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 š“ and šµ, the notion š“ā‰„šµ (š“>šµ, š“ā‰¤šµ, š“<šµ) means š“āˆ’šµ is positive semidefinite (positive definite, negative semidefinite, negative definite) matrix. šœ†max(ā‹…) (šœ†min(ā‹…)) represents the maximum (minimum) eigenvalue of the corresponding matrix, respectively. ā€–ā‹…ā€– denotes Euclidean norm for vectors or the spectral norm of matrices. Moreover, let (Ī©,ā„±,{ā„±š‘”}š‘”ā‰„0,š‘ƒ) be a complete probability space with a filtration {ā„±š‘”}š‘”ā‰„0 satisfying the usual conditions, that is, the filtration contains all P-null sets and is right continuous. Let PC([āˆ’šœ,0],ā„š‘›) denote the set of piecewise right continuous function šœ™āˆ¶[āˆ’šœ,0]ā†’ā„š‘› with the norm defined by ā€–šœ™ā€–šœ=supāˆ’šœā‰¤šœƒā‰¤0ā€–šœ™(šœƒ)ā€–, where šœ is a known positive constant, PC(š›æ)={šœ‘āˆ£šœ‘āˆˆPC([āˆ’šœ,0],ā„š‘›),ā€–šœ‘ā€–šœā‰¤š›æ}, PCš‘ā„±0([āˆ’šœ,0],ā„š‘›) denote the family of all ā„±0-measurable PC([āˆ’šœ,0],ā„š‘›)-valued stochastic process šœ‘={šœ‘(š‘ )āˆ¶āˆ’šœā‰¤š‘ ā‰¤0} with supāˆ’šœā‰¤š‘ ā‰¤0š”¼{ā€–šœ‘(š‘ )ā€–2}<āˆž, where š”¼{ā‹…} represents the mathematical expectation operator with respect to the probability measure š‘ƒ, PCš‘ā„±0(š›æ)={šœ‘āˆ£šœ‘āˆˆPCš‘ā„±0([āˆ’šœ,0],ā„š‘›),supāˆ’šœā‰¤š‘ ā‰¤0š”¼{ā€–šœ‘(š‘ )ā€–2}ā‰¤š›æ, ā„’ denote the well-known ā„’-operator given by the ItĆ“'s formula.

In this paper, we consider a class of ItĆ“ impulsive stochastic differential systems with time delay ī€·dš‘„(š‘”)=š‘“š‘”,š‘„(š‘”),š‘„š‘”ī€øī€·dš‘”+š‘”š‘”,š‘„(š‘”),š‘„š‘”ī€ødšœ”(š‘”),š‘”ā‰„š‘”0,š‘”ā‰ š‘”š‘˜,š‘„ī€·š‘”š‘˜ī€ø=š»š‘˜ī€·š‘„ī€·š‘”āˆ’š‘˜š‘„ī€·š‘”ī€øī€ø,š‘˜=1,2,ā€¦,0ī€ø[],+šœƒ=šœ‘(šœƒ),šœƒāˆˆāˆ’šœ,0(2.1) where the initial value šœ‘āˆˆPCš‘ā„±0(š›æ), the fixed impulsive time moments š‘”š‘˜ satisfy 0ā‰¤š‘”0<š‘”1<š‘”2<ā‹Æ<š‘”š‘˜<ā‹Æ(š‘”š‘˜ā†’āˆž as š‘˜ā†’āˆž). š‘„(š‘”)āˆˆā„š‘› is the system state, š‘“āˆˆš¶(ā„Ć—ā„š‘›Ć—ā„š‘›,ā„š‘›), š‘”āˆˆš¶(ā„Ć—ā„š‘›Ć—ā„š‘›,ā„š‘›Ć—š‘š). šœ”(š‘”)āˆˆā„š‘š is an standard Brownian motion defined on the complete probability space (Ī©,ā„±,{ā„±š‘”}š‘”ā‰„0,š‘ƒ). Besides, we assume that š»š‘˜(0)=0,(š‘˜=1,2,ā€¦), š‘“(š‘”,0,0)=0, š‘”(š‘”,0,0)=0 and ā€–ā€–š»š‘˜ī€·š‘„ī€·š‘”āˆ’š‘˜ā€–ā€–ī€øī€øā‰¤š›¾š‘˜ā€–ā€–š‘„ī€·š‘”āˆ’š‘˜ī€øā€–ā€–,š›¾š‘˜ā‰„0,š‘˜=1,2,ā€¦.(2.2)

In the following, we will divide three cases to consider the mean square stability of system (2.1). We denote by š’©inf(š›½) and š’©sup(š›½) the class of impulsive time sequences that satisfy infš‘˜{š‘”š‘˜āˆ’š‘”š‘˜āˆ’1}ā‰„š›½ and supš‘˜{š‘”š‘˜āˆ’š‘”š‘˜āˆ’1}ā‰¤š›½, respectively.

We need the following lemma and definitions.

