Stability of Stochastic Reaction-Diffusion Systems with Markovian Switching and Impulsive Perturbations

Abstract

This paper is devoted to investigating mean square stability of a class of stochastic reaction-diffusion systems with Markovian switching and impulsive perturbations. Based on Lyapunov functions and stochastic analysis method, some new criteria are established. Moreover, a class of semilinear stochastic impulsive reaction-diffusion differential equations with Markovian switching is discussed and a numerical example is presented to show the effectiveness of the obtained results.

1. Introduction

Markovian jump systems, introduced by Krasovskiĭ and Lidskiĭ [1] in 1961, have received increasing attention, see [2–15] and references therein. Shi and Boukas [3] have probed 𝐻_∞ control for Markovian jumping linear systems with parametric uncertainty. Zhang et al. [4–6] have discussed markovian jump linear systems with partly unknown transition probability. Mao et al. [7–13] have established a number of stability criteria for stochastic differential equations with Markovian switching. However, impulsive perturbations have not been included in the above results.

In fact, impulsive effects widely exist in many fields, such as medicine and biology, economics, mechanics, electronics, and telecommunications [16–19]. Recently, impulsive stochastic differential equations have attracted more and more researchers [20–27]. L. Xu and D. Xu [20] have investigated mean square exponential stability of impulsive control stochastic systems with time-varying delay. Li [23] has obtained the attracting set for impulsive stochastic difference equations with continuous time. Pan and Cao [24] have considered exponential stability of impulsive stochastic functional differential equations. Zhang et al. [25] have studied stability of impulsive stochastic differential equations in terms of two measures via perturbing Lyapunov functions. Moreover, Markovian Jump Systems with impulsive perturbations have been investigated [28–31]. Zhang et al. [28] have established several criteria for stochastic stability analysis of neutral-type impulsive neural networks with mixed time-varying delays and Markovian jumping. Zhu and Cao [30] have obtained several sufficient conditions on stability of Markovian jump neural networks with impulse control and time-varying delays.

Besides impulsive and stochastic effects, reaction diffusion phenomena cannot be ignored in real systems [32–42]. Kao et al. [34] have discussed exponential stability of impulsive stochastic fuzzy reaction-diffusion Cohen-Grossberg neural networks with mixed delays. Wang et al. [40] have probed stochastic exponential stability of the delayed reaction-diffusion recurrent neural networks with Markovian jumping parameters. However, to the best of our knowledge, there are few considering the Markovian jump systems with impulsive perturbations and reaction-diffusion effects.

Motivated by the above discussions, in this paper, we consider mean square stability of a class of impulsive stochastic reaction-diffusion differential systems with Markovian switching. In Section 2, model description and preliminaries are presented. In Section 3, by utilizing Lyapunov function and stochastic analysis, we obtain some new conditions ensuring mean square stability of impulsive stochastic reaction-diffusion differential equations with Markovian switching. Moreover, mean square stability of a class of semilinear stochastic impulsive reaction-diffusion systems has also been discussed. In Section 4, an example is provided. Section 5 is conclusions.

