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

Authors: Yanbo Li; Yonggui Kao
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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 Section2, 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(2.1)du(t,x)=[D(t,x,u)Δu+f(t,x,u,γ(t))]dt+σ(t,x,u,γ(t))dw(t),t≥0,t≠tk,x∈G,(2.2)u(tk,x)=Hk(u(tk-,x),γ(tk)),x∈G,
with boundary condition
(2.3)∂u∂N|∂G=0,t≥0
and initial condition
(2.4)u(0,x,i0)=φ(x),x∈G,i0∈𝕊,
where u(t,x)=(u1(t,x),…,un(t,x))⊺,x=(x1,…,xm)⊺∈G⊂Rm,G is a bounded set with smooth boundary ∂G,∂/∂N is the outward normal derivative. tk is the impulsive moment satisfying 0=t0<t1<t2<⋯, and limk→∞tk=+∞,u(tk+,x) and u(tk-,x) denote the right-hand limit and left-hand limit of u(t,x) at tk, respectively. u(t,x)∈PC[[0,+∞)×G,Rn]={u(t,x):[0,+∞)×G→Rn∣u(t,x) is continuous for all t≥0 but points tk,u(tk+,x) and u(tk-,x) exist, furthermore, u(tk+,x)=u(tk,x),k={1,2,…}. D(t,x,u) is a n×n matrix, Δu={Δu1,…,Δun}⊺,Δui=∑l=1m‍(∂2ui/∂xi2),i=1,2,…,n.f and σ are continuous, in addition, f(t,x,u,γ(t)) = {f1(t,x,u,γ(t)),…,fn(t,x,u,γ(t))}⊺, σ(t,x,u,γ(t)) = {σ1(t,x,u,γ(t)),…,σn(t,x,u,γ(t))}⊺. Hk(u(tk-,x),γ(t)) = {H1k(u(tk-,x),γ(t)),…,Hnk(u(tk-,x),γ(t))}⊺ represents the impulsive perturbation of u at time tk. w(t) is a one-dimensional standard Brownian motion on a complete probability space (Ω,F,{Ft}t≥0,P) with a natural filtration {Ft}t≥0. {γ(t),t≥0} is a left-continuous Markov process on the probability space (Ω,F,(Ft)t∈I,P) and takes values in the finite space 𝕊={1,2,…,N~} with generator Λ=(πij)(i,j∈𝕊) given by
(2.5)P{γ(t+Δ)=j∣γ(t)=i}={πijΔ+o(Δ),ifi≠j,1+πiiΔ+o(Δ),ifi=j,
where Δ>0 and limδ→0o(Δ)/Δ=0, πij≥0 is the transition rate from i to j if i≠j and πii=-∑j≠i‍πij. We suppose that the Markov chain γ(·) is independent of the Brownian motion W(·). Moreover, we assume that Hk(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),…,un(t,x))⊺∈Rn, we define ∥u(t,x)∥G=(∫G‍|u(t,x)|2dx)1/2 where |u(t,x)|2=∑i=1n‍ui2(t,x). For simplicity, we denote ∥u(t,x)∥G by u-(t) throughout this paper.Letu(t,x;0,φ,i0) stand for the solution of system (2.1)–(2.4) through (0,φ,i0).Definition 2.1.
The trivial solutionu=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
(2.6)E{∥u(t,x;0,φ,i0)∥G2}<ε,t≥0,
when φ satisfies E{∥φ∥G2}≤δ.Definition 2.2.
The functionV(t,y,γ(t)):[0,+∞)×R+×𝕊→R+ belongs to class v01,2 if(1)
fork=1,2,…, the function V is once continuously differentiable in t and twice in y on (tk-1,tk)×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 eachk=1,2,…, there exist finite limits
(2.7)V(t,q,γ(t))→V(tk-,y,γ(tk)),if(t,q,γ(t))→(tk-,y,γ(tk)),V(t,q,γ(t))→V(tk+,y,γ(tk)),if(t,q,γ(t))→(tk+,y,γ(tk)).