Lemma 2.1 (Chaplygin Comparison Theorem, see Shi et al. [12]). Assume that š‘“,š¹āˆˆš¶(šŗ), š‘”āŠ‚ā„2 and š‘“(š‘”,š‘„)<š¹(š‘”,š‘„),(š‘”,š‘„)āˆˆšŗ.(2.3) If šœ™(š‘”) (š‘”āˆˆš‘ˆ1) and Ī¦(š‘”) (š‘”āˆˆš‘ˆ2) are the solutions of Cauchy problems š‘„Ģ‡š‘„=š‘“(š‘”,š‘„),(š‘”,š‘„)āˆˆšŗ,(šœ)=šœ‰,Ģ‡š‘„=š¹(š‘”,š‘„),(š‘”,š‘„)āˆˆšŗ,š‘„(šœ)=šœ‰,(2.4) respectively, then for š‘”āˆˆ(šœ,āˆž)āˆ©š‘ˆ1āˆ©š‘ˆ2, šœ™(š‘”)<Ī¦(š‘”)(2.5) and for š‘”āˆˆ(āˆ’āˆž,šœ)āˆ©š‘ˆ1āˆ©š‘ˆ2, ā€ƒ šœ™(š‘”)>Ī¦(š‘”).(2.6)

Definition 2.2. For a given class š’© of admissible impulsive time sequence, the solution of (2.1) is called mean squarely stable if for any šœ€>0, there exists a scalar š›æ>0, such that the initial function šœ‘āˆˆPCš‘ā„±0(š›æ) implies š”¼{ā€–š‘„(š‘”)ā€–2}<šœ€, š‘”ā‰„š‘”0 for all admissible time sequence in š’©.

Definition 2.3 (see Yang et al. [9]). The function š‘‰āˆ¶[š‘”0āˆ’šœ,āˆž)Ɨā„š‘›ā†’ā„+ belongs to class š’±(1,2) if
(1)the function š‘‰(š‘”,š‘„) is continuously differentiable in š‘” and twice continuously differentiable in š‘„ on each of the sets [š‘”š‘˜āˆ’1,š‘”š‘˜)Ɨā„š‘›,(š‘˜=1,2,ā€¦) and for all š‘”ā‰„š‘”0, š‘‰(š‘”,0)ā‰”0, (2)š‘‰(š‘”,š‘„) is locally Lipschitaian in š‘„, (3)for each š‘˜=1,2,ā€¦, there exist finite limits limī€·š‘”(š‘”,š‘¦)ā†’āˆ’š‘˜ī€ø,š‘„š‘‰ī€·š‘”(š‘”,š‘¦)=š‘‰āˆ’š‘˜ī€ø,š‘„,limī€·š‘”(š‘”,š‘¦)ā†’+š‘˜ī€ø,š‘„š‘‰ī€·š‘”(š‘”,š‘¦)=š‘‰+š‘˜ī€ø,š‘‰ī€·š‘”,š‘„+š‘˜ī€øī€·š‘”,š‘„=š‘‰š‘˜ī€ø.,š‘„(2.7)

3. Main Results

Theorem 3.1. Assume that there exist scalars šœ†2>šœ†1>0, šœ†šœ>0, š›½>0, šœ†ā‰¤0, šœŒ>0 matrix š‘ƒ>0 and Lyapunov-Krasovskii functional š‘‰(š‘”,š‘„(š‘”))āˆˆš’±(1,2), such that
(C1)šœ†1ā€–š‘„(š‘”)ā€–2ā‰¤š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†2ā€–š‘„š‘”ā€–2šœ, (C2)š”¼ā„’š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†š”¼š‘‰(š‘”,š‘„(š‘”))+šœ†šœš”¼š‘‰(š‘”,š‘„š‘”),š‘”āˆˆ[š‘”š‘˜āˆ’1,š‘”š‘˜),š‘˜=1,2,ā€¦, whenever š”¼š‘‰(š‘”,š‘„š‘”)ā‰¤(šœ‡+šœŒ)š”¼š‘‰(š‘”,š‘„(š‘”)),(C3)šœ‡=supš‘˜āˆˆā„•{šœ†ī…žš‘˜=(šœ†2/šœ†1)š›¾2š‘˜}>1 and šœ†+(šœ‡+šœŒ)šœ†šœā‰¤āˆ’((ln(šœ‡+šœŒ))/š›½),
then the trivial solution of system (2.1) is mean squarely stable over š’©inf(šœ+š›½).