2. Model Description and Preliminaries

In this section, we investigate the impulsive stochastic reaction diffusion equations with Markovian switching described by ()=𝐷()(𝑡)(𝑡)(𝑡)𝑑𝑢𝑡,𝑥𝑡,𝑥,𝑢Δ𝑢+𝑓𝑡,𝑥,𝑢,𝛾𝑑𝑡+𝜎𝑡,𝑥,𝑢,𝛾𝑑𝑤,𝑡≥0,𝑡≠𝑡_𝑘𝑢𝑡,𝑥∈𝐺,_𝑘,𝑥=𝐻_𝑘𝑢𝑡⁻_𝑘,𝑥,𝛾𝑡_𝑘,𝑥∈𝐺, (2.1)(2.2) with boundary condition 𝜕𝑢𝜕𝑁_𝜕𝐺=0,𝑡≥0 (2.3) and initial condition 𝑢0,𝑥,𝑖₀=𝜑(𝑥),𝑥∈𝐺,𝑖₀∈𝕊, (2.4) where 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥))^⊺,𝑥=(𝑥₁,…,𝑥_𝑚)^⊺∈G⊂𝑅^𝑚,𝐺 is a bounded set with smooth boundary 𝜕𝐺,𝜕/𝜕𝑁 is the outward normal derivative. 𝑡_𝑘 is the impulsive moment satisfying 0=𝑡₀<𝑡₁<𝑡₂<⋯, and lim_𝑘→∞𝑡_𝑘=+∞,𝑢(𝑡⁺_𝑘,𝑥) and 𝑢(𝑡⁻_𝑘,𝑥) denote the right-hand limit and left-hand limit of 𝑢(𝑡,𝑥) at 𝑡_𝑘, respectively. 𝑢(𝑡,𝑥)∈𝑃𝐶[[0,+∞)×𝐺,𝑅^𝑛]={𝑢(𝑡,𝑥)∶[0,+∞)×𝐺→𝑅^𝑛∣𝑢(𝑡,𝑥) is continuous for all 𝑡≥0 but points 𝑡_𝑘,𝑢(𝑡⁺_𝑘,𝑥) and 𝑢(𝑡⁻_𝑘,𝑥) exist, furthermore, 𝑢(𝑡⁺_𝑘,𝑥)=𝑢(𝑡_𝑘,𝑥),𝑘={1,2,…}. 𝐷(𝑡,𝑥,𝑢) is a 𝑛×𝑛 matrix, Δ𝑢={Δ𝑢₁,…,Δ𝑢_𝑛}^⊺,Δ𝑢_𝑖=∑^𝑚_𝑙=1(𝜕²𝑢_𝑖/𝜕𝑥²_𝑖),𝑖=1,2,…,𝑛.𝑓 and 𝜎 are continuous, in addition, 𝑓(𝑡,𝑥,𝑢,𝛾(𝑡)) = {𝑓₁(𝑡,𝑥,𝑢,𝛾(𝑡)),…,𝑓_𝑛(𝑡,𝑥,𝑢,𝛾(𝑡))}^⊺, 𝜎(𝑡,𝑥,𝑢,𝛾(𝑡)) = {𝜎₁(𝑡,𝑥,𝑢,𝛾(𝑡)),…,𝜎_𝑛(𝑡,𝑥,𝑢,𝛾(𝑡))}^⊺. 𝐻_𝑘(𝑢(𝑡⁻_𝑘,𝑥),𝛾(𝑡)) = {𝐻_1𝑘(𝑢(𝑡⁻_𝑘,𝑥),𝛾(𝑡)),…,𝐻_𝑛𝑘(𝑢(𝑡⁻_𝑘,𝑥),𝛾(𝑡))}^⊺ represents the impulsive perturbation of 𝑢 at time 𝑡_𝑘. 𝑤(𝑡) is a one-dimensional standard Brownian motion on a complete probability space (Ω,𝐹,{𝐹_𝑡}_𝑡≥0,𝑃) with a natural filtration {𝐹_𝑡}_𝑡≥0. {𝛾(𝑡),𝑡≥0} is a left-continuous Markov process on the probability space (Ω,𝐹,(𝐹_𝑡)_𝑡∈𝐼,𝑃) and takes values in the finite space 𝕊={1,2,…,𝑁} with generator Λ=(𝜋_𝑖𝑗)(𝑖,𝑗∈𝕊) given by 𝑃𝛾()(𝑡)=𝜋𝑡+Δ=𝑗∣𝛾=𝑖^𝑖𝑗Δ+𝑜(Δ),if𝑖≠𝑗,1+𝜋_𝑖𝑖Δ+𝑜(Δ),if𝑖=𝑗, (2.5) where Δ>0 and lim_𝛿→0𝑜(Δ)/Δ=0, 𝜋_𝑖𝑗≥0 is the transition rate from 𝑖 to 𝑗 if 𝑖≠𝑗 and 𝜋_𝑖𝑖=−∑_𝑗≠𝑖𝜋_𝑖𝑗. We suppose that the Markov chain 𝛾(⋅) is independent of the Brownian motion 𝑊(⋅). Moreover, we assume that 𝐻_𝑘(0,𝛾₀)=𝑓(𝑡,𝑥,0,𝛾₀)=𝜎(𝑡,𝑥,0,𝛾₀)=0, then system (2.1) admits a trivial solution 𝑢=0. For 𝑢(𝑡,𝑥)=(𝑢₁(𝑡,𝑥),…,𝑢_𝑛(𝑡,𝑥))^⊺∈𝑅^𝑛, we define ‖𝑢(𝑡,𝑥)‖_𝐺=(∫_𝐺|𝑢(𝑡,𝑥)|²𝑑𝑥)^1/2 where |𝑢(𝑡,𝑥)|²=∑^𝑛_𝑖=1𝑢²_𝑖(𝑡,𝑥). For simplicity, we denote ‖𝑢(𝑡,𝑥)‖_𝐺 by 𝑢(𝑡) throughout this paper.

Let 𝑢(𝑡,𝑥;0,𝜑,𝑖₀) stand for the solution of system (2.1)–(2.4) through (0,𝜑,𝑖₀).

