean square stable over 𝒩inf(𝜏+𝛽). For 𝑉(𝑡,𝑥(𝑡))𝒱(1,2), by Itô formula, for 𝑡𝑡𝑘,𝑘=1,2,, we have d𝑉(𝑡,𝑥(𝑡))=𝑉(𝑡,𝑥(𝑡))d𝑡+𝑉𝑥(𝑡,𝑥(𝑡))𝑔(𝑡,𝑥(𝑡))d𝜔(𝑡),(3.1) where 𝑉(𝑡,𝑥(𝑡))=𝑉𝑡(𝑡,𝑥(𝑡))+𝑉𝑥(𝑡,𝑥(𝑡))𝑓+(1/2)tr(𝑔𝑇𝑉𝑥𝑥𝑔).
For 𝑡[𝑡𝑘1,𝑡𝑘), 𝑘=1,2,, integrate (3.1) from 𝑡𝑘1 to 𝑡, we have 𝑡𝑉(𝑡,𝑥(𝑡))=𝑉𝑘1𝑡,𝑥𝑘1+𝑡𝑡𝑘1𝑉(𝑠,𝑥(𝑠))d𝑠+𝑡𝑡𝑘1𝑉𝑥(𝑠,𝑥(𝑠))𝑔(𝑠,𝑥(𝑠))d𝜔(𝑠).(3.2) Taking the mathematical expectation of both sides of the above equation, we obtain 𝑡𝔼𝑉(𝑡,𝑥(𝑡))=𝔼𝑉𝑘1𝑡,𝑥𝑘1+𝑡𝑡𝑘1𝔼𝑉(𝑠,𝑥(𝑠))d𝑠.(3.3) So for 𝑠[𝑡,𝑡+Δ𝑡] with 𝑡+Δ𝑡[𝑡𝑘1,𝑡𝑘) and Δ𝑡>0, if 𝔼𝑉(𝑠,𝑥𝑠)(𝜇+𝜌)𝔼𝑉(𝑠,𝑥(𝑠)), then we have by (C2) 𝔼𝑉(𝑡+Δ𝑡,𝑥(𝑡+Δ𝑡))𝔼𝑉(𝑡,𝑥(𝑡))=𝑡𝑡+Δ𝑡𝔼𝑉(𝑠,𝑥(𝑠))d𝑠𝑡𝑡+Δ𝑡𝜆𝔼𝑉(𝑠,𝑥(𝑠))+𝜆𝜏𝔼𝑉𝑠,𝑥𝑠d𝑠.(3.4)
In what follows, we first prove that for 𝑡[𝑡0𝜏,𝑡1), 𝔼𝑉(𝑡,𝑥(𝑡))𝜆1𝜀2.(3.5) Obviously, for 𝑡[𝑡0𝜏,𝑡0], by (C1) and 𝑥𝑡0PC𝑏0(𝛿), we obtain 𝔼𝑉(𝑡,𝑥(𝑡))𝜆2𝔼𝑥𝑡02𝜏𝜆2𝛿2𝜆1𝜀𝜇+𝜌2<𝜆1𝜀2.(3.6) Now it needs only to prove that for 𝑡(𝑡0,𝑡1), (3.5) holds. Otherwise, there exists 𝑠(𝑡0,𝑡1), such that 𝔼𝑉(𝑠,𝑥(𝑠))>𝜆1𝜀2.(3.7) Set 𝑠1𝑡=inf𝑡0,𝑡1𝔼𝑉(𝑡,𝑥(𝑡))>𝜆1𝜀2,(3.8) then by (3.6), (3.7), and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)) on [𝑡0,𝑡1), we know that 𝑠1(𝑡0,𝑡1),  𝑠𝔼𝑉1𝑠,𝑥1=𝜆1𝜀2,(3.9) and for 𝑡[𝑡0𝜏,𝑠1], (3.5) holds. Set 𝑠2𝑡=sup𝑡0,𝑠1𝜆𝔼𝑉(𝑡,𝑥(𝑡))1𝜀𝜇+𝜌2,(3.10) then by (3.6) and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)), we have 𝑠2[𝑡0,𝑠1), 𝑠𝔼𝑉2𝑠,𝑥2=𝜆1𝜀𝜇+𝜌2,(3.11) and for 𝑡[𝑠2,𝑠1], 𝔼𝑉𝑡,𝑥𝑡𝜆1𝜀2(𝜇+𝜌)𝔼𝑉(𝑡,𝑥(𝑡)),(3.12) which implies with (3.4) and (C3) that for 𝑡[𝑠2,𝑠1], 𝐷+𝔼𝑉(𝑡,𝑥(𝑡))𝜆𝔼𝑉(𝑡,𝑥(𝑡))+𝜆𝜏𝔼𝑉𝑡,𝑥𝑡𝜆+𝜆𝜏(𝜇+𝜌)𝔼𝑉(𝑡,𝑥(𝑡))0.(3.13) This is a contradiction with (3.9) and (3.11).