Proof. For any given šœ€>0, choose āˆš0<š›æā‰¤šœ†1/(šœ‡+šœŒ)šœ†2šœ€. We assume that the initial function šœ‘āˆˆPCš‘ā„±0(š›æ) and denote the solution š‘„(š‘”,š‘”0,šœ‘) of system (2.1) through (š‘”0,šœ‘) by š‘„(š‘”). In the following, we will prove that š‘„(š‘”) is mean square stable over š’©inf(šœ+š›½). For š‘‰(š‘”,š‘„(š‘”))āˆˆš’±(1,2), by ItĆ“ formula, for š‘”ā‰ š‘”š‘˜,š‘˜=1,2,ā€¦, we have dš‘‰(š‘”,š‘„(š‘”))=ā„’š‘‰(š‘”,š‘„(š‘”))dš‘”+š‘‰š‘„(š‘”,š‘„(š‘”))š‘”(š‘”,š‘„(š‘”))dšœ”(š‘”),(3.1) where ā„’š‘‰(š‘”,š‘„(š‘”))=š‘‰š‘”(š‘”,š‘„(š‘”))+š‘‰š‘„(š‘”,š‘„(š‘”))š‘“+(1/2)tr(š‘”š‘‡š‘‰š‘„š‘„š‘”).
For š‘”āˆˆ[š‘”š‘˜āˆ’1,š‘”š‘˜), š‘˜=1,2,ā€¦, integrate (3.1) from š‘”š‘˜āˆ’1 to š‘”, we have ī€·š‘”š‘‰(š‘”,š‘„(š‘”))=š‘‰š‘˜āˆ’1ī€·š‘”,š‘„š‘˜āˆ’1+ī€œī€øī€øš‘”š‘”š‘˜āˆ’1ī€œā„’š‘‰(š‘ ,š‘„(š‘ ))dš‘ +š‘”š‘”š‘˜āˆ’1š‘‰š‘„(š‘ ,š‘„(š‘ ))š‘”(š‘ ,š‘„(š‘ ))dšœ”(š‘ ).(3.2) Taking the mathematical expectation of both sides of the above equation, we obtain ī€·š‘”š”¼š‘‰(š‘”,š‘„(š‘”))=š”¼š‘‰š‘˜āˆ’1ī€·š‘”,š‘„š‘˜āˆ’1+ī€œī€øī€øš‘”š‘”š‘˜āˆ’1š”¼ā„’š‘‰(š‘ ,š‘„(š‘ ))dš‘ .(3.3) So for š‘ āˆˆ[š‘”,š‘”+Ī”š‘”] with š‘”+Ī”š‘”āˆˆ[š‘”š‘˜āˆ’1,š‘”š‘˜) and Ī”š‘”>0, if š”¼š‘‰(š‘ ,š‘„š‘ )ā‰¤(šœ‡+šœŒ)š”¼š‘‰(š‘ ,š‘„(š‘ )), then we have by (C2) ī€œš”¼š‘‰(š‘”+Ī”š‘”,š‘„(š‘”+Ī”š‘”))āˆ’š”¼š‘‰(š‘”,š‘„(š‘”))=š‘”š‘”+Ī”š‘”ā‰¤ī€œš”¼ā„’š‘‰(š‘ ,š‘„(š‘ ))dš‘ š‘”š‘”+Ī”š‘”ī€·šœ†š”¼š‘‰(š‘ ,š‘„(š‘ ))+šœ†šœī€·š”¼š‘‰š‘ ,š‘„š‘ ī€øī€ødš‘ .(3.4)
In what follows, we first prove that for š‘”āˆˆ[š‘”0āˆ’šœ,š‘”1), š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†1šœ€2.(3.5) Obviously, for š‘”āˆˆ[š‘”0āˆ’šœ,š‘”0], by (C1) and š‘„š‘”0āˆˆPCš‘ā„±0(š›æ), we obtain š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†2š”¼ī‚†ā€–ā€–š‘„š‘”0ā€–ā€–2šœī‚‡ā‰¤šœ†2š›æ2ā‰¤šœ†1šœ€šœ‡+šœŒ2<šœ†1šœ€2.(3.6) Now it needs only to prove that for š‘”āˆˆ(š‘”0,š‘”1), (3.5) holds. Otherwise, there exists š‘ āˆˆ(š‘”0,š‘”1), such that š”¼š‘‰(š‘ ,š‘„(š‘ ))>šœ†1šœ€2.(3.7) Set š‘ 1ī€½ī€·š‘”=infš‘”āˆˆ0,š‘”1ī€øāˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))>šœ†1šœ€2ī€¾,(3.8) then by (3.6), (3.7), and the continuity of š”¼š‘‰(š‘”,š‘„(š‘”)) on [š‘”0,š‘”1), we know that š‘ 1āˆˆ(š‘”0,š‘”1),ā€ƒ ī€·š‘ š”¼š‘‰1ī€·š‘ ,š‘„1ī€øī€ø=šœ†1šœ€2,(3.9) and for š‘”āˆˆ[š‘”0āˆ’šœ,š‘ 1], (3.5) holds. Set š‘ 2ī‚»ī€ŗš‘”=supš‘”āˆˆ0,š‘ 1ī€øšœ†āˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤1šœ€šœ‡+šœŒ2ī‚¼,(3.10) then by (3.6) and the continuity of š”¼š‘‰(š‘”,š‘„(š‘”)), we have š‘ 2āˆˆ[š‘”0,š‘ 1), ī€·š‘ š”¼š‘‰2ī€·š‘ ,š‘„2=šœ†ī€øī€ø1šœ€šœ‡+šœŒ2,(3.11) and for š‘”āˆˆ[š‘ 2,š‘ 1], ī€·š”¼š‘‰š‘”,š‘„š‘”ī€øā‰¤šœ†1šœ€2ā‰¤(šœ‡+šœŒ)š”¼š‘‰(š‘”,š‘„(š‘”)),(3.12) which implies with (3.4) and (C3) that for š‘”āˆˆ[š‘ 2,š‘ 1], š·+š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†š”¼š‘‰(š‘”,š‘„(š‘”))+šœ†šœī€·š”¼š‘‰š‘”,š‘„š‘”ī€øā‰¤ī€·šœ†+šœ†šœī€ø(šœ‡+šœŒ)š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤0.(3.13) This is a contradiction with (3.9) and (3.11).