Definition 2.1. The trivial solution 𝑢=0 is said to be mean square stable if for any 𝜀>0, there exists 𝛿=𝛿(𝜀) such that for all 𝑖₀∈𝕊, we have 𝐸𝑢𝑡,𝑥;0,𝜑,𝑖₀²_𝐺<𝜀,𝑡≥0, (2.6) when 𝜑 satisfies 𝐸{‖𝜑‖²_𝐺}≤𝛿.

Definition 2.2. The function 𝑉(𝑡,𝑦,𝛾(𝑡))∶[0,+∞)×𝑅⁺×𝕊→𝑅⁺ belongs to class 𝑣₀^1,2 if(1) for 𝑘=1,2,…, the function 𝑉 is once continuously differentiable in 𝑡 and twice in 𝑦 on (𝑡_𝑘−1,𝑡_𝑘)×𝑅⁺×𝕊, and, in addition, 𝑉(𝑡,0,𝛾₀)=0 holds for 𝑡≥0;(2)𝑉(𝑡,𝑦,𝛾(𝑡)) is locally Lipschitzian in 𝑦;(3) for each 𝑘=1,2,…, there exist finite limits 𝑉𝑡,𝑞,𝛾(𝑡)→𝑉𝑡⁻_𝑘,𝑦,𝛾𝑡_𝑘,if𝑡,𝑞,𝛾(𝑡)→𝑡⁻_𝑘,𝑦,𝛾𝑡_𝑘,𝑉𝑡,𝑞,𝛾(𝑡)→𝑉𝑡⁺_𝑘,𝑦,𝛾𝑡_𝑘,if𝑡,𝑞,𝛾(𝑡)→𝑡⁺_𝑘,𝑦,𝛾𝑡_𝑘. (2.7)

3. Main Results

In this section, we will discuss mean square stability of the trivial solution of system (2.1)–(2.4). Assume 𝐻_𝑘(𝑢(𝑡⁻_𝑘,𝑥),𝛾(𝑡_𝑘)) satisfies ‖𝐻_𝑘(𝑢(𝑡⁻_𝑘,𝑥),𝛾(𝑡_𝑘))‖²_𝐺≤Γ²_𝑘‖𝑢(𝑡⁻_𝑘,𝑥)‖²_𝐺,Γ_𝑘≥0,𝛾(𝑡)=𝑖∈𝕊,𝑘=1,2,….

Theorem 3.1. If there exist constants 𝛼>0,𝛽>0,𝜅>0 and a Lyapunov function 𝑉(𝑡,𝑢(𝑡),𝑖) such that for 𝛾(𝑡)=𝑖,𝑖∈𝕊, we have the following.(A1)𝛼𝑢(𝑡)≤𝑉(𝑡,𝑢(𝑡),𝑖)≤𝜅𝑢(𝑡). (A2)ℒ𝑉(𝑡,𝑢(𝑡),𝑖)≤𝛽𝑉(𝑡,𝑢(𝑡),𝑖),𝑡∈[𝑡_𝑘−1,𝑡_𝑘).  Here the operator ℒ𝑉(𝑡,𝑢(𝑡),𝑖) is defined as ℒ𝑉(𝑡,𝑢(𝑡),𝑖)=𝜕𝑉+𝜕𝑡𝜕𝑉𝜕𝑢_𝐺2𝑢^⊺𝐷(𝑡,𝑥,𝑢)Δ𝑢𝑑𝑥+_𝐺2𝑢^⊺𝑓(𝑡,𝑥,𝑢,𝑖)𝑑𝑥+2trace_𝐺𝑢^⊺𝜎()𝜕𝑡,𝑥,𝑢,𝑖𝑑𝑥²𝑉𝜕𝑢²_𝐺𝑢^⊺𝜎()+𝑁𝑡,𝑥,𝑢,𝑖𝑑𝑥_𝑗=1𝜋_𝑖𝑗𝑉𝑡,𝑥,𝑗. (3.1)(A3)𝜆>1, where 𝜆=inf{𝜆_𝑘∣𝜆_𝑘=𝛼/𝜅𝛾²,𝑘=1,2,…}.(A4)𝛽(𝑡_𝑘−𝑡_𝑘−1)<ln𝜆,𝑘=1,2,….
Then, the trivial solution 𝑢=0 of system (2.1)–(2.4) is stable in mean square.

