Mean Square Stability of Impulsive Stochastic Differential Systems

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 𝐴≥𝐵 (𝐴>𝐵, 𝐴≤𝐵, 𝐴<𝐵) means 𝐴−𝐵 is positive semidefinite (positive definite, negative semidefinite, negative definite) matrix. 𝜆_max(⋅) (𝜆_min(⋅)) represents the maximum (minimum) eigenvalue of the corresponding matrix, respectively. ‖⋅‖ denotes Euclidean norm for vectors or the spectral norm of matrices. Moreover, let (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions, that is, the filtration contains all P-null sets and is right continuous. Let PC([−𝜏,0],ℝ^𝑛) denote the set of piecewise right continuous function 𝜙∶[−𝜏,0]→ℝ^𝑛 with the norm defined by ‖𝜙‖_𝜏=sup_{−𝜏≤𝜃≤0}‖𝜙(𝜃)‖, where 𝜏 is a known positive constant, PC(𝛿)={𝜑∣𝜑∈PC([−𝜏,0],ℝ^𝑛),‖𝜑‖_𝜏≤𝛿}, PC^𝑏ℱ₀([−𝜏,0],ℝ^𝑛) denote the family of all ℱ₀-measurable PC([−𝜏,0],ℝ^𝑛)-valued stochastic process 𝜑={𝜑(𝑠)∶−𝜏≤𝑠≤0} with sup_{−𝜏≤𝑠≤0}𝔼{‖𝜑(𝑠)‖²}<∞, where 𝔼{⋅} represents the mathematical expectation operator with respect to the probability measure 𝑃, PC^𝑏ℱ₀(𝛿)={𝜑∣𝜑∈PC^𝑏ℱ₀([−𝜏,0],ℝ^𝑛),sup_{−𝜏≤𝑠≤0}𝔼{‖𝜑(𝑠)‖²}≤𝛿, ℒ denote the well-known ℒ-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)=𝑓𝑡,𝑥𝑥(𝑡𝑡)_𝑘,𝑥_𝑡d𝑡+𝑔𝑡,𝑥(𝑡),𝑥_𝑡d𝜔(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑥𝑡=𝐻_𝑘𝑥𝑡^−𝑘,𝑘=1,2,…,₀+𝜃=𝜑(𝜃),𝜃∈[−𝜏,0],(2.1) where the initial value 𝜑∈PC^𝑏ℱ₀(𝛿), the fixed impulsive time moments 𝑡_𝑘 satisfy 0≤𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯(𝑡_𝑘→∞ as 𝑘→∞). 𝑥(𝑡)∈ℝ^𝑛 is the system state, 𝑓∈𝐶(ℝ×ℝ^𝑛×ℝ^𝑛,ℝ^𝑛), 𝑔∈𝐶(ℝ×ℝ^𝑛×ℝ^𝑛,ℝ^𝑛×𝑚). 𝜔(𝑡)∈ℝ^𝑚 is an standard Brownian motion defined on the complete probability space (Ω,ℱ,{ℱ_𝑡}_𝑡≥0,𝑃). Besides, we assume that 𝐻_𝑘(0)=0,(𝑘=1,2,…), 𝑓(𝑡,0,0)=0, 𝑔(𝑡,0,0)=0 and ‖‖𝐻_𝑘𝑥𝑡^−𝑘‖‖≤𝛾_𝑘‖‖𝑥𝑡^−𝑘‖‖,𝛾_𝑘≥0,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 𝒩_inf(𝛽) and 𝒩_sup(𝛽) the class of impulsive time sequences that satisfy inf_𝑘{𝑡_𝑘−𝑡_𝑘−1}≥𝛽 and sup_𝑘{𝑡_𝑘−𝑡_𝑘−1}≤𝛽, respectively.

We need the following lemma and definitions.