Now, we assume that, for 𝑡[𝑡𝑚1,𝑡𝑚), 𝑚=1,2,,𝑘, (3.5) holds. For 𝑚=𝑘+1, we will show that (3.5) holds. To this end, we first prove that for 𝜃[𝜏,0], 𝑡𝔼𝑉𝑘𝑡+𝜃,𝑥𝑘𝜆+𝜃1𝜀𝜇+𝜌2.(3.14) Noticing [𝑡𝑘𝜏,𝑡𝑘)[𝑡𝑘1,𝑡𝑘), we assume that there exists some 𝑠[𝑡𝑘𝜏,𝑡𝑘], such that 𝔼𝑉(𝑠,𝑥(𝑠𝜆))>1𝜀𝜇+𝜌2,(3.15) then there are two cases to be considered.
(i)For all 𝑡[𝑡𝑘1,𝑠], 𝔼𝑉(𝑡,𝑥(𝑡))>(𝜆1/(𝜇+𝜌))𝜀2. Hence, for 𝑡[𝑡𝑘1,𝑠], (3.12) and (3.13) hold, which follows by (C3), (3.5), and Lemma (2.1), 𝔼𝑉(𝑠,𝑥(𝑠))exp𝜆+(𝜇+𝜌)𝜆𝜏𝑠𝑡𝑘1𝑡𝔼𝑉𝑘1𝑡,𝑥𝑘1𝜆1𝜀2exp𝜆+(𝜇+𝜌)𝜆𝜏𝑡𝑘𝑡𝑘1𝜆𝜏1𝜀𝜇+𝜌2,(3.16) this is a contradiction with the assumption.(ii)There exists some 𝑡[𝑡𝑘1,𝑠), such that 𝔼𝑉(𝑡,𝑥(𝑡))(𝜆1/(𝜇+𝜌))𝜀2. Set 𝑠1𝑡=sup𝑡𝑘1𝜆,𝑠𝔼𝑉(𝑡,𝑥(𝑡))1𝜀𝜇+𝜌2,(3.17) then 𝑠1[𝑡𝑘1,𝑠), 𝑠𝔼𝑉1𝑠,𝑥1=𝜆1𝜀𝜇+𝜌2(3.18) and for 𝑡[𝑠1,𝑠], (3.12) and (3.13) hold, which is a contradiction with (3.15) and (3.18), that is, (3.14) holds.
By (2.1), (2.2), and (3.14), we have 𝑡𝔼𝑉𝑘𝑡,𝑥𝑘𝑡=𝔼𝑉𝑘,𝐻𝑘𝑥𝑡𝑘𝜆2𝔼𝐻𝑘𝑥𝑡𝑘2𝜏𝜆2𝛾2𝑘𝔼𝑥𝑡𝑘2𝜏𝜆2𝛾2𝑘𝜆1sup𝜏𝜃0𝑡𝔼𝑉𝑘𝑡+𝜃,𝑥𝑘+𝜃𝜆𝑘𝜆1𝜀𝜇+𝜌2<𝜆1𝜀2.(3.19)
Now we will prove that (3.5) holds for 𝑡[𝑡𝑘,𝑡𝑘+1). Otherwise, there exists some 𝑡(𝑡𝑘,𝑡𝑘+1), such that (3.7) holds. Let 𝑠1𝑡=inf𝑡𝑘,𝑡𝑘+1𝔼𝑉(𝑡,𝑥(𝑡))>𝜆1𝜀2.(3.20) Then by (3.14), (3.19), and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)) on [𝑡𝑘,𝑡𝑘+1), we know that 𝑠1(𝑡𝑘,𝑡𝑘+1) and 𝔼𝑉(𝑠1,𝑥(𝑠1))=𝜆1𝜀2. If there exists 𝑡[𝑡𝑘,𝑠1], such that 𝔼𝑉(𝑡,𝑥(𝑡))(𝜆1/𝜇)𝜀2, then let 𝑠2𝑡=sup𝑡𝑘,𝑠1𝜆𝔼𝑉(𝑡,𝑥(𝑡))1𝜀𝜇+𝜌2.(3.21) Otherwise, let 𝑠2=𝑡𝑘. Then for 𝑡[𝑠2,𝑠1], we obtain (3.12) and (3.13), which follows a contradiction.
By mathematical induction, (3.5) holds for any 𝑚=1,2,, which implies that system (2.1) is mean squarely stable.
If substituting condition
(C'1) 𝜆1𝑥(𝑡)2𝑉(𝑡,𝑥(𝑡))𝜆2𝑥(𝑡)2
for (C1) in Theorem (3.1), then we have the following result.