Proof. For any 𝜀>0, there must exist a scalar 𝛿=𝛿(𝜀)>0 such that 𝛿<(𝛼/𝜅𝜆)𝜀. Next we will prove that 𝐸{‖𝑢(𝑡,𝑥;0,𝜑,𝑖₀)‖²_𝐺}<𝜀 if 𝜑 satisfies 𝐸{‖𝜑‖²_𝐺}≤𝛿.
Let 𝑢=𝑢(𝑡,𝑥;0,𝜑,𝑖₀). Multiplying both sides of (2.1) by 𝑢^⊺, we obtain 12𝑑|𝑢|²=𝑢^⊺𝐷(𝑡,𝑥,𝑢)Δ𝑢+𝑢^⊺𝑓𝑡,𝑥,𝑢,𝛾(𝑡)𝑑𝑡+𝑢^⊺𝜎𝑡,𝑥,𝑢,𝛾(𝑡)𝑑𝑤(𝑡). (3.2) By integrating the above equality with respect to 𝑥 on 𝐺, we then have 12𝑑_𝐺|𝑢|²𝑑𝑥=_𝐺𝑢^⊺𝐷(𝑡,𝑥,𝑢)Δ𝑢𝑑𝑥+_𝐺𝑢^⊺𝑓(𝑡)+𝑡,𝑥,𝑢,𝛾𝑑𝑥𝑑𝑡_𝐺𝑢^⊺𝜎𝑡,𝑥,𝑢,𝛾(𝑡)𝑑𝑥𝑑𝑤(𝑡). (3.3) Namely, 𝑑𝑢(𝑡)=_𝐺2𝑢^⊺𝐷(𝑡,𝑥,𝑢)Δ𝑢𝑑𝑥+_𝐺𝑢^⊺𝑓(𝑡)+𝑡,𝑥,𝑢,𝛾𝑑𝑥𝑑𝑡_𝐺2𝑢^⊺𝜎𝑡,𝑥,𝑢,𝛾(𝑡)𝑑𝑥𝑑𝑤(𝑡). (3.4) Applying Itô formula, we further compute, when 𝑡≠𝑡_𝑘, 𝑑𝑉(𝑡,𝑢(𝑡),𝑖)=ℒ𝑉(𝑡,𝑢(𝑡),𝑖)+𝜕𝑉𝜕𝑢_𝐺2𝑢^⊺𝜎(𝑡,𝑥,𝑢,𝑖)𝑑𝑥𝑑𝑤(𝑡), (3.5) where 𝛾(𝑡)=𝑖 and ℒ𝑉(𝑡,𝑢(𝑡))=(,𝑖𝜕𝑉𝑡,𝑢(𝑡),𝑖)+(𝜕𝑡𝜕𝑉𝑡,𝑢(𝑡),𝑖)𝜕𝑢_𝐺2𝑢^⊺𝐷(𝑡,𝑥,𝑢)Δ𝑢𝑑𝑥+_𝐺2𝑢^⊺𝑓(𝑡,𝑥,𝑢,𝑖)𝑑𝑥+2trace_𝐺𝑢^⊺𝜎()𝜕𝑡,𝑥,𝑢,𝑖𝑑𝑥²𝑉𝜕𝑢²_𝐺𝑢^⊺𝜎()+𝑁𝑡,𝑥,𝑢,𝑖𝑑𝑥_𝑗=1𝜋_𝑖𝑗𝑉𝑡,𝑥,𝑗. (3.6) For 𝑡∈[𝑡_𝑘,𝑡_𝑘+1), integrating (3.5) with respect to 𝑡 from 𝑡_𝑘 to 𝑡, one has 𝑉𝑡,𝑢(𝑡),𝛾(𝑡)=𝑉𝑡_𝑘,𝑢𝑡_𝑘,𝛾𝑡_𝑘+^𝑡_𝑡𝑘ℒ𝑉𝑠,𝑢(𝑠)(𝑠)+,𝛾𝑑𝑠_𝑡𝑡𝑘𝜕𝑉𝜕𝑢_𝐺2𝑢^⊺𝜎𝑡,𝑥,𝑢,𝛾(𝑡)𝑑𝑥𝑑𝑤(𝑡). (3.7) Taking the mathematical expectation of both sides of (3.7), we obtain 𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)=𝐸𝑉𝑡_𝑘,𝑢𝑡_𝑘,𝛾𝑡_𝑘+^𝑡_𝑡𝑘𝐸ℒ𝑉𝑠,𝑢(𝑠),𝛾(𝑠)𝑑𝑠. (3.8) Choosing small enough Δ𝑡>0 such that 𝑡+Δ𝑡∈[𝑡_𝑘,𝑡_𝑘+1), it is easy to see that 𝐸𝑉𝑡+Δ𝑡,𝑢(𝑡+Δ𝑡),𝛾(𝑡+Δ𝑡)=𝐸𝑉𝑡_𝑘,𝑢𝑡_𝑘,𝛾𝑡_𝑘+_𝑡^𝑡+Δ𝑡_𝑘𝐸ℒ𝑉𝑠,𝑢(𝑠),𝛾(𝑠)𝑑𝑠. (3.9) We thus derive from (3.8) and (3.9) that 𝐸𝑉𝑡+Δ𝑡,𝑢(𝑡+Δ𝑡),𝛾(𝑡+Δ𝑡)−𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)=_𝑡^𝑡+Δ𝑡𝐸ℒ𝑉𝑠,𝑢(𝑠)(𝑠≤,𝛾)𝑑𝑠_{𝑡𝑡+Δ𝑡}𝛽𝐸𝑉𝑠,𝑢(𝑠),𝛾(𝑠)𝑑𝑠. (3.10) So 𝐷⁺𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝛽𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡). (3.11) Next, we will first prove 𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝜅𝜆𝛿,0≤𝑡<𝑡₁. (3.12) Obviously, 𝐸𝑉0,𝑢(0),𝛾₀≤𝐸𝜅𝑢(0)=𝜅𝐸𝜑²_𝐺≤𝜅𝛿<𝜅𝜆𝛿. (3.13) If inequality (3.12) does not hold, there must exist some 𝑠∈(0,𝑡₁) such that 𝐸𝑉𝑠,𝑢(𝑠),𝛾(𝑠)>𝜅𝜆𝛿>𝜅𝛿≥𝐸𝑉0,𝑢(0),𝛾₀. (3.14) Let 𝑠₁=inf{𝑠∈(0,𝑡₁)∣𝐸𝑉(𝑠,𝑢(𝑠),𝛾(𝑠)>𝜅𝜆𝛿}. Since 𝐸𝑉(𝑡,𝑢(𝑡),𝛾(𝑡)) is continuous on [0,𝑠₁], there exist 𝑠̂∈(0,𝑠₁) such that 𝐸𝑉𝑠,̂𝑢(𝑠)̂,𝛾(𝑠)̂=𝜅𝜆𝛿,𝐸𝑉𝑡,𝑢(𝑡)(𝑡)[𝑠)̂,𝐷,𝛾≤𝜅𝜆𝛿,𝑡∈0,₊𝐸𝑉𝑠,̂𝑢(𝑠)̂,𝛾(𝑠)̂>0. (3.15) From 𝐸𝑉(𝑠,̂𝑢(𝑠)̂,𝛾(𝑠)̂)=𝜅𝜆𝛿>𝜅𝛿, if 𝐸𝑉(0,𝑢(0),𝛾₀)≤𝜅𝛿, we know that there is 𝑠₂∈[0,𝑠)̂ such that 𝐸𝑉𝑠₂,𝑢𝑠₂,𝛾𝑠₂=𝜅𝛿,𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≥𝜅𝛿,𝑡∈𝑠₂,𝑠̂,𝐷⁺𝐸𝑉𝑠₂,𝑢𝑠₂,𝛾𝑠₂>0. (3.16) On