Lemma 2.1 (Chaplygin Comparison Theorem, see Shi et al. [12]). Assume that 𝑓,𝐹∈𝐶(𝐺), 𝑔⊂ℝ² and 𝑓(𝑡,𝑥)<𝐹(𝑡,𝑥),(𝑡,𝑥)∈𝐺.(2.3) If 𝜙(𝑡) (𝑡∈𝑈₁) and Φ(𝑡) (𝑡∈𝑈₂) are the solutions of Cauchy problems 𝑥𝑥̇̇==𝐹𝑓((𝑡𝑡𝑥𝑥,,𝑥𝑥((𝜏𝜏)))),,==((𝜉𝜉𝑡𝑡,,,,𝑥𝑥))∈∈𝐺𝐺,,(2.4) respectively, then for 𝑡∈(𝜏,∞)∩𝑈₁∩𝑈₂, 𝜙(𝑡)<Φ(𝑡)(2.5) and for 𝑡∈(−∞,𝜏)∩𝑈₁∩𝑈₂,   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 𝜑∈PC^𝑏ℱ₀(𝛿) implies 𝔼{‖𝑥(𝑡)‖²}<𝜀, 𝑡≥𝑡₀ for all admissible time sequence in 𝒩.

Definition 2.3 (see Yang et al. [9]). The function 𝑉∶[𝑡₀−𝜏,∞)×ℝ^𝑛→ℝ⁺ 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 [𝑡_𝑘−1,𝑡_𝑘)×ℝ^𝑛,(𝑘=1,2,…) and for all 𝑡≥𝑡₀, 𝑉(𝑡,0)≡0, (2)𝑉(𝑡,𝑥) is locally Lipschitaian in 𝑥, (3)for each 𝑘=1,2,…, there exist finite limits lim_{𝑡(𝑡,𝑦)→−𝑘,𝑥}𝑉(𝑡,𝑦)=𝑉𝑉𝑡^−𝑘𝑡,𝑥,lim_{𝑡(𝑡,𝑦)→+𝑘,𝑥}𝑉(𝑡,𝑦)=𝑉𝑡^+𝑘,𝑥_+𝑘,,𝑥=𝑉𝑡_𝑘,𝑥.(2.7)

3. Main Results

Theorem 3.1. Assume that there exist scalars 𝜆₂>𝜆₁>0, 𝜆_𝜏>0, 𝛽>0, 𝜆≤0, 𝜌>0 matrix 𝑃>0 and Lyapunov-Krasovskii functional 𝑉(𝑡,𝑥(𝑡))∈𝒱^(1,2), such that
(C1)𝜆₁‖𝑥(𝑡)‖²≤𝑉(𝑡,𝑥(𝑡))≤𝜆₂‖𝑥_𝑡‖^2𝜏, (C2)𝔼ℒ𝑉(𝑡,𝑥(𝑡))≤𝜆𝔼𝑉(𝑡,𝑥(𝑡))+𝜆_𝜏𝔼𝑉(𝑡,𝑥_𝑡),𝑡∈[𝑡_𝑘−1,𝑡_𝑘),𝑘=1,2,…, whenever 𝔼𝑉(𝑡,𝑥_𝑡)≤(𝜇+𝜌)𝔼𝑉(𝑡,𝑥(𝑡)),(C3)𝜇=sup_𝑘∈ℕ{𝜆^𝑘=(𝜆₂/𝜆₁)𝛾^2𝑘}>1 and 𝜆+(𝜇+𝜌)𝜆_𝜏≤−((ln(𝜇+𝜌))/𝛽),
then the trivial solution of system (2.1) is mean squarely stable over 𝒩_inf(𝜏+𝛽).

Proof. For any given 𝜀>0, choose 0<𝛿≤√𝜆₁/(𝜇+𝜌)𝜆₂𝜀. We assume that the initial function 𝜑∈PC^𝑏ℱ₀(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 𝒩_inf(𝜏+𝛽). For 𝑉(𝑡,𝑥(𝑡))∈𝒱^(1,2), by Itô formula, for 𝑡≠𝑡_𝑘,𝑘=1,2,…, we have d𝑉(𝑡,𝑥(𝑡))=ℒ𝑉(𝑡,𝑥(𝑡))d𝑡+𝑉_𝑥(𝑡,𝑥(𝑡))𝑔(𝑡,𝑥(𝑡))d𝜔(𝑡),(3.1) where ℒ𝑉(𝑡,𝑥(𝑡))=𝑉_𝑡(𝑡,𝑥(𝑡))+𝑉_𝑥(𝑡,𝑥(𝑡))𝑓+(1/2)tr(𝑔^𝑇𝑉_𝑥𝑥𝑔).