Theorem 3.2. Assume that there exist scalars 𝜆2>𝜆1>0, 𝜆𝜏>0, 𝜆0, 𝜌>0, matrix 𝑃>0 and Lyapunov-Krasovskii functional 𝑉(𝑡,𝑥(𝑡))𝒱(1,2), such that conditions (C'1), (C2), and (C3) hold, then the trivial solution of system (2.1) is mean square stable over 𝒩inf(𝛽).

Proof. The proof is similar to Theorem (3.1), so we omit it. The proof is complete.

Remark 3.3. Comparing the results in Theorems (3.1) and (3.2), we find the influence of the time delay on the mean square stability of system (2.1).

Remark 3.4. When 𝜇>1, the impulses which may be destabilizing, so we require the impulses should not happen so frequently.
When 𝜇=1, we have the following results.

Theorem 3.5. Assume that there exist scalars 𝜆2>𝜆1>0, 𝜆𝜏>0, 𝜆0, matrix 𝑃>0 and Lyapunov-Krasovskii functional 𝑉(𝑡,𝑥(𝑡))𝒱(1,2), such that condition (C1) and (C'2)𝔼𝑉(𝑡,𝑥(𝑡))𝜆𝔼𝑉(𝑡,𝑥(𝑡))+𝜆𝜏𝔼𝑉(𝑡,𝑥𝑡), 𝑡[𝑡𝑘1,𝑡𝑘),𝑘=1,2, whenever 𝔼𝑉(𝑡,𝑥𝑡)𝔼𝑉(𝑡,𝑥𝑡),(C'3)𝜇=sup𝑘{𝜆𝑘=(𝜆2/𝜆1)𝛾2𝑘}=1, 𝜆+𝜆𝜏0hold, then the trivial solution of system (2.1) is mean squarely stable over any impulsive sequences.