Now, we assume that, for š‘”āˆˆ[š‘”š‘šāˆ’1,š‘”š‘š), š‘š=1,2,ā€¦,š‘˜, (3.5) holds. For š‘š=š‘˜+1, we will show that (3.5) holds. To this end, we first prove that for šœƒāˆˆ[āˆ’šœ,0], ī€·š‘”š”¼š‘‰āˆ’š‘˜ī€·š‘”+šœƒ,š‘„āˆ’š‘˜ā‰¤šœ†+šœƒī€øī€ø1šœ€šœ‡+šœŒ2.(3.14) Noticing [š‘”š‘˜āˆ’šœ,š‘”š‘˜)āŠ‚[š‘”š‘˜āˆ’1,š‘”š‘˜), we assume that there exists some š‘ āˆˆ[š‘”š‘˜āˆ’šœ,š‘”š‘˜], such that š”¼š‘‰(š‘ āˆ’,š‘„(š‘ āˆ’šœ†))>1šœ€šœ‡+šœŒ2,(3.15) then there are two cases to be considered.
(i)For all š‘”āˆˆ[š‘”š‘˜āˆ’1,š‘ ], š”¼š‘‰(š‘”āˆ’,š‘„(š‘”āˆ’))>(šœ†1/(šœ‡+šœŒ))šœ€2. Hence, for š‘”āˆˆ[š‘”š‘˜āˆ’1,š‘ ], (3.12) and (3.13) hold, which follows by (C3), (3.5), and Lemma (2.1), š”¼š‘‰(š‘ āˆ’,š‘„(š‘ āˆ’))ā‰¤expī€½ī€·šœ†+(šœ‡+šœŒ)šœ†šœī€øī€·š‘ āˆ’š‘”š‘˜āˆ’1ī€·š‘”ī€øī€¾š”¼š‘‰š‘˜āˆ’1ī€·š‘”,š‘„š‘˜āˆ’1ī€øī€øā‰¤šœ†1šœ€2expī€½ī€·šœ†+(šœ‡+šœŒ)šœ†šœš‘”ī€øī€·š‘˜āˆ’š‘”š‘˜āˆ’1ā‰¤šœ†āˆ’šœī€øī€¾1šœ€šœ‡+šœŒ2,(3.16) this is a contradiction with the assumption.(ii)There exists some š‘”āˆˆ[š‘”š‘˜āˆ’1,š‘ ), such that š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤(šœ†1/(šœ‡+šœŒ))šœ€2. Set š‘ 1ī‚»ī€ŗš‘”=supš‘”āˆˆš‘˜āˆ’1ī€øšœ†,š‘ āˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤1šœ€šœ‡+šœŒ2ī‚¼,(3.17) then š‘ 1āˆˆ[š‘”š‘˜āˆ’1,š‘ ), ī€·š‘ š”¼š‘‰1ī€·š‘ ,š‘„1=šœ†ī€øī€ø1šœ€šœ‡+šœŒ2(3.18) and for š‘”āˆˆ[š‘ 1,š‘ ], (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 ī€·š‘”š”¼š‘‰š‘˜ī€·š‘”,š‘„š‘˜ī€·š‘”ī€øī€ø=š”¼š‘‰š‘˜,š»š‘˜ī€·š‘„ī€·š‘”āˆ’š‘˜ī€øī€øī€øā‰¤šœ†2š”¼ī‚†ā€–ā€–š»š‘˜ī€·š‘„ī€·š‘”āˆ’š‘˜ā€–ā€–ī€øī€ø2šœī‚‡ā‰¤šœ†2š›¾2š‘˜š”¼ī‚†ā€–ā€–š‘„ī€·š‘”āˆ’š‘˜ī€øā€–ā€–2šœī‚‡ā‰¤šœ†2š›¾2š‘˜šœ†1supāˆ’šœā‰¤šœƒā‰¤0ī€·š‘”š”¼š‘‰š‘˜ī€·š‘”+šœƒ,š‘„āˆ’š‘˜+šœƒī€øī€øā‰¤šœ†ī…žš‘˜šœ†1šœ€šœ‡+šœŒ2<šœ†1šœ€2.(3.19)
Now we will prove that (3.5) holds for š‘”āˆˆ[š‘”š‘˜,š‘”š‘˜+1). Otherwise, there exists some š‘”āˆˆ(š‘”š‘˜,š‘”š‘˜+1), such that (3.7) holds. Let š‘ 1ī€½ī€·š‘”=infš‘”āˆˆš‘˜,š‘”š‘˜+1ī€øāˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))>šœ†1šœ€2ī€¾.(3.20) Then by (3.14), (3.19), and the continuity of š”¼š‘‰(š‘”,š‘„(š‘”)) on [š‘”š‘˜,š‘”š‘˜+1), we know that š‘ 1āˆˆ(š‘”š‘˜,š‘”š‘˜+1) and š”¼š‘‰(š‘ 1,š‘„(š‘ 1))=šœ†1šœ€2. If there exists š‘”āˆˆ[š‘”š‘˜,š‘ 1], such that š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤(šœ†1/šœ‡)šœ€2, then let š‘ 2ī‚»ī€ŗš‘”=supš‘”āˆˆš‘˜,š‘ 1ī€øšœ†āˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤1šœ€šœ‡+šœŒ2ī‚¼.(3.21) Otherwise, let š‘ 2=š‘”š‘˜. Then for š‘”āˆˆ[š‘ 2,š‘ 1], we obtain (3.12) and (3.13), which follows a contradiction.
By mathematical induction, (3.5) holds for any š‘š=1,2,ā€¦, which implies that system (2.1) is mean squarely stable.
If substituting condition
(C'1) šœ†1ā€–š‘„(š‘”)ā€–2ā‰¤š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†2ā€–š‘„(š‘”)ā€–2
for (C1) in Theorem (3.1), then we have the following result.