For 𝑡∈[𝑡_𝑘−1,𝑡_𝑘), 𝑘=1,2,…, integrate (3.1) from 𝑡_𝑘−1 to 𝑡, we have 𝑉(𝑡,𝑥(𝑡))=𝑉𝑡_𝑘−1,𝑥𝑡_𝑘−1+^𝑡𝑡_𝑘−1ℒ𝑉(𝑠,𝑥(𝑠))d𝑠+^𝑡𝑡_𝑘−1𝑉_𝑥(𝑠,𝑥(𝑠))𝑔(𝑠,𝑥(𝑠))d𝜔(𝑠).(3.2) Taking the mathematical expectation of both sides of the above equation, we obtain 𝔼𝑉(𝑡,𝑥(𝑡))=𝔼𝑉𝑡_𝑘−1,𝑥𝑡_𝑘−1+^𝑡𝑡_𝑘−1𝔼ℒ𝑉(𝑠,𝑥(𝑠))d𝑠.(3.3) So for 𝑠∈[𝑡,𝑡+Δ𝑡] with 𝑡+Δ𝑡∈[𝑡_𝑘−1,𝑡_𝑘) and Δ𝑡>0, if 𝔼𝑉(𝑠,𝑥_𝑠)≤(𝜇+𝜌)𝔼𝑉(𝑠,𝑥(𝑠)), then we have by (C2) 𝔼𝑉(𝑡+Δ𝑡,𝑥(𝑡+Δ𝑡))−𝔼𝑉(𝑡,𝑥(𝑡))=≤^{𝑡𝑡+Δ𝑡}𝔼ℒ𝑉(𝑠,𝑥(𝑠))d𝑠_{𝑡𝑡+Δ𝑡}𝜆𝔼𝑉(𝑠,𝑥(𝑠))+𝜆_𝜏𝔼𝑉𝑠,𝑥_𝑠d𝑠.(3.4)
In what follows, we first prove that for 𝑡∈[𝑡₀−𝜏,𝑡₁), 𝔼𝑉(𝑡,𝑥(𝑡))≤𝜆₁𝜀².(3.5) Obviously, for 𝑡∈[𝑡₀−𝜏,𝑡₀], by (C1) and 𝑥_𝑡0∈PC^𝑏ℱ₀(𝛿), we obtain 𝔼𝑉(𝑡,𝑥(𝑡))≤𝜆₂𝔼‖‖𝑥_𝑡0‖‖^2𝜏≤𝜆²𝛿²≤𝜆¹𝜇+𝜌𝜀²<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 𝑠₁=inf𝑡∈𝑡₀,𝑡₁∶𝔼𝑉(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),  𝔼𝑉𝑠₁,𝑥𝑠₁=𝜆₁𝜀²,(3.9) and for 𝑡∈[𝑡₀−𝜏,𝑠₁], (3.5) holds. Set 𝑠₂=sup𝑡∈𝑡₀,𝑠₁∶𝔼𝑉(𝑡,𝑥(𝑡))≤𝜆¹𝜇+𝜌𝜀²,(3.10) then by (3.6) and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)), we have 𝑠₂∈[𝑡₀,𝑠₁), 𝔼𝑉𝑠₂,𝑥𝑠₂=𝜆¹𝜇+𝜌𝜀²,(3.11) and for 𝑡∈[𝑠₂,𝑠₁], 𝔼𝑉𝑡,𝑥_𝑡≤𝜆₁𝜀²≤(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))+𝜆_𝜏𝔼𝑉𝑡,𝑥_𝑡≤𝜆+𝜆_𝜏(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∈[𝑡_𝑚−1,𝑡_𝑚), 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], 𝔼𝑉𝑡^−𝑘+𝜃,𝑥𝑡^−𝑘+𝜃≤𝜆¹𝜇+𝜌𝜀².(3.14) Noticing [𝑡_𝑘−𝜏,𝑡_𝑘)⊂[𝑡_𝑘−1,𝑡_𝑘), we assume that there exists some 𝑠∈[𝑡_𝑘−𝜏,𝑡_𝑘], such that 𝔼𝑉(𝑠⁻,𝑥(𝑠⁻))>𝜆¹𝜇+𝜌𝜀²,(3.15) then there are two cases to be considered.