Proof. For any given 𝜀>0, choose 0<𝛿𝜆1/𝜆2𝜀. We assume that the initial function 𝜑PC𝑏0(𝛿). In what follows, we first prove that for 𝑡𝑡0, (3.5) holds.
Obviously, for 𝑡[𝑡0𝜏,𝑡0], by (C1) and 𝑥𝑡0PC𝑏0(𝛿), we obtain 𝔼𝑉(𝑡,𝑥(𝑡))𝜆2𝔼𝑥𝑡02𝜏𝜆2𝛿2𝜆1𝜀2.(3.22) Now we should prove that (3.5) holds. Otherwise, there exists 𝑠(𝑡0,𝑡1), such that (3.7) holds. By (3.22) and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)) on [𝑡0,𝑡1), we know there exist 𝑡[𝑡0,𝑡1) and small scalar 𝜌>0, such that 𝔼𝑉𝑡,𝑥𝑡=𝜆1𝜀2(3.23) and for every 𝑡1,𝑡2[𝑡,𝑡+𝜌], 𝑡1<𝑡2, 𝔼𝑉𝑡1,𝑥𝑡1<𝔼𝑉𝑡2,𝑥𝑡2.(3.24) Let 𝑠=inf{𝑡[𝑡0,𝑡1)𝔼𝑉(𝑡,𝑥(𝑡))=𝜆1𝜀2,𝔼𝑉(𝑢,𝑥(𝑢))>𝜆1𝜀2,𝑢(𝑡,𝑡+𝜌1][𝑡0,𝑡1),𝔼𝑉(𝑡1,𝑥(𝑡1))<𝔼𝑉(𝑡2,𝑥(𝑡2)),forevery𝑡1,𝑡2[𝑡,𝑡+𝜌1],𝑡1<𝑡2}, where 𝜌1>0 is some scalar. Then [𝑠,𝑠+𝜌1][𝑡0,t1) and for 𝑡[𝑠,𝑠+𝜌1], 𝔼𝑉𝑡,𝑥𝑡𝔼𝑉(𝑡,𝑥(𝑡)),(3.25) which implies with (C'2) and (C'3) that for 𝑡[𝑠,𝑠+𝜌1], 𝐷+𝔼𝑉(𝑡,𝑥(𝑡))𝜆𝔼𝑉(𝑡,𝑥(𝑡))+𝜆𝜏𝔼𝑉𝑡,𝑥𝑡𝜆+𝜆𝜏𝔼𝑉(𝑡,𝑥(𝑡))0.(3.26) This is a contradiction with the fact 𝔼𝑉(𝑠+𝜌1,𝑥(𝑠+𝜌1))>𝔼𝑉(𝑠,𝑥(𝑠)), that is, for 𝑡[𝑡0𝜏,𝑡1), (3.5) holds.
Now, we assume that, for 𝑡[𝑡𝑚1,𝑡𝑚), 𝑚=1,2,,𝑘, (3.5) holds. For 𝑚=𝑘+1, we will show that (3.5) holds. To this end, we first prove that 𝑡𝔼𝑉𝑘𝑡,𝑥𝑘𝜆1𝜀2.(3.27) In fact, by (2.1), (2.2), (C1), and (C'3) 𝑡𝔼𝑉𝑘𝑡,𝑥𝑘𝑡=𝔼𝑉𝑘,𝐻𝑘𝑥𝑡𝑘𝜆2𝔼𝐻𝑘𝑥𝑡𝑘2𝜏𝜆2𝛾2𝑘𝔼𝑥𝑡𝑘2𝜏𝜆2𝛾2𝑘𝜆1𝑡𝔼𝑉𝑘𝑡,𝑥𝑘𝜆1𝜀2.(3.28)
Secondly, we assume that there exists 𝑠(𝑡𝑘,𝑡𝑘+1), such that (3.7) holds. By (3.27) and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)) on [𝑡𝑘,𝑡𝑘+1), we know that there exist 𝑡[𝑡𝑘,𝑡𝑘+1), 𝜌2>0 such that for every 𝑡1,𝑡2[𝑡,𝑡+𝜌2], 𝑡<𝑡2, (3.23) and (3.24) hold.
Let 𝑠=inf{𝑡[𝑡𝑘,𝑡𝑘+1)𝔼𝑉(𝑡,𝑥(𝑡))=𝜆1𝜀2,𝔼𝑉(𝑢,𝑥(𝑢))>𝜆1𝜀2,𝑢(𝑡,𝑡+𝜌2][𝑡𝑘,𝑡𝑘+1),𝔼𝑉(𝑡1,𝑥(𝑡1))<𝔼𝑉(𝑡2,𝑥(𝑡2)),forevery𝑡1,𝑡2[𝑡,𝑡+𝜌2],𝑡1<𝑡2}, where 𝜌2>0 is some scalar.
Then for 𝑡[𝑠,𝑠+𝜌2], (3.25) and (3.26) hold. This is a contraction, that is, (3.5) holds for 𝑡[𝑡𝑘,𝑡𝑘+1). By mathematical induction, (3.5) holds for any 𝑚=1,2,, which implies that system (2.1) is mean squarely stable.

Remark 3.6. When 𝜇=1, both the continuous dynamics and discrete dynamics are stable under the conditions in Theorem (3.5), so the impulse system can be mean squarely stable regardless of how often or how seldom impulses occur.
When 𝜇<1, we have the following results.