Theorem 3.2. Assume that there exist scalars šœ†2>šœ†1>0, šœ†šœ>0, šœ†ā‰¤0, šœŒ>0, matrix š‘ƒ>0 and Lyapunov-Krasovskii functional š‘‰(š‘”,š‘„(š‘”))āˆˆš’±(1,2), such that conditions (C'1), (C2), and (C3) hold, then the trivial solution of system (2.1) is mean square stable over š’©inf(š›½).

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 šœ‡>1, the impulses which may be destabilizing, so we require the impulses should not happen so frequently.
When šœ‡=1, we have the following results.

Theorem 3.5. Assume that there exist scalars šœ†2>šœ†1>0, šœ†šœ>0, šœ†ā‰¤0, matrix š‘ƒ>0 and Lyapunov-Krasovskii functional š‘‰(š‘”,š‘„(š‘”))āˆˆš’±(1,2), such that condition (C1) and (C'2)š”¼ā„’š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†š”¼š‘‰(š‘”,š‘„(š‘”))+šœ†šœš”¼š‘‰(š‘”,š‘„š‘”), š‘”āˆˆ[š‘”š‘˜āˆ’1,š‘”š‘˜),š‘˜=1,2,ā€¦ whenever š”¼š‘‰(š‘”,š‘„š‘”)ā‰¤š”¼š‘‰(š‘”,š‘„š‘”),(C'3)šœ‡=supš‘˜āˆˆā„•{šœ†ī…žš‘˜=(šœ†2/šœ†1)š›¾2š‘˜}=1, šœ†+šœ†šœā‰¤0hold, then the trivial solution of system (2.1) is mean squarely stable over any impulsive sequences.