(i)For all 𝑡∈[𝑡_𝑘−1,𝑠], 𝔼𝑉(𝑡⁻,𝑥(𝑡⁻))>(𝜆₁/(𝜇+𝜌))𝜀². Hence, for 𝑡∈[𝑡_𝑘−1,𝑠], (3.12) and (3.13) hold, which follows by (C3), (3.5), and Lemma (2.1), 𝔼𝑉(𝑠⁻,𝑥(𝑠⁻))≤≤𝜆e₁xp𝜆+(𝜇+𝜌)𝜆_𝜏𝑠−𝑡_𝑘−1𝔼𝑉𝑡_𝑘−1,𝑥𝑡_𝑘−1≤𝜆𝜀²exp𝜆+(𝜇+𝜌)𝜆_𝜏𝑡_𝑘−𝑡_𝑘−1−𝜏₁𝜇+𝜌𝜀²,(3.16) this is a contradiction with the assumption.(ii)There exists some 𝑡∈[𝑡_𝑘−1,𝑠), such that 𝔼𝑉(𝑡,𝑥(𝑡))≤(𝜆₁/(𝜇+𝜌))𝜀². Set 𝑠₁=sup𝑡∈𝑡_𝑘−1,𝑠∶𝔼𝑉(𝑡,𝑥(𝑡))≤𝜆¹𝜇+𝜌𝜀²,(3.17) then 𝑠₁∈[𝑡_𝑘−1,𝑠), 𝔼𝑉𝑠₁,𝑥𝑠₁=𝜆¹𝜇+𝜌𝜀²(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 𝔼𝑉𝑡_𝑘,𝑥𝑡_𝑘=≤𝔼𝜆𝑉₂𝑡_𝑘,𝐻_𝑘𝑥𝑡^−𝑘≤𝜆₂𝔼‖‖𝐻_𝑘𝑥𝑡^−𝑘‖‖^2𝜏≤𝜆²𝛾^2𝑘𝔼‖‖𝑥𝑡^−𝑘‖‖^2𝜏𝛾^2𝑘𝜆₁sup_{−𝜏≤𝜃≤0}𝔼𝑉𝑡_𝑘+𝜃,𝑥𝑡^−𝑘+𝜃≤𝜆^𝑘𝜆₁𝜇+𝜌𝜀²<𝜆₁𝜀².(3.19)
Now we will prove that (3.5) holds for 𝑡∈[𝑡_𝑘,𝑡_𝑘+1). Otherwise, there exists some 𝑡∈(𝑡_𝑘,𝑡_𝑘+1), such that (3.7) holds. Let 𝑠₁=inf𝑡∈𝑡_𝑘,𝑡_𝑘+1∶𝔼𝑉(𝑡,𝑥(𝑡))>𝜆₁𝜀².(3.20) Then by (3.14), (3.19), and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)) on [𝑡_𝑘,𝑡_𝑘+1), we know that 𝑠₁∈(𝑡_𝑘,𝑡_𝑘+1) and 𝔼𝑉(𝑠₁,𝑥(𝑠₁))=𝜆₁𝜀². If there exists 𝑡∈[𝑡_𝑘,𝑠₁], such that 𝔼𝑉(𝑡,𝑥(𝑡))≤(𝜆₁/𝜇)𝜀², then let 𝑠₂=sup𝑡∈𝑡_𝑘,𝑠₁∶𝔼𝑉(𝑡,𝑥(𝑡))≤𝜆¹𝜇+𝜌𝜀².(3.21) Otherwise, let 𝑠₂=𝑡_𝑘. 