Theorem 3.7. Assume that there exist scalars 𝜆2>𝜆1>0, 𝜆𝜏>0, 𝜆0, matrix 𝑃>0, and Lyapunov-Krasovskii functional 𝑉(𝑡,𝑥(𝑡))𝒱(1,2), such that (C1), (C2), and
(C”3) 𝜇=sup𝑘{𝜆𝑘=(𝜆2/𝜆1)𝛾2𝑘}<1
hold, then
(i)if 0<𝜇𝜆+𝜆𝜏𝜆𝜏ln𝜇, system (2.1) is mean squarely stable over impulsive time sequences 𝒩sup(𝜇ln𝜇/(𝜇𝜆+𝜆𝜏)); (ii)if 𝜇𝜆+𝜆𝜏0, system (2.1) is mean squarely stable over any impulsive time sequences.

Proof. We prove (i) and omit the proof of (ii).
Because 𝜇<1 and 0<𝜇𝜆+𝜆𝜏𝜆𝜏ln𝜇, then there exist a sufficiently small 𝜌0>0, such that 𝜇+𝜌0<1,𝜆𝜇+𝜌0+𝜆𝜏>0,ln𝜇𝜆+𝜇1𝜆𝜏ln𝜇+𝜌0𝜆+𝜇+𝜌01𝜆𝜏.(3.29) For any given 𝜀>0, choose 0<𝛿((𝜇+𝜌0)𝜆1)/𝜆2𝜀. We assume the initial function 𝜑PC𝑏0(𝛿). For 𝑡[𝑡0𝜏,𝑡0], by (C1), (3.29), and 𝑥𝑡0PC𝑏0(𝛿), we obtain 𝔼𝑉(𝑡,𝑥(𝑡))𝜆2𝔼𝑥𝑡02𝜏𝜆2𝛿2𝜆1𝜇+𝜌0𝜀2<𝜆1𝜀2.(3.30)
Now we will prove that (3.5) holds. Otherwise, there exists 𝑠(𝑡0,𝑡1), such that (3.7) holds. Set 𝑡𝑡=inf𝑡0,𝑡1𝔼𝑉(𝑡,𝑥(𝑡))𝜆1𝜀2,(3.31) then by (3.7), (3.30), and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)) on [𝑡0,𝑡1), we know that 𝑡(𝑡0,𝑡1), 𝔼𝑉(𝑡,𝑥(𝑡))=𝜆1𝜀2. Set 𝑡𝑡=sup𝑡0,𝑡𝔼𝑉(𝑡,𝑥(𝑡))𝜆1𝜇+𝜌0𝜀2,(3.32) then by (3.30) and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)), we have 𝑡[𝑡0,𝑡), 𝔼𝑉(𝑡,𝑥(𝑡))=𝜆1(𝜇+𝜌0)𝜀2 and for 𝑡[𝑡,𝑡], 𝔼𝑉𝑡,𝑥𝑡𝜆1𝜀21𝜌0+𝜇𝔼𝑉(𝑡,𝑥(𝑡)).(3.33) Conditions (C2) and (C”3) imply that for 𝑡[𝑡,𝑡], 𝐷+𝔼𝑉(𝑡,𝑥(𝑡))𝜆𝔼𝑉(𝑡,𝑥(𝑡))+𝜆𝜏𝔼𝑉𝑡,𝑥𝑡𝜆𝜆+𝜏𝜇+𝜌0𝔼𝑉(𝑡,𝑥(𝑡)).(3.34) By Lemma (2.1), (3.29), (3.26), and 𝑡1𝑡0(ln𝜇/(𝜆+(𝜆𝜏/𝜇))), we have 𝑡𝔼𝑉𝑡,𝑥𝜆exp𝜆+𝜏𝜇+𝜌0𝑡𝑡𝔼𝑉𝑡,𝑥𝑡𝜆<𝑒𝑥𝑝𝜆+𝜏𝜇+𝜌0𝑡1𝑡0𝜇+𝜌0𝜆1𝜀2𝜆1𝜀2,(3.35) this is a contradiction with the fact 𝔼𝑉(𝑡,𝑥(𝑡))=𝜆1𝜀2.
Now, we assume that, for 𝑡[𝑡𝑚1,𝑡𝑚), 𝑚=1,2,,𝑘, (3.5) holds. For 𝑚=𝑘+1, we will show that (3.5) holds. To this end, we first prove that 𝑡𝔼𝑉𝑘𝑡,𝑥𝑘𝜇+𝜌0𝜆1𝜀2.(3.36) In fact, by (2.1), (2.2), (C1), and (C”3) 𝑡𝔼𝑉𝑘𝑡,𝑥𝑘𝑡=𝔼𝑉𝑘,𝐻𝑘𝑥𝑡𝑘𝜆2𝔼𝐻𝑘𝑥𝑡𝑘2𝜏𝜆2𝛾2𝑘𝔼𝑥𝑡𝑘2𝜏𝜆2𝛾2𝑘𝜆1𝑡𝔼𝑉𝑘𝑡,𝑥𝑘𝜇+𝜌0𝜆1𝜀2.(3.37)
Secondly, we assume that there exists 𝑠(𝑡𝑘,𝑡𝑘+1), such that (3.7) holds. Set 𝑡𝑡=inf𝑡𝑘,𝑡𝑘+1𝔼𝑉(𝑡,𝑥(𝑡))𝜆1𝜀2,𝑡𝑡=sup𝑡𝑘,𝑡𝔼𝑉(𝑡,𝑥(𝑡))𝜆1𝜇+𝜌0𝜀2,(3.38) then by (3.37) and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)) on [𝑡𝑘,𝑡𝑘+1), we have 𝑡(𝑡𝑘,𝑡𝑘+1), 𝑡[𝑡𝑘,𝑡) and 𝔼𝑉(𝑡,𝑥(𝑡))=𝜆1𝜀2, 𝔼𝑉(𝑡,𝑥(𝑡))=(𝜇+𝜌0)𝜆1𝜀2.
On the other hand, for 𝑡[𝑡,𝑡], (3.33) and (3.34) hold, which lead to a contradiction, that is, (3.5) holds for 𝑡[𝑡𝑘,𝑡𝑘+1). By mathematical induction, (3.5) holds for any 𝑚=1,2,, which implies that system (2.1) is mean squarely stable.