Proof. For any given šœ€>0, choose āˆš0<š›æā‰¤šœ†1/šœ†2šœ€. We assume that the initial function šœ‘āˆˆPCš‘ā„±0(š›æ). In what follows, we first prove that for š‘”ā‰„š‘”0, (3.5) holds.
Obviously, for š‘”āˆˆ[š‘”0āˆ’šœ,š‘”0], by (C1) and š‘„š‘”0āˆˆPCš‘ā„±0(š›æ), we obtain š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†2š”¼ī‚†ā€–ā€–š‘„š‘”0ā€–ā€–2šœī‚‡ā‰¤šœ†2š›æ2ā‰¤šœ†1šœ€2.(3.22) Now we should prove that (3.5) holds. Otherwise, there exists š‘ āˆˆ(š‘”0,š‘”1), such that (3.7) holds. By (3.22) and the continuity of š”¼š‘‰(š‘”,š‘„(š‘”)) on [š‘”0,š‘”1), we know there exist š‘”āˆˆ[š‘”0,š‘”1) and small scalar šœŒ>0, such that ī€·š”¼š‘‰ī€·š‘”,š‘„š‘”ī€øī€ø=šœ†1šœ€2(3.23) and for every š‘”1,š‘”2āˆˆ[š‘”,š‘”+šœŒ], š‘”1<š‘”2, ī€·š”¼š‘‰š‘”1ī€·,š‘„š‘”1ī€·ī€øī€ø<š”¼š‘‰š‘”2ī€·,š‘„š‘”2.ī€øī€ø(3.24) Let š‘ =inf{š‘”āˆˆ[š‘”0,š‘”1)āˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))=šœ†1šœ€2,š”¼š‘‰(š‘¢,š‘„(š‘¢))>šœ†1šœ€2,š‘¢āˆˆ(š‘”,š‘”+šœŒ1]āŠ‚[š‘”0,š‘”1),š”¼š‘‰(š‘”1,š‘„(š‘”1))<š”¼š‘‰(š‘”2,š‘„(š‘”2)),foreveryš‘”1,š‘”2āˆˆ[š‘”,š‘”+šœŒ1],š‘”1<š‘”2}, where šœŒ1>0 is some scalar. Then [š‘ ,š‘ +šœŒ1]āŠ‚[š‘”0,t1) and for š‘”āˆˆ[š‘ ,š‘ +šœŒ1], ī€·š”¼š‘‰š‘”,š‘„š‘”ī€øā‰¤š”¼š‘‰(š‘”,š‘„(š‘”)),(3.25) which implies with (C'2) and (C'3) that for š‘”āˆˆ[š‘ ,š‘ +šœŒ1], š·+š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†š”¼š‘‰(š‘”,š‘„(š‘”))+šœ†šœī€·š”¼š‘‰š‘”,š‘„š‘”ī€øā‰¤ī€·šœ†+šœ†šœī€øš”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤0.(3.26) This is a contradiction with the fact š”¼š‘‰(š‘ +šœŒ1,š‘„(š‘ +šœŒ1))>š”¼š‘‰(š‘ ,š‘„(š‘ )), that is, for š‘”āˆˆ[š‘”0āˆ’šœ,š‘”1), (3.5) holds.
Now, we assume that, for š‘”āˆˆ[š‘”š‘šāˆ’1,š‘”š‘š), š‘š=1,2,ā€¦,š‘˜, (3.5) holds. For š‘š=š‘˜+1, we will show that (3.5) holds. To this end, we first prove that ī€·š‘”š”¼š‘‰š‘˜ī€·š‘”,š‘„š‘˜ī€øī€øā‰¤šœ†1šœ€2.(3.27) In fact, by (2.1), (2.2), (C1), and (C'3) ī€·š‘”š”¼š‘‰š‘˜ī€·š‘”,š‘„š‘˜ī€·š‘”ī€øī€ø=š”¼š‘‰š‘˜,š»š‘˜ī€·š‘„ī€·š‘”āˆ’š‘˜ī€øī€øī€øā‰¤šœ†2š”¼ī‚†ā€–ā€–š»š‘˜ī€·š‘„ī€·š‘”āˆ’š‘˜ā€–ā€–ī€øī€ø2šœī‚‡ā‰¤šœ†2š›¾2š‘˜š”¼ī‚†ā€–ā€–š‘„ī€·š‘”āˆ’š‘˜ī€øā€–ā€–2šœī‚‡ā‰¤šœ†2š›¾2š‘˜šœ†1ī€·š‘”š”¼š‘‰š‘˜ī€·š‘”,š‘„āˆ’š‘˜ī€øī€øā‰¤šœ†1šœ€2.(3.28)
Secondly, we assume that there exists š‘ āˆˆ(š‘”š‘˜,š‘”š‘˜+1), such that (3.7) holds. By (3.27) and the continuity of š”¼š‘‰(š‘”,š‘„(š‘”)) on [š‘”š‘˜,š‘”š‘˜+1), we know that there exist š‘”āˆˆ[š‘”š‘˜,š‘”š‘˜+1), šœŒ2>0 such that for every š‘”1,š‘”2āˆˆ[š‘”,š‘”+šœŒ2], š‘”<š‘”2, (3.23) and (3.24) hold.
Let š‘ =inf{š‘”āˆˆ[š‘”š‘˜,š‘”š‘˜+1)āˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))=šœ†1šœ€2,š”¼š‘‰(š‘¢,š‘„(š‘¢))>šœ†1šœ€2,š‘¢āˆˆ(š‘”,š‘”+šœŒ2]āŠ‚[š‘”š‘˜,š‘”š‘˜+1),š”¼š‘‰(š‘”1,š‘„(š‘”1))<š”¼š‘‰(š‘”2,š‘„(š‘”2)),foreveryš‘”1,š‘”2āˆˆ[š‘”,š‘”+šœŒ2],š‘”1<š‘”2}, where šœŒ2>0 is some scalar.
Then for š‘”āˆˆ[š‘ ,š‘ +šœŒ2], (3.25) and (3.26) hold. This is a contraction, that is, (3.5) holds for š‘”āˆˆ[š‘”š‘˜,š‘”š‘˜+1). By mathematical induction, (3.5) holds for any š‘š=1,2,ā€¦, which implies that system (2.1) is mean squarely stable.