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) 𝜆₁‖𝑥(𝑡)‖²≤𝑉(𝑡,𝑥(𝑡))≤𝜆₂‖𝑥(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, 𝜆_𝜏>0, 𝜆≤0, 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 𝒩_inf(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, 𝜆_𝜏>0, 𝜆≤0, matrix 𝑃>0 and Lyapunov-Krasovskii functional 𝑉(𝑡,𝑥(𝑡))∈𝒱^(1,2), such that condition (C1) and (C'2)𝔼ℒ𝑉(𝑡,𝑥(𝑡))≤𝜆𝔼𝑉(𝑡,𝑥(𝑡))+𝜆_𝜏𝔼𝑉(𝑡,𝑥_𝑡), 𝑡∈[𝑡_𝑘−1,𝑡_𝑘),𝑘=1,2,… whenever 𝔼𝑉(𝑡,𝑥_𝑡)≤𝔼𝑉(𝑡,𝑥_𝑡),(C'3)𝜇=sup_𝑘∈ℕ{𝜆^𝑘=(𝜆₂/𝜆₁)𝛾^2𝑘}=1, 𝜆+𝜆_𝜏≤0hold, then the trivial solution of system (2.1) is mean squarely stable over any impulsive sequences.

Proof. For any given 𝜀>0, choose 0<𝛿≤√𝜆₁/𝜆₂𝜀. We assume that the initial function 𝜑∈PC^𝑏ℱ₀(𝛿). In what follows, we first prove that for 𝑡≥𝑡₀, (3.5) holds.
Obviously, for 𝑡∈[𝑡₀−𝜏,𝑡₀], by (C1) and 𝑥_𝑡0∈PC^𝑏ℱ₀(𝛿), we obtain 𝔼𝑉(𝑡,𝑥(𝑡))≤𝜆₂𝔼‖‖𝑥_𝑡0‖‖^2𝜏≤𝜆²𝛿²≤𝜆₁𝜀².(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 𝔼𝑉𝑡,𝑥𝑡=𝜆₁𝜀²(3.23) and for every 𝑡₁,𝑡₂∈[𝑡,𝑡+𝜌], 𝑡₁<𝑡₂, 𝔼𝑉𝑡₁,𝑥𝑡₁<𝔼𝑉𝑡₂,𝑥𝑡₂.(3.24) Let 𝔼𝑠𝑉=(inf{𝑡∈[𝑡₀,𝑡₁)∶𝔼𝑉(𝑡,𝑥(𝑡))=𝜆₁𝜀²,𝔼𝑉(𝑢,𝑥(𝑢))>𝜆₁𝜀²,𝑢∈(𝑡,𝑡+𝜌₁]⊂[𝑡₀,𝑡₁),𝔼𝑉(𝑡₁,𝑥(𝑡₁))<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], 𝔼𝑉𝑡,𝑥_𝑡≤𝔼𝑉(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))+𝜆_𝜏𝔼𝑉𝑡,𝑥_𝑡≤𝜆+𝜆_𝜏𝔼𝑉(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∈[𝑡₀−𝜏,𝑡₁), (3.5) holds.