4. Application and Numerical Example

As an application, we consider the stochastic impulsive Hopfield neural network with delays in Yang et al. [9] as follows: 𝑥d𝑥(𝑡)=𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑔𝑡d𝑡+𝜎𝑡,𝑥(𝑡),𝑥𝑡d𝜔(𝑡),𝑡𝑡0,𝑡𝑡𝑘,𝑥𝑡𝑘=𝐻𝑘𝑥𝑡𝑘𝑥𝑡,𝑘=1,2,,0[],+𝜃=𝜑(𝜃),𝑠𝜏,0(4.1) where the initial value 𝜑(𝑠)PC𝑏0(𝛿), 𝑥(𝑡)=(𝑥1(𝑡),𝑥2(𝑡),,𝑥𝑛(𝑡))𝑇𝑛 is the state vector, 𝐶=diag(𝑐1,𝑐2,,𝑐𝑛), 𝑐𝑖>0 is the neuron-charging time constant, 𝐴=(𝑎𝑖𝑗)𝑛×𝑛 are, respectively, the connection weight matrix, the discretely delayed connection weight matrix. 𝑓(𝑥(𝑡))=(𝑓1(𝑥1(𝑡)),𝑓2(𝑥2(𝑡)),,𝑓𝑛(𝑥𝑛(𝑡)))𝑇𝑛 and 𝑔(𝑥𝑡)=(𝑔1(𝑥1𝑡),𝑔2(𝑥2𝑡),,𝑔𝑛(𝑥𝑛𝑡))𝑛, where 𝑓𝑖(𝑥𝑖(𝑡)) and 𝑔𝑖(𝑥𝑖𝑡) denote, respectively, the measures of response or activation to its incoming potentials of the unit 𝑖 at time 𝑡 and time 𝑡𝜏𝑖. We also assume that 𝐻𝑘(0)=0, (𝑘=1,2,), 𝑓(0)=0, 𝑔(0)=0, and 𝜎(𝑡,0,0)=0, then system (4.1) admits an equilibrium solution 𝑥(𝑡)0. Moreover, we assume that 𝐻() satisfies (2.2), and 𝑓(), 𝑔(), 𝜎() satisfy 𝑔𝑥𝑓(𝑥(𝑡))𝐹𝑥(𝑡),𝑡𝐺𝑥𝑡𝜎,(4.2)tr𝑇𝑡,𝑥(𝑡),𝑥𝑡𝜎𝑡,𝑥(𝑡),𝑥𝑡𝐾𝑥(𝑡)2+𝐾𝜏𝑥𝑡2,(4.3) where 𝐹, 𝐺, 𝐾, and 𝐾𝜏 are known constant matrices with appropriate dimensions.