## 3. Main Results

In this section, we will discuss mean square stability of the trivial solution of system (2.1)–(2.4). Assume Hk(u(tk-,x),γ(tk)) satisfies ∥Hk(u(tk-,x),γ(tk))∥G2≤Γk2∥u(tk-,x)∥G2,Γk≥0,γ(t)=i∈𝕊,k=1,2,….Theorem 3.1.
If there exist constantsα>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∈[tk-1,tk).Here the operatorℒV(t,u-(t),i) is defined as
(3.1)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,i)dx)∂2V∂u-2(∫G‍u⊺σ(t,x,u,i)dx)}+∑j=1N~‍πijV(t,x,j).(A3)
λ>1, where λ=inf{λk∣λk=α/κγ2,k=1,2,…}.(A4)
β(tk-tk-1)<lnλ,k=1,2,….
Then, the trivial solutionu=0 of system (2.1)–(2.4) is stable in mean square.Proof.
For anyε>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)∥G2}<ε if φ satisfies E{∥φ∥G2}≤δ.
Letu=u(t,x;0,φ,i0). Multiplying both sides of (2.1) by u⊺, we obtain
(3.2)12d|u|2=[u⊺D(t,x,u)Δu+u⊺f(t,x,u,γ(t))]dt+u⊺σ(t,x,u,γ(t))dw(t).
By integrating the above equality with respect to x on G, we then have
(3.3)12d∫G‍|u|2dx=[∫G‍u⊺D(t,x,u)Δudx+∫G‍u⊺f(t,x,u,γ(t))dx]dt+∫G‍u⊺σ(t,x,u,γ(t))dxdw(t).
Namely,
(3.4)du-(t)=[∫G‍2u⊺D(t,x,u)Δudx+∫G‍u⊺f(t,x,u,γ(t))dx]dt+∫G‍2u⊺σ(t,x,u,γ(t))dxdw(t).
Applying Itô formula, we further compute, when t≠tk,
(3.5)dV(t,u-(t),i)=ℒV(t,u-(t),i)+∂V∂u-∫G‍2u⊺σ(t,x,u,i)dxdw(t),
where γ(t)=i and
(3.6)ℒV(t,u-(t),i)=∂V(t,u-(t),i)∂t+∂V(t,u-(t),i)∂u-[∫G‍2u⊺D(t,x,u)Δudx+∫G‍2u⊺f(t,x,u,i)dx]+2trace{(∫G‍u⊺σ(t,x,u,i)dx)∂2V∂u-2(∫G‍u⊺σ(t,x,u,i)dx)}+∑j=1N~‍πijV(t,x,j).
For t∈[tk,tk+1), integrating (3.5) with respect to t from tk to t, one has
(3.7)V(t,u-(t),γ(t))=V(tk,u-(tk),γ(tk))+∫tkt‍ℒV(s,u-(s),γ(s))ds+∫tkt‍∂V∂u-∫G‍2u⊺σ(t,x,u,γ(t)dxdw(t)).
Taking the mathematical expectation of both sides of (3.7), we obtain
(3.8)EV(t,u-(t),γ(t))=EV(tk,u-(tk),γ(tk))+∫tkt‍EℒV(s,u-(s),γ(s))ds.
Choosing small enough Δt>0 such that t+Δt∈[tk,tk+1), it is easy to see that
(3.9)EV(t+Δt,u-(t+Δt),γ(t+Δt))=EV(tk,u-(tk),γ(tk))+∫tkt+Δt‍EℒV(s,u-(s),γ(s))ds.
We thus derive from (3.8) and (3.9) that
(3.10)EV(t+Δt,u-(t+Δt),γ(t+Δt))-EV(t,u-(t),γ(t))=∫tt+Δt‍EℒV(s,u-(s),γ(s))ds≤∫tt+Δt‍βEV(s,u-(s),γ(s))ds.
So
(3.11)D+EV(t,u-(t),γ(t))≤βEV(t,u-(t),γ(t)).
Next, we will first prove
(3.12)EV(t,u-(t),γ(t))≤κλδ,0≤t<t1.
Obviously,
(3.13)EV(0,u-(0),γ0)≤Eκu-(0)=κE∥φ∥G2≤κδ<κλδ.