Now, we assume that, for 𝑡∈[𝑡_𝑚−1,𝑡_𝑚), 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 𝔼𝑉𝑡_𝑘,𝑥𝑡_𝑘≤𝜆₁𝜀².(3.27) In fact, by (2.1), (2.2), (C1), and (C'3) 𝔼𝑉𝑡_𝑘,𝑥𝑡_𝑘=≤𝔼𝜆₂𝑉𝑡_𝑘,𝐻_𝑘𝑥𝑡^−𝑘≤𝜆₂𝔼‖‖𝐻_𝑘𝑥𝑡^−𝑘‖‖^2𝜏𝛾^2𝑘𝔼‖‖𝑥𝑡^−𝑘‖‖^2𝜏≤𝜆²𝛾^2𝑘𝜆₁≤𝜆₁𝔼𝑉𝑡_𝑘,𝑥𝑡^−𝑘𝜀².(3.28)
Secondly, we assume that there exists 𝑠∈(𝑡_𝑘,𝑡_𝑘+1), such that (3.7) holds. By (3.27) and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)) on [𝑡_𝑘,𝑡_𝑘+1), we know that there exist 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), 𝜌₂>0 such that for every 𝑡₁,𝑡₂∈[𝑡,𝑡+𝜌₂], 𝑡<𝑡₂, (3.23) and (3.24) hold.
Let 𝑠𝔼𝑉=(inf{𝑡∈[𝑡_𝑘,𝑡_𝑘+1)∶𝔼𝑉(𝑡,𝑥(𝑡))=𝜆₁𝜀²,𝔼𝑉(𝑢,𝑥(𝑢))>𝜆₁𝜀²,𝑢∈(𝑡,𝑡+𝜌₂]⊂[𝑡_𝑘,𝑡_𝑘+1),𝔼𝑉(𝑡₁,𝑥(𝑡₁))<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∈[𝑡_𝑘,𝑡_𝑘+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, 𝜆_𝜏>0, 𝜆≤0, matrix 𝑃>0, and Lyapunov-Krasovskii functional 𝑉(𝑡,𝑥(𝑡))∈𝒱^(1,2), such that (C1), (C2), and
(C”3) 𝜇=sup_𝑘∈ℕ{𝜆^𝑘=(𝜆₂/𝜆₁)𝛾^2𝑘}<1
hold, then
(i)if 0<𝜇𝜆+𝜆_𝜏≤−𝜆_𝜏ln𝜇, system (2.1) is mean squarely stable over impulsive time sequences 𝒩_sup(−𝜇ln𝜇/(𝜇𝜆+𝜆_𝜏)); (ii)if 𝜇𝜆+𝜆_𝜏≤0, 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+𝜆_𝜏≤−𝜆_𝜏ln𝜇, then there exist a sufficiently small 𝜌₀>0, such that 𝜇+−𝜌₀ln<𝜇1,𝜆𝜇+𝜌₀+𝜆_𝜏>0,𝜆+𝜇⁻¹𝜆_𝜏≤−ln𝜇+𝜌⁰𝜆+𝜇+𝜌₀⁻¹𝜆_𝜏.(3.29) For any given 𝜀>0, choose 0<𝛿≤√((𝜇+𝜌₀)𝜆₁)/𝜆₂𝜀. We assume the initial function 𝜑∈PC^𝑏ℱ₀(𝛿). For 𝑡∈[𝑡₀−𝜏,𝑡₀], by (C1), (3.29), and 𝑥_𝑡0∈PC^𝑏ℱ₀(𝛿), we obtain 𝔼𝑉(𝑡,𝑥(𝑡))≤𝜆₂𝔼‖‖𝑥_𝑡0‖‖^2𝜏≤𝜆²𝛿²≤𝜆₁𝜇+𝜌₀𝜀²<𝜆₁𝜀².(3.30)
Now we will prove that (3.5) holds. Otherwise, there exists 𝑠∈(𝑡₀,𝑡₁), such that (3.7) holds. Set 𝑡^∗=inf𝑡∈𝑡₀,𝑡₁∶𝔼𝑉(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 𝑡=sup𝑡∈𝑡₀,𝑡^∗∶𝔼𝑉(𝑡,𝑥(𝑡))≤𝜆₁𝜇+𝜌₀𝜀²,(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^∗], 𝔼𝑉𝑡,𝑥_𝑡≤𝜆₁𝜀²≤1𝜌₀+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))+𝜆_𝜏𝔼𝑉𝑡,𝑥_𝑡≤𝜆+𝜆^𝜏𝜇+𝜌₀𝔼𝑉(𝑡,𝑥(𝑡)).(3.34) By Lemma (2.1), (3.29), (3.26), and 𝑡₁−𝑡₀≤(−ln𝜇/(𝜆+(𝜆_𝜏/𝜇))), we have 𝔼𝑉𝑡^∗,𝑥𝑡^∗≤exp𝜆+𝜆^𝜏𝜇+𝜌₀𝑡^∗<𝑒𝑥𝑝𝜆+𝜆_𝜏−𝑡𝔼𝑉𝑡,𝑥𝑡𝜇+𝜌₀𝑡¹−𝑡₀𝜇+𝜌₀𝜆₁𝜀²≤𝜆₁𝜀²,(3.35) this is a contradiction with the fact 𝔼𝑉(𝑡^∗,𝑥(𝑡^∗))=𝜆₁𝜀².