Remark 3.6. When šœ‡=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 šœ‡<1, we have the following results.

Theorem 3.7. Assume that there exist scalars šœ†2>šœ†1>0, šœ†šœ>0, šœ†ā‰¤0, matrix š‘ƒ>0, and Lyapunov-Krasovskii functional š‘‰(š‘”,š‘„(š‘”))āˆˆš’±(1,2), such that (C1), (C2), and
(Cā€3) šœ‡=supš‘˜āˆˆā„•{šœ†ī…žš‘˜=(šœ†2/šœ†1)š›¾2š‘˜}<1
hold, then
(i)if 0<šœ‡šœ†+šœ†šœā‰¤āˆ’šœ†šœlnšœ‡, system (2.1) is mean squarely stable over impulsive time sequences š’©sup(āˆ’šœ‡lnšœ‡/(šœ‡šœ†+šœ†šœ)); (ii)if šœ‡šœ†+šœ†šœā‰¤0, system (2.1) is mean squarely stable over any impulsive time sequences.

Proof. We prove (i) and omit the proof of (ii).
Because šœ‡<1 and 0<šœ‡šœ†+šœ†šœā‰¤āˆ’šœ†šœlnšœ‡, then there exist a sufficiently small šœŒ0>0, such that šœ‡+šœŒ0ī€·<1,šœ†šœ‡+šœŒ0ī€ø+šœ†šœ>0,āˆ’lnšœ‡šœ†+šœ‡āˆ’1šœ†šœā‰¤ī€·āˆ’lnšœ‡+šœŒ0ī€øī€·šœ†+šœ‡+šœŒ0ī€øāˆ’1šœ†šœ.(3.29) For any given šœ€>0, choose āˆš0<š›æā‰¤((šœ‡+šœŒ0)šœ†1)/šœ†2šœ€. We assume the initial function šœ‘āˆˆPCš‘ā„±0(š›æ). For š‘”āˆˆ[š‘”0āˆ’šœ,š‘”0], by (C1), (3.29), and š‘„š‘”0āˆˆPCš‘ā„±0(š›æ), we obtain š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†2š”¼ī‚†ā€–ā€–š‘„š‘”0ā€–ā€–2šœī‚‡ā‰¤šœ†2š›æ2ā‰¤šœ†1ī€·šœ‡+šœŒ0ī€øšœ€2<šœ†1šœ€2.(3.30)
Now we will prove that (3.5) holds. Otherwise, there exists š‘ āˆˆ(š‘”0,š‘”1), such that (3.7) holds. Set š‘”āˆ—ī€½ī€·š‘”=infš‘”āˆˆ0,š‘”1ī€øāˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))ā‰„šœ†1šœ€2ī€¾,(3.31) then by (3.7), (3.30), and the continuity of š”¼š‘‰(š‘”,š‘„(š‘”)) on [š‘”0,š‘”1), we know that š‘”āˆ—āˆˆ(š‘”0,š‘”1), š”¼š‘‰(š‘”āˆ—,š‘„(š‘”āˆ—))=šœ†1šœ€2. Set ī€½ī€ŗš‘”š‘”=supš‘”āˆˆ0,š‘”āˆ—ī€øāˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†1ī€·šœ‡+šœŒ0ī€øšœ€2ī€¾,(3.32) then by (3.30) and the continuity of š”¼š‘‰(š‘”,š‘„(š‘”)), we have š‘”āˆˆ[š‘”0,š‘”āˆ—), š”¼š‘‰(š‘”,š‘„(š‘”))=šœ†1(šœ‡+šœŒ0)šœ€2 and for š‘”āˆˆ[š‘”,š‘”āˆ—], ī€·š”¼š‘‰š‘”,š‘„š‘”ī€øā‰¤šœ†1šœ€2ā‰¤1šœŒ0+šœ‡š”¼š‘‰(š‘”,š‘„(š‘”)).(3.33) Conditions (C2) and (Cā€3) imply that for š‘”āˆˆ[š‘”,š‘”āˆ—], š·+š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†š”¼š‘‰(š‘”,š‘„(š‘”))+šœ†šœī€·š”¼š‘‰š‘”,š‘„š‘”ī€øā‰¤ī‚µšœ†šœ†+šœšœ‡+šœŒ0ī‚¶š”¼š‘‰(š‘”,š‘„(š‘”)).(3.34) By Lemma (2.1), (3.29), (3.26), and š‘”1āˆ’š‘”0ā‰¤(āˆ’lnšœ‡/(šœ†+(šœ†šœ/šœ‡))), we have ī€·š‘”š”¼š‘‰āˆ—ī€·š‘”,š‘„āˆ—šœ†ī€øī€øā‰¤expī‚»ī‚µšœ†+šœšœ‡+šœŒ0ī‚¶ī€·š‘”āˆ—āˆ’š‘”ī€øī‚¼ī€·š”¼š‘‰ī€·š‘”,š‘„š‘”šœ†ī€øī€ø<š‘’š‘„š‘ī‚»ī‚µšœ†+šœšœ‡+šœŒ0ī‚¶ī€·š‘”1āˆ’š‘”0ī€øī‚¼ī€·šœ‡+šœŒ0ī€øšœ†1šœ€2ā‰¤šœ†1šœ€2,(3.35) this is a contradiction with the fact š”¼š‘‰(š‘”āˆ—,š‘„(š‘”āˆ—))=šœ†1šœ€2.
Now, we assume that, for š‘”āˆˆ[š‘”š‘šāˆ’1,š‘”š‘š), š‘š=1,2,ā€¦,š‘˜, (3.5) holds. For š‘š=š‘˜+1, we will show that (3.5) holds. To this end, we first prove that ī€·š‘”š”¼š‘‰š‘˜ī€·š‘”,š‘„š‘˜ā‰¤ī€·ī€øī€øšœ‡+šœŒ0ī€øšœ†1šœ€2.(3.36) In fact, by (2.1), (2.2), (C1), and (Cā€3) ī€·š‘”š”¼š‘‰š‘˜ī€·š‘”,š‘„š‘˜ī€·š‘”ī€øī€ø=š”¼š‘‰š‘˜,š»š‘˜ī€·š‘„ī€·š‘”āˆ’š‘˜ī€øī€øī€øā‰¤šœ†2š”¼ī‚†ā€–ā€–š»š‘˜ī€·š‘„ī€·š‘”āˆ’š‘˜ā€–ā€–ī€øī€ø2šœī‚‡ā‰¤šœ†2š›¾2š‘˜š”¼ī‚†ā€–ā€–š‘„ī€·š‘”āˆ’š‘˜ī€øā€–ā€–2šœī‚‡ā‰¤šœ†2š›¾2š‘˜šœ†1ī€·š‘”š”¼š‘‰š‘˜ī€·š‘”,š‘„āˆ’š‘˜ā‰¤ī€·ī€øī€øšœ‡+šœŒ0ī€øšœ†1šœ€2.(3.37)
Secondly, we assume that there exists š‘ āˆˆ(š‘”š‘˜,š‘”š‘˜+1), such that (3.7) holds. Set š‘”āˆ—ī€½ī€·š‘”=infš‘”āˆˆš‘˜,š‘”š‘˜+1ī€øāˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))ā‰„šœ†1šœ€2ī€¾,ī€½ī€ŗš‘”š‘”=supš‘”āˆˆš‘˜,š‘”āˆ—ī€øāˆ¶š”¼š‘‰(š‘”,š‘„(š‘”))ā‰¤šœ†1ī€·šœ‡+šœŒ0ī€øšœ€2ī€¾,(3.38) then by (3.37) and the continuity of š”¼š‘‰(š‘”,š‘„(š‘”)) on [š‘”š‘˜,š‘”š‘˜+1), we have š‘”āˆ—āˆˆ(š‘”š‘˜,š‘”š‘˜+1), š‘”āˆˆ[š‘”š‘˜,š‘”āˆ—) and š”¼š‘‰(š‘”āˆ—,š‘„(š‘”āˆ—))=šœ†1šœ€2, š”¼š‘‰(š‘”,š‘„(š‘”))=(šœ‡+šœŒ0)šœ†1šœ€2.
On the other hand, for š‘”āˆˆ[š‘”,š‘”āˆ—], (3.33) and (3.34) hold, which lead to a contradiction, that is, (3.5) holds for š‘”āˆˆ[š‘”š‘˜,š‘”š‘˜+1). By mathematical induction, (3.5) holds for any š‘š=1,2,ā€¦, 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: ī€ŗī€·š‘„dš‘„(š‘”)=āˆ’š¶š‘„(š‘”)+š“š‘“(š‘„(š‘”))+šµš‘”š‘”ī€·ī€øī€»dš‘”+šœŽš‘”,š‘„(š‘”),š‘„š‘”ī€ødšœ”(š‘”),š‘”ā‰„š‘”0,š‘”ā‰ š‘”š‘˜,š‘„ī€·š‘”š‘˜ī€ø=š»š‘˜ī€·š‘„ī€·š‘”āˆ’š‘˜š‘„ī€·š‘”ī€øī€ø,š‘˜=1,2,ā€¦,0ī€ø[],+šœƒ=šœ‘(šœƒ),š‘ āˆˆāˆ’šœ,0(4.1) where the initial value šœ‘(š‘ )āˆˆPCš‘ā„±0(š›æ), š‘„(š‘”)=(š‘„1(š‘”),š‘„2(š‘”),ā€¦,š‘„š‘›(š‘”))š‘‡āˆˆā„š‘› is the state vector, š¶=diag(š‘1,š‘2,ā€¦,š‘š‘›), š‘š‘–>0 is the neuron-charging time constant, š“=(š‘Žš‘–š‘—)š‘›Ć—š‘› are, respectively, the connection weight matrix, the discretely delayed connection weight matrix. š‘“(š‘„(š‘”))=(š‘“1(š‘„1(š‘”)),š‘“2(š‘„2(š‘”)),ā€¦,š‘“š‘›(š‘„š‘›(š‘”)))š‘‡āˆˆā„š‘› and š‘”(š‘„š‘”)=(š‘”1(š‘„1š‘”),š‘”2(š‘„2š‘”),ā€¦,š‘”š‘›(š‘„š‘›š‘”))ā„š‘›, where š‘“š‘–(š‘„š‘–(š‘”)) and š‘”š‘–(š‘„š‘–š‘”) denote, respectively, the measures of response or activation to its incoming potentials of the unit š‘– at time š‘” and time š‘”āˆ’šœš‘–. We also assume that š»š‘˜(0)=0, (š‘˜=1,2,ā€¦), š‘“(0)=0, š‘”(0)=0, and šœŽ(š‘”,0,0)=0, then system (4.1) admits an equilibrium solution š‘„(š‘”)ā‰”0. Moreover, we assume that š»(ā‹…) satisfies (2.2), and š‘“(ā‹…), š‘”(ā‹…), šœŽ(ā‹…) satisfy ā€–ā€–š‘”ī€·š‘„ā€–š‘“(š‘„(š‘”))ā€–ā‰¤ā€–š¹š‘„(š‘”)ā€–,š‘”ī€øā€–ā€–ā‰¤ā€–ā€–šŗš‘„š‘”ā€–ā€–ī€ŗšœŽ,(4.2)trš‘‡ī€·š‘”,š‘„(š‘”),š‘„š‘”ī€øšœŽī€·š‘”,š‘„(š‘”),š‘„š‘”ā‰¤ā€–ī€øī€»ā€–š¾š‘„(š‘”)2+ā€–ā€–š¾šœš‘„š‘”ā€–ā€–2,(4.3) where š¹, šŗ, š¾, and š¾šœ are known constant matrices with appropriate dimensions.