If inequality (3.12) does not hold, there must exist some s∈(0,t1) such that
(3.14)EV(s,u-(s),γ(s))>κλδ>κδ≥EV(0,u-(0),γ0).
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 s^∈(0,s1) such that
(3.15)EV(s^,u-(s^),γ(s^))=κλδ,EV(t,u-(t),γ(t))≤κλδ,t∈[0,s^),D+EV(s^,u-(s^),γ(s^))>0.
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
(3.16)EV(s2,u-(s2),γ(s2))=κδ,EV(t,u-(t),γ(t))≥κδ,t∈[s2,s^],D+EV(s2,u-(s2),γ(s2))>0.
On the other hand, noticing D+EV(t,u-(t),γ(t))≤βEV(t,u-(t),γ(t)), we obtain(3.17)D+EV(t,u-(t),γ(t))EV(t,u-(t),γ(t))≤β.
Integrating both sides of (3.17) on t∈[s2,s^] gives
(3.18)∫s2s^‍D+EV(s,u-(s),γ(s))EV(s,u-(s),γ(s))ds≤∫s2s^‍βds<βt1<lnλ.
However,
(3.19)∫s2s^‍D+EV(s,u-(s),γ(s))EV(s,u-(s),γ(s))ds=∫EV(s2,u-(s2),γ(s2))EV(s^,u-(s^),γ(s^))‍dηη=∫κδκλδ‍dηη=ln(κλδ)-ln(κδ)=lnλ,
which is a contradiction. Therefore,
(3.20)EV(t,u-(t),γ(t))≤κλδ,0≤t<t1.
Furthermore,
(3.21)EV(t1,u-(t1),γ(t1))≤Eκu-(t1)=κE∥H1(u(t1-,x))∥G2≤κΓ12E∥u(t1-,x)∥Ω2≤κΓ12αEV(t1,u-(t1-,γ(t1)))≤κλδλ1≤κδ.
Now we assume that
(3.22)EV(t,u-(t),γ(t))≤κλδ,tm-1≤t<tmEV(tm,u-(tm),γ(tm),γ(tm))≤κδ
and then prove
(3.23)EV(t,u-(t),γ(t))≤κλδ,tm≤t<tm+1,EV(tm+1,u-(tm+1),γ(tm+1))≤κδ.
If not, there must exist some τ∈(tm,tm+1) such that
(3.24)EV(τ,u¯(τ),γ(τ))>κλδ>κδ≥EV(tm,u-(tm),γ(tm),γ(tm)).
Let
(3.25)τ1=inf{τ∈(tm,tm+1)∣EV(τ,u-(τ),γ(τ))>κλδ}.
Since EV(t,u-(t),γ(t)) is continuous in [tm,τ1], there exists τ2∈(tm,τ1) satisfying
(3.26)EV(τ2,u-(τ2),γ(τ2))=κλδ,EV(t,u-(t),γ(t))≤κλδ,t∈[tm,τ2],D+EV(τ2,u-(τ2),γ(τ2))>0.
Because of EV(τ2,u-(τ2),γ(τ2))=κλδ>κδ and EV(tm,u-(tm),γ(tm))≤κε, there is τ3∈[tm,τ2) such that
(3.27)EV(τ3,u-(τ3),γ(τ3))=κδ,EV(t,u-(t),γ(t))≤κδ,t∈[τ3,τ2].D+EV(τ3,u-(τ3),γ(τ3))>0.
Noticing D+EV(t,u-(t),γ(t))≤βEV(t,u-(t),γ(t)), we obtain
(3.28)D+EV(t,u-(t),γ(t))EV(t,u-(t),γ(t))≤β.
Integrating both sides of (3.28) on t∈[τ3,τ2], we claim that
(3.29)∫τ3τ2‍D+EV(s,u-(s),γ(s))EV(s,u-(s),γ(s))ds≤∫τ3τ2‍βds<β(tm+1-tm)<lnλ.
However,
(3.30)∫τ3τ2‍D+EV(s,u-(s),γ(s))EV(s,u-(s),γ(s))ds=∫EV(τ3,u-(τ3),γ(τ3))EV(τ2,u-(τ2),γ(τ2))‍dηη=∫κδκλδ‍dηη=ln(κλδ)-ln(κδ)=lnλ.