Now, we assume that, for 𝑡∈[𝑡_𝑚−1,𝑡_𝑚), 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 𝔼𝑉𝑡_𝑘,𝑥𝑡_𝑘≤𝜇+𝜌₀𝜆₁𝜀².(3.36) In fact, by (2.1), (2.2), (C1), and (C”3) 𝔼𝑉𝑡_𝑘,𝑥𝑡_𝑘=≤𝔼𝜆₂𝑉𝑡_𝑘,𝐻_𝑘𝑥𝑡^−𝑘≤𝜆₂𝔼‖‖𝐻_𝑘𝑥𝑡^−𝑘‖‖^2𝜏𝛾^2𝑘𝔼‖‖𝑥𝑡^−𝑘‖‖^2𝜏≤𝜆²𝛾^2𝑘𝜆₁≤𝜇+𝜌₀𝔼𝑉𝑡_𝑘,𝑥𝑡^−𝑘𝜆₁𝜀².(3.37)
Secondly, we assume that there exists 𝑠∈(𝑡_𝑘,𝑡_𝑘+1), such that (3.7) holds. Set 𝑡^∗=inf𝑡∈𝑡_𝑘,𝑡_𝑘+1∶𝔼𝑉(𝑡,𝑥(𝑡))≥𝜆₁𝜀²,𝑡=sup𝑡∈𝑡_𝑘,𝑡^∗∶𝔼𝑉(𝑡,𝑥(𝑡))≤𝜆₁𝜇+𝜌₀𝜀²,(3.38) then by (3.37) and the continuity of 𝔼𝑉(𝑡,𝑥(𝑡)) on [𝑡_𝑘,𝑡_𝑘+1), we have 𝑡^∗∈(𝑡_𝑘,𝑡_𝑘+1), 𝑡∈[𝑡_𝑘,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∈[𝑡_𝑘,𝑡_𝑘+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: d𝑥(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓𝑥(𝑥𝑡(_𝑘𝑡))+𝐵𝑔𝑥_𝑡d𝑡+𝜎𝑡,𝑥(𝑡),𝑥_𝑡d𝜔(𝑡),𝑡≥𝑡₀,𝑡≠𝑡_𝑘,𝑥𝑡=𝐻_𝑘𝑥𝑡^−𝑘,𝑘=1,2,…,₀+𝜃=𝜑(𝜃),𝑠∈[−𝜏,0],(4.1) where the initial value 𝜑(𝑠)∈PC^𝑏ℱ₀(𝛿), 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈ℝ^𝑛 is the state vector, 𝐶=diag(𝑐₁,𝑐₂,…,𝑐_𝑛), 𝑐_𝑖>0 is the neuron-charging time constant, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 are, respectively, the connection weight matrix, the discretely delayed connection weight matrix. 𝑓(𝑥(𝑡))=(𝑓₁(𝑥₁(𝑡)),𝑓₂(𝑥₂(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡)))^𝑇∈ℝ^𝑛 and 𝑔(𝑥_𝑡)=(𝑔₁(𝑥_1𝑡),𝑔₂(𝑥_2𝑡),…,𝑔_𝑛(𝑥_𝑛𝑡))ℝ^𝑛, where 𝑓_𝑖(𝑥_𝑖(𝑡)) and 𝑔_𝑖(𝑥_𝑖𝑡) denote, respectively, the measures of response or activation to its incoming potentials of the unit 𝑖 at time 𝑡 and time 