the other hand, noticing 𝐷⁺𝐸𝑉(𝑡,𝑢(𝑡),𝛾(𝑡))≤𝛽𝐸𝑉(𝑡,𝑢(𝑡),𝛾(𝑡)), we obtain𝐷⁺𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝛽. (3.17) Integrating both sides of (3.17) on 𝑡∈[𝑠₂,𝑠]̂ gives _𝑠𝑠₂̂𝐷⁺𝐸𝑉𝑠,𝑢(𝑠),𝛾(𝑠)𝐸𝑉𝑠,𝑢(𝑠)(𝑠)𝑠̂,𝛾𝑑𝑠≤_𝑠2𝛽𝑑𝑠<𝛽𝑡₁<ln𝜆. (3.18) However, _𝑠𝑠₂̂𝐷⁺𝐸𝑉𝑠,𝑢(𝑠),𝛾(𝑠)𝐸𝑉𝑠,𝑢(𝑠),𝛾(𝑠)𝑑𝑠=^𝐸𝑉(𝑠̂_,𝑢(𝑠̂_),𝛾(𝑠̂_{𝑠))𝐸𝑉2,𝑢𝑠2𝑠,𝛾2}𝑑𝜂𝜂=^𝜅𝜆𝛿_𝜅𝛿𝑑𝜂𝜂=ln(𝜅𝜆𝛿)−ln(𝜅𝛿)=ln𝜆, (3.19) which is a contradiction. Therefore, 𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝜅𝜆𝛿,0≤𝑡<𝑡₁. (3.20) Furthermore, 𝐸𝑉𝑡₁,𝑢𝑡₁,𝛾𝑡₁≤𝐸𝜅𝑢𝑡₁=𝜅𝐸𝐻₁(𝑢(𝑡⁻₁,𝑥))²_𝐺≤𝜅Γ²₁𝐸𝑢𝑡⁻₁,𝑥²_Ω≤𝜅Γ²₁𝛼𝐸𝑉𝑡₁,𝑢𝑡⁻₁,𝛾𝑡₁≤𝜅𝜆𝛿𝜆₁≤𝜅𝛿. (3.21) Now we assume that 𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝜅𝜆𝛿,𝑡_𝑚−1≤𝑡<𝑡_𝑚𝐸𝑉𝑡_𝑚,𝑢𝑡_𝑚,𝛾𝑡_𝑚,𝛾𝑡_𝑚≤𝜅𝛿 (3.22) and then prove 𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝜅𝜆𝛿,𝑡_𝑚≤𝑡<𝑡_𝑚+1,𝐸𝑉𝑡_𝑚+1,𝑢𝑡_𝑚+1,𝛾𝑡_𝑚+1≤𝜅𝛿. (3.23) If not, there must exist some 𝜏∈(𝑡_𝑚,𝑡_𝑚+1) such that 𝐸𝑉𝜏,𝑢(𝜏),𝛾(𝜏)>𝜅𝜆𝛿>𝜅𝛿≥𝐸𝑉𝑡_𝑚,𝑢𝑡_𝑚,𝛾𝑡_𝑚,𝛾𝑡_𝑚. (3.24) Let 𝜏₁=inf𝜏∈𝑡_𝑚,𝑡_𝑚+1∣𝐸𝑉𝜏,𝑢(𝜏),𝛾(𝜏)>𝜅𝜆𝛿. (3.25) Since 𝐸𝑉(𝑡,𝑢(𝑡),𝛾(𝑡)) is continuous in [𝑡_𝑚,𝜏₁], there exists 𝜏₂∈(𝑡_𝑚,𝜏₁) satisfying 𝐸𝑉𝜏₂,𝑢𝜏₂,𝛾𝜏₂=𝜅𝜆𝛿,𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝜅𝜆𝛿,𝑡∈𝑡_𝑚,𝜏₂,𝐷⁺𝐸𝑉𝜏₂,𝑢𝜏₂,𝛾𝜏₂>0. (3.26) Because of 𝐸𝑉(𝜏₂,𝑢(𝜏₂),𝛾(𝜏₂))=𝜅𝜆𝛿>𝜅𝛿 and 𝐸𝑉(𝑡_𝑚,𝑢(𝑡_𝑚),𝛾(𝑡_𝑚))≤𝜅𝜀, there is 𝜏₃∈[𝑡_𝑚,𝜏₂) such that 𝐸𝑉𝜏₃,𝑢𝜏₃,𝛾𝜏₃=𝜅𝛿,𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝜅𝛿,𝑡∈𝜏₃,𝜏₂.𝐷⁺𝐸𝑉𝜏₃,𝑢𝜏₃,𝛾𝜏₃>0. (3.27) Noticing 𝐷⁺𝐸𝑉(𝑡,𝑢(𝑡),𝛾(𝑡))≤𝛽𝐸𝑉(𝑡,𝑢(𝑡),𝛾(𝑡)), we obtain 𝐷⁺𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝛽. (3.28) Integrating both sides of (3.28) on 𝑡∈[𝜏₃,𝜏₂], we claim that ^𝜏2_𝜏3𝐷⁺𝐸𝑉𝑠,𝑢(𝑠),𝛾(𝑠)𝐸𝑉𝑠,𝑢(𝑠),𝛾(𝑠)𝑑𝑠≤^𝜏2_𝜏3𝛽𝑑𝑠<𝛽𝑡_𝑚+1−𝑡_𝑚<ln𝜆. (3.29) However, ^𝜏2_𝜏3𝐷⁺𝐸𝑉𝑠,𝑢(𝑠),𝛾(𝑠)𝐸𝑉𝑠,𝑢(𝑠),𝛾(𝑠)𝑑𝑠=^{𝐸𝑉(𝜏2,𝑢(𝜏2),𝛾(𝜏2}_𝜏⁾⁾_{𝐸𝑉3,𝑢𝜏3𝜏,𝛾3}𝑑𝜂𝜂=^𝜅𝜆𝛿_𝜅𝛿𝑑𝜂𝜂=ln(𝜅𝜆𝛿)−ln(𝜅𝛿)=ln𝜆. (3.30) This leads to a contradiction. Then, we have 𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝜅𝜆𝛿,𝑡_𝑚≤𝑡<𝑡_𝑚+1. (3.31) Moreover, 𝐸𝑉𝑡_𝑚+1,𝑢𝑡_𝑚+1,𝛾𝑡_𝑚+1≤𝐸𝜅𝑢𝑡_𝑚+1=𝜅𝐸𝐻_𝑚+1𝑢𝑡⁻_𝑚+1,𝑥²_𝐺≤𝜅Γ²_𝑚+1𝐸𝑢𝑡⁻_𝑚+1,𝑥²_𝐺≤𝜅Γ²_𝑚+1𝛼𝐸𝑉𝑡_𝑚+1,𝑢𝑡⁻_𝑚+1,𝛾𝑡_𝑚+1≤𝜅𝜆𝛿𝜆_𝑚+1≤𝜅𝛿. (3.32) Therefore, 𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝜅𝜆𝛿,𝑡≥0, (3.33) which results in 𝛼𝐸𝑢(𝑡)≤𝐸𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤𝜅𝜆𝛿,𝑡≥0 (3.34) Namely, 𝐸𝑢𝑡,𝑥;0,𝜑²_𝐺<𝜀,𝑡≥0. (3.35) This ends the proof of Theorem 3.1.