This leads to a contradiction. Then, we have
(3.31)EV(t,u-(t),γ(t))≤κλδ,tm≤t<tm+1.
Moreover,
(3.32)EV(tm+1,u-(tm+1),γ(tm+1))≤Eκu-(tm+1)=κE∥Hm+1(u(tm+1-,x))∥G2≤κΓm+12E∥u(tm+1-,x)∥G2≤κΓm+12αEV(tm+1,u-(tm+1-),γ(tm+1))≤κλδλm+1≤κδ.
Therefore,
(3.33)EV(t,u-(t),γ(t))≤κλδ,t≥0,
which results in
(3.34)αEu-(t)≤EV(t,u-(t),γ(t))≤κλδ,t≥0
Namely,
(3.35)E{∥u(t,x;0,φ)∥G2}<ε,t≥0.
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:(3.36)du(t,x)=[C(γ(t))Δu(t,x)+A(γ(t))u(t,x)]dt+σ(t,x,u,γ(t))dw(t),t≥0,t≠tk,x∈G,(3.37)u(tk,x)=Hk(u(tk-,x),γ(tk)),x∈G,
with boundary condition
(3.38)∂u∂N|∂G=0,t≥0
and initial condition
(3.39)u(0,x,i0)=φ(x),x∈G,i0∈𝕊
where C(γ(t))=diag{c1(γ(t)),…,cn(γ(t))}=diag{ci1,…,cin} with cij≥0 for i∈𝕊,j=1,2,…,n. A(γ(t))=Ai=(alj(γ(t)))n×n=(alj(i))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{λk∣λk=1/Γk2,k=1,2,…},(A6)
2maxi{λmax(Ai)}(tk-tk-1)<lnλ,k=1,2…,i∈𝕊.
Then, the trivial solutionu=0 of system (3.36)–(3.39) is stable in mean square.Proof.
Construct a Lyapunov functionV(t,u-(t),γ(t))=u-(t), and compute the operator ℒV(t,u-(t),γ(t)) that
(3.40)ℒV(t,u-(t),γ(t))=∫G‍2u⊺C(γ(t))Δudx+∫G‍2u⊺A(γ(t))u(t,x)dx.
By Green formula, we get
(3.41)∫G‍2u⊺CiΔudx=2∫G‍∑j=1n‍ujcijΔujdx=2∑j=1n‍∫G‍ujcijΔujdx=2∑j=1n‍∫∂Ω‍ujcij∂uj∂Nds-2∑j=1n‍∫G‍cij∇uj·∇ujdx.
It follows from boundary condition that
(3.42)∑j=1n‍∫∂Ω‍ujcij∂uj∂Nds=0.
Thus,
(3.43)∫G‍2u⊺CiΔudx=-2∑j=1n‍∫G‍cij∇uj·∇ujdx≤0.
Therefore,
(3.44)LV(t,u-(t),γ(t))≤2∫G‍u⊺A(γ(t))u(t,x)dx≤2maxi{λmax(Ai)}u-(t).
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 have(4.1)(du1(t,x)du2(t,x))=(3.4005.5)(Δu1(t,x)Δu2(t,x))+(2.6001.7)(u1(t,x)u2(t,x))+(2u1(t,x)-u2(t,x))dw(t),t≥0,t≠tk,(u1(tk,x)u2(tk,x))=e-0.1k(0.5-0.20.30.6)(u1(tk-,x)u2(tk-,x)),
when r(t)=2(4.2)(du1(t,x)du2(t,x))=(2.4002.5)(Δu1(t,x)Δu2(t,x))+(1.6002.7)(u1(t,x)u2(t,x))+(2u1(t,x)-u2(t,x))dw(t),t≥0,t≠tk,(u1(tk,x)u2(tk,x))=e-0.1k(0.6-0.30.30.4)(u1(tk-,x)u2(tk-,x)),
where t0=0,tk=tk-1+0.1,(k=1,2,…). By simple calculation, we obtain Γk=0.6e-0.1k,λ=2.6>1,2λmax(Ai)(tk-tk-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.



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