Corollary 4.1. Assume that there exist positive scalars 𝜀1, 𝜀2, 𝛽, symmetric matrix 𝑃>0 and 𝜇=sup𝑘{𝜆𝑘=(𝜆max(𝑃)/𝜆min(𝑃))𝛾2𝑘}. Then the following results hold:
(i)if 𝜇>1, 𝜆+𝜇𝜆𝜏<ln𝜇/𝛽, then system (4.1) is mean squarely stable over impulsive time sequence 𝒩inf(𝜏+𝛽);(ii)if 𝜇=1, 𝜆+𝜆𝜏0, then system (4.1) is mean squarely stable over any impulsive time sequence;(iii)if 𝜇<1 and 0<𝜇𝜆+𝜆𝜏𝜆𝜏ln𝜇, then system (4.1) is mean squarely stable over impulsive time sequence 𝒩sup(𝜇ln𝜇/(𝜇𝜆+𝜆𝜏));(iv)if 𝜇<1, 𝜇𝜆+𝜆𝜏0, then system (4.1) is mean squarely stable over any impulsive time sequence, where 𝜆=𝜆max2𝐶+𝜀1𝐴𝐴𝑇𝑃+𝜀2𝐵𝐵𝑇𝑃+𝑃1𝜀11𝐹𝑇𝐹+𝜆max(𝑃)𝐾𝑇𝐾,𝜆𝜏=𝜆max𝑃1𝜀21𝐺𝑇𝐺+𝜆max(𝑃)𝐾𝑇𝜏𝐾𝜏.(4.4)

Remark 4.2. Obviously, for this application, we extended and improved the according results in Yang et al. [9].
By Corollary (4.1), we consider the numerical example in Yang et al. [9]. d𝑥1(𝑡)d𝑥2(=𝑥𝑡)10.50012.21𝑥(𝑡)2(+𝑡)1.20.20.62.4sin𝑥1(𝑡)arctan𝑥2(+×𝑡)1.60.30.51.8sin𝑥11𝑡2arctan𝑥21𝑡2d𝑡+2𝑥1(𝑡)𝑥21𝑡3𝑥11𝑡2𝑥2(𝑡)d𝜔1(𝑡)d𝜔2(𝑡),𝑡𝑡0,𝑡𝑡𝑘,𝑥1𝑡𝑘𝑥2𝑡𝑘=𝑒0.1𝑘𝑥0.50.150.120.61𝑡𝑘𝑥2𝑡𝑘,𝑘=1,2,,(4.5) where 𝑡0=0.
Similar to the result, we can verify that the point (0,0)𝑇 is an equilibrium point and can obtain by calculation that 𝑃=0.56000.68,𝐾𝑇𝐾=4001,(4.6) and 𝜀𝑖=1(𝑖=1,2),𝜆max(𝑃)=0.68,𝜆min(𝑃)=0.56,𝛾𝑘=0.620exp(0.1𝑘),𝐾𝑇𝜏𝐾𝜏=𝐼,𝐹=𝐺=𝐼,𝜇=0.4668<1,𝜆=12.0443,𝜆𝜏=3, and, hence, we have 𝜇𝜆+𝜆𝜏=2.6223, which implies by (iv) in Corollary (4.1) that the above system is mean squarely stable over any impulsive time sequence.

5. Conclusion

In this paper, mean square stability of a class of impulsive stochastic differential equations with time delay has been considered. By Lyapunov-Krasovakii function and stochastic analysis, we obtain some new criteria ensuring mean square stability of the system (2.1). Some related results in Chen and Zheng [3] and Yang et al. [9] have been improved.

Acknowledgment

This work is supported by Distinguished Expert Science Foundation of Naval Aeronautical and Astronautical University.

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