As an application, we consider a class of semilinear impulsive stochastic reaction-diffusion equations with Markovian switching as follows: ()=𝐶𝛾(𝑡)()𝛾(𝑡)𝑢((𝑡)(𝑡)𝑑𝑢𝑡,𝑥Δ𝑢𝑡,𝑥+𝐴𝑡,𝑥)𝑑𝑡+𝜎𝑡,𝑥,𝑢,𝛾𝑑𝑤,𝑡≥0,𝑡≠𝑡_𝑘𝑢𝑡,𝑥∈𝐺,_𝑘,𝑥=𝐻_𝑘𝑢𝑡⁻_𝑘,𝑥,𝛾𝑡_𝑘,𝑥∈𝐺, (3.36)(3.37) with boundary condition 𝜕𝑢𝜕𝑁_𝜕𝐺=0,𝑡≥0 (3.38) and initial condition 𝑢0,𝑥,𝑖₀=𝜑(𝑥),𝑥∈𝐺,𝑖₀∈𝕊 (3.39) where 𝐶(𝛾(𝑡))=diag{𝑐₁(𝛾(𝑡)),…,𝑐_𝑛(𝛾(𝑡))}=diag{𝑐_𝑖1,…,𝑐_𝑖𝑛} with 𝑐_𝑖𝑗≥0 for 𝑖∈𝕊,𝑗=1,2,…,𝑛. 𝐴(𝛾(𝑡))=𝐴_𝑖=(𝑎_𝑙𝑗(𝛾(𝑡)))_𝑛×𝑛=(𝑎^(𝑖)_𝑙𝑗)_𝑛×𝑛,𝑖∈𝕊, is matrices. The remainder of system (3.36)–(3.39) is the same as that defined in system (2.1)–(2.4).