𝑡−𝜏_𝑖. We also assume that 𝐻_𝑘(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 tr𝜎_𝑇‖𝑓(𝑥(𝑡))‖≤‖𝐹𝑥(𝑡)‖,‖‖𝑔𝑥_𝑡‖‖≤‖‖𝐺𝑥_𝑡‖‖,(4.2)𝑡,𝑥(𝑡),𝑥_𝑡𝜎𝑡,𝑥(𝑡),𝑥_𝑡≤‖𝐾𝑥(𝑡)‖²+‖‖𝐾_𝜏𝑥_𝑡‖‖²,(4.3) where 𝐹, 𝐺, 𝐾, and 𝐾_𝜏 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=sup_𝑘∈ℕ{𝜆^𝑘=(𝜆_max(𝑃)/𝜆_min(𝑃))𝛾^2𝑘}. Then the following results hold:
(i)if 𝜇>1, 𝜆+𝜇𝜆_𝜏<−ln𝜇/𝛽, then system (4.1) is mean squarely stable over impulsive time sequence 𝒩_inf(𝜏+𝛽);(ii)if 𝜇=1, 𝜆+𝜆_𝜏≤0, then system (4.1) is mean squarely stable over any impulsive time sequence;(iii)if 𝜇<1 and 0<𝜇𝜆+𝜆_𝜏≤−𝜆_𝜏ln𝜇, then system (4.1) is mean squarely stable over impulsive time sequence 𝒩_sup(−𝜇ln𝜇/(𝜇𝜆+𝜆_𝜏));(iv)if 𝜇<1, 𝜇𝜆+𝜆_𝜏≤0, then system (4.1) is mean squarely stable over any impulsive time sequence, where 𝜆=𝜆_max−2𝐶𝜆+𝜀₁𝐴𝐴^𝑇𝑃+𝜀₂𝐵𝐵^𝑇𝑃+𝑃⁻¹𝜀¹⁻¹𝐹^𝑇𝐹+𝜆_max(𝑃)𝐾^𝑇𝐾_𝜏,=𝜆_max𝑃⁻¹𝜀²⁻¹𝐺^𝑇𝐺+𝜆_max(𝑃)𝐾^𝑇𝜏𝐾_𝜏.(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]. d𝑥¹d𝑥₂(𝑡)(𝑡)=⎧⎪⎨⎪⎩−100.5−102.2𝑥¹𝑥(𝑡)₂(𝑡)+10..26−20.4.2sin𝑥¹arctan𝑥(₂𝑡)×⎡⎢⎢⎢⎣sin𝑥₁(𝑡)+−10.6.510..83arctan𝑥₂𝑡−12𝑡−12⎤⎥⎥⎥⎦⎫⎪⎬⎪⎭d𝑡+⎡⎢⎢⎢⎣2𝑥¹(𝑡)𝑥₂𝑡−1𝑥₁3𝑡−12−𝑥²(𝑡)⎤⎥⎥⎥⎦d𝜔¹d𝜔₂(𝑡)(𝑡),𝑡≥𝑡⁰𝑥₁,𝑡≠𝑡_𝑘,𝑥₂𝑡_𝑘𝑡_𝑘=𝑒^−0.1𝑘00..152−00..615𝑥¹𝑥₂𝑡^−𝑘𝑡^−𝑘,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=0.0560.068,𝐾^𝑇𝐾=4001,(4.6) and 𝜀0_𝑖.4=6681<(𝑖1=,𝜆1=,2−),1𝜆2.0_max44(𝑃3),𝜆=_𝜏0.68,𝜆_min(𝑃)=0.56,𝛾_𝑘=0.620exp(−0.1𝑘),𝐾^𝑇𝜏𝐾_𝜏=𝐼,𝐹=𝐺=𝐼,𝜇==3, and, hence, we have 𝜇𝜆+𝜆_𝜏=−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.