Corollary 4.1. Assume that there exist positive scalars šœ€1, šœ€2, š›½, symmetric matrix š‘ƒ>0 and šœ‡=supš‘˜āˆˆā„•{šœ†ī…žš‘˜=(šœ†max(š‘ƒ)/šœ†min(š‘ƒ))š›¾2š‘˜}. Then the following results hold:
(i)if šœ‡>1, šœ†+šœ‡šœ†šœ<āˆ’lnšœ‡/š›½, then system (4.1) is mean squarely stable over impulsive time sequence š’©inf(šœ+š›½);(ii)if šœ‡=1, šœ†+šœ†šœā‰¤0, then system (4.1) is mean squarely stable over any impulsive time sequence;(iii)if šœ‡<1 and 0<šœ‡šœ†+šœ†šœā‰¤āˆ’šœ†šœlnšœ‡, then system (4.1) is mean squarely stable over impulsive time sequence š’©sup(āˆ’šœ‡lnšœ‡/(šœ‡šœ†+šœ†šœ));(iv)if šœ‡<1, šœ‡šœ†+šœ†šœā‰¤0, then system (4.1) is mean squarely stable over any impulsive time sequence, where šœ†=šœ†maxī€·āˆ’2š¶+šœ€1š“š“š‘‡š‘ƒ+šœ€2šµšµš‘‡š‘ƒ+š‘ƒāˆ’1ī€·šœ€1āˆ’1š¹š‘‡š¹+šœ†max(š‘ƒ)š¾š‘‡š¾,šœ†ī€øī€øšœ=šœ†maxī€·š‘ƒāˆ’1ī€·šœ€2āˆ’1šŗš‘‡šŗ+šœ†max(š‘ƒ)š¾š‘‡šœš¾šœ.ī€øī€ø(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]. īƒ¬dš‘„1(š‘”)dš‘„2(īƒ­=āŽ§āŽŖāŽØāŽŖāŽ©īƒ¬š‘„š‘”)āˆ’10.500āˆ’12.2īƒ­īƒ¬1š‘„(š‘”)2(īƒ­+īƒ¬š‘”)1.2āˆ’0.20.62.4īƒ­īƒ¬sinš‘„1(š‘”)arctanš‘„2(īƒ­+īƒ¬īƒ­Ć—āŽ”āŽ¢āŽ¢āŽ¢āŽ£š‘”)1.60.3āˆ’0.51.8sinš‘„1ī‚€1š‘”āˆ’2ī‚arctanš‘„2ī‚€1š‘”āˆ’2ī‚āŽ¤āŽ„āŽ„āŽ„āŽ¦āŽ«āŽŖāŽ¬āŽŖāŽ­āŽ”āŽ¢āŽ¢āŽ¢āŽ£dš‘”+2š‘„1(š‘”)š‘„2ī‚€1š‘”āˆ’3ī‚š‘„1ī‚€1š‘”āˆ’2ī‚āˆ’š‘„2āŽ¤āŽ„āŽ„āŽ„āŽ¦īƒ¬(š‘”)dšœ”1(š‘”)dšœ”2(īƒ­š‘”),š‘”ā‰„š‘”0,š‘”ā‰ š‘”š‘˜,īƒ¬š‘„1ī€·š‘”š‘˜ī€øš‘„2ī€·š‘”š‘˜ī€øīƒ­=š‘’āˆ’0.1š‘˜īƒ¬š‘„0.5āˆ’0.150.120.6īƒ­īƒ¬1ī€·š‘”āˆ’š‘˜ī€øš‘„2ī€·š‘”āˆ’š‘˜ī€øīƒ­,š‘˜=1,2,ā€¦,(4.5) where š‘”0=0.
Similar to the result, we can verify that the point (0,0)š‘‡ is an equilibrium point and can obtain by calculation that īƒ©īƒŖš‘ƒ=0.56000.68,š¾š‘‡īƒ©īƒŖš¾=4001,(4.6) and šœ€š‘–=1(š‘–=1,2),šœ†max(š‘ƒ)=0.68,šœ†min(š‘ƒ)=0.56,š›¾š‘˜=0.620exp(āˆ’0.1š‘˜),š¾š‘‡šœš¾šœ=š¼,š¹=šŗ=š¼,šœ‡=0.4668<1,šœ†=āˆ’12.0443,šœ†šœ=3, and, hence, we have šœ‡šœ†+šœ†šœ=āˆ’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.