Theorem 3.2. Assume that(A5)𝜆>1, where 𝜆=inf{𝜆_𝑘∣𝜆_𝑘=1/Γ²_𝑘,𝑘=1,2,…},(A6)2max_𝑖{𝜆_max(𝐴_𝑖)}(𝑡_𝑘−𝑡_𝑘−1)<ln𝜆,𝑘=1,2…,𝑖∈𝕊.
Then, the trivial solution 𝑢=0 of system (3.36)–(3.39) is stable in mean square.

Proof. Construct a Lyapunov function 𝑉(𝑡,𝑢(𝑡),𝛾(𝑡))=𝑢(𝑡), and compute the operator ℒ𝑉(𝑡,𝑢(𝑡),𝛾(𝑡)) that ℒ𝑉𝑡,𝑢(𝑡),𝛾(𝑡)=_𝐺2𝑢^⊺𝐶𝛾(𝑡)Δ𝑢𝑑𝑥+_𝐺2𝑢^⊺𝐴𝛾(𝑡)𝑢(𝑡,𝑥)𝑑𝑥. (3.40) By Green formula, we get _𝐺2𝑢^⊺𝐶_𝑖Δ𝑢𝑑𝑥=2_𝐺^𝑛_𝑗=1𝑢_𝑗𝑐_𝑖𝑗Δ𝑢_𝑗𝑑𝑥=2^𝑛_𝑗=1𝐺𝑢_𝑗𝑐_𝑖𝑗Δ𝑢_𝑗𝑑𝑥=2^𝑛_𝑗=1𝜕Ω𝑢_𝑗𝑐_𝑖𝑗𝜕𝑢_𝑗𝜕𝑁𝑑𝑠−2^𝑛_𝑗=1𝐺𝑐_𝑖𝑗∇𝑢_𝑗⋅∇𝑢_𝑗𝑑𝑥. (3.41) It follows from boundary condition that ^𝑛_𝑗=1𝜕Ω𝑢_𝑗𝑐_𝑖𝑗𝜕𝑢_𝑗𝜕𝑁𝑑𝑠=0. (3.42) Thus, _𝐺2𝑢^⊺𝐶_𝑖Δ𝑢𝑑𝑥=−2^𝑛_𝑗=1𝐺𝑐_𝑖𝑗∇𝑢_𝑗⋅∇𝑢_𝑗𝑑𝑥≤0. (3.43) Therefore, L𝑉𝑡,𝑢(𝑡),𝛾(𝑡)≤2_𝐺𝑢^⊺𝐴𝛾(𝑡)𝑢(𝑡,𝑥)𝑑𝑥≤2max_𝑖𝜆_max𝐴_𝑖𝑢(𝑡). (3.44) According to Theorem 3.1, we find that the trivial solution of system (3.36)–(3.39) is stable in mean square.

4. Example

Consider the following two dimension Markovian jumping impulsive stochastic reaction diffusion systems with two modes. The parameters are given as follows: Let |𝐺|=1/8, when 𝑟(𝑡)=1, we have𝑑𝑢₁(𝑡,𝑥)𝑑𝑢₂()=𝑡,𝑥3.4005.5Δ𝑢₁(𝑡,𝑥)Δ𝑢₂()+𝑢𝑡,𝑥2.6001.7¹()𝑢𝑡,𝑥₂()+𝑡,𝑥2𝑢₁(𝑡,𝑥)−𝑢₂()(𝑡)𝑡,𝑥𝑑𝑤,𝑡≥0,𝑡≠𝑡_𝑘,𝑢₁𝑡_𝑘𝑢,𝑥₂𝑡_𝑘,𝑥=𝑒^−0.1𝑘0.5−0.2𝑢0.30.6¹𝑡⁻_𝑘𝑢,𝑥₂𝑡⁻_𝑘,,𝑥 (4.1) when 𝑟(𝑡)=2𝑑𝑢₁(𝑡,𝑥)𝑑𝑢₂()=𝑡,𝑥2.4002.5Δ𝑢₁(𝑡,𝑥)Δ𝑢₂()+𝑢𝑡,𝑥1.6002.7¹()𝑢𝑡,𝑥₂()+𝑡,𝑥2𝑢₁(𝑡,𝑥)−𝑢₂()(𝑡)𝑡,𝑥𝑑𝑤,𝑡≥0,𝑡≠𝑡_𝑘,𝑢₁𝑡_𝑘𝑢,𝑥₂𝑡_𝑘,𝑥=𝑒^−0.1𝑘0.6−0.3𝑢0.30.4¹𝑡⁻_𝑘𝑢,𝑥₂𝑡⁻_𝑘,,𝑥 (4.2) where 𝑡₀=0,𝑡_𝑘=𝑡_𝑘−1+0.1,(𝑘=1,2,…). By simple calculation, we obtain Γ_𝑘=0.6𝑒^−0.1𝑘,𝜆=2.6>1,2𝜆_max(𝐴_𝑖)(𝑡_𝑘−𝑡_𝑘−1)<ln𝜆=0.856. From Theorem 3.2, the trivial solution of this system is stable in mean square.

5. Conclusion

In this paper, we discuss mean square stability of stochastic reaction diffusion equations with Markovian switching and impulsive perturbations, by means of Lyapunov function and stochastic analysis. As an application, we investigate a class of semilinear impulsive stochastic reaction-diffusion equations with Markovian switching and establish the stability criterion. Finally, we provide an example to demonstrate the effectiveness and efficiency of the obtained results.

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and constructive suggestions. This research is supported by the National Natural Science Foundations of China (60974025, 60939003), National 863 Plan Project (2008AA04Z401, 2009AA043404), the Natural Science Foundation of Shandong Province (no. Y 2007G30), the Natural Science Foundation of Guangxi Autonomous Region (no. 2012GXNSFBA053003), the Scientific and Technological Project of Shandong Province (no. 2007GG3WZ04016), the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF.2001120), the China Postdoctoral Science Foundation (20100481000) and the Shandong Provincial Key Laboratory of Industrial Control Technique (Qingdao University).