# Stability Analysis for Impulsive Stochastic Reaction-Diffusion Differential System and Its Application to Neural Networks

Authors: Yanke Du; Yanlu Li; Rui Xu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

This paper is concerned with the stability of impulsive stochastic reaction-diffusion differential systems with mixed time delays. First, an equivalent relation between the solution of a stochastic reaction-diffusion differential system with time delays and impulsive effects and that of corresponding system without impulses is established. Then, some stability criteria for the stochastic reaction-diffusion differential system with time delays and impulsive effects are derived. Finally, the stability criteria are applied to impulsive stochastic reaction-diffusion Cohen-Grossberg neural networks with mixed time delays, and sufficient conditions are obtained for the exponentialp-stability of the zero solution to the neural networks. An example is given to illustrate the effectiveness of our theoretical results. The systems we studied in this paper are more general, and some existing results are improved and extended.


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## Body

## 1. Introduction

In recent years, impulsive dynamical systems have attracted considerable attention due to its wide applications in the areas of economics, physics, population dynamics, engineering, biology, and so on. These systems arise because they are subject to abrupt state changes at certain moments of time, and these changes may be related to such phenomena as shocks, harvesting, or other faults. Meanwhile, time delays are frequently encountered in real world, which can cause instability and oscillations in a system. A large number of stability criteria of impulsive delay systems have been reported (see [1–6] and references therein).Stochastic effects are common phenomena due to disturbances or uncertainties in a system. A lot of dynamical systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment switching. Hence, considerable attention has been paid to the study of stochastic systems, and various interesting results have been reported in the literatures; for example, see [2–4, 7, 8] and references therein. In particular, Li et al. [8] considered the following impulsive stochastic differential system with time delay:
(1)dyi(t,x)=Fi(t,y1(t),…,yn(t),y1(t-τ(t)),…,yn(t-τ(t)))dt+Gi(t,y1(t),…,yn(t),y1(t-τ(t)),…,yn(t-τ(t)))dwi(t),t≠tk,yi(tk+)-yi(tk)=Iki(y1(tk),…,yn(tk)),kkkkkkkkt=tk,k∈N,i=1,2,…,n.
They showed the stability results of system (1) by transforming (1) into an equivalent system without impulses.Generally speaking, diffusion effects cannot be avoided in systems modeling many real world phenomena. As a representation example in neural networks, when electrons are moving in an asymmetric electromagnetic field, it inevitably leads to diffusion phenomena. In [5, 9–11], the stabilities of the equilibrium points of some types of neural networks with reaction-diffusion terms have been investigated. On the other hand, distributed delay systems can characterize the cumulative effects of the past values of the dynamic and are often used to model the time lag phenomena in thermodynamics, ecology, epidemiology, and neural networks. For example, neural networks usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths. It is desired to model them by introducing continuously distributed delays over a certain duration of time such that the distant past has less influence compared to the recent behavior of the state.However, in [8], the authors neglected the effects of diffusion and distributed delays. To the best of our knowledge, there are few results about the stability of impulsive stochastic reaction-diffusion deferential systems (ISRDDSs) with mixed time delays. Motivated by [8] and the previous discussions, we are concerned with the stability of the following ISRDDS with time-varying discrete delays and distributed delays:
(2)du(t,x)=∇·(D(t,x,u(t,x))∘∇u(t,x))dt+F(∫t-r(t)t‍t,u(t,x),u(t-τ1(t),x),…,kkkkkkkku(t-τp(t),x),∫t-r(t)t‍u(s,x)ds)dt+G(∫t-r(t)t‍∫t-r(t)t‍t,u(t,x),u(t-τ1(t),x),…,u(t-τp(t),x),∫t-r(t)t‍u(s,x)ds)dw(t),t≠tk,u(tk+,x)-u(tk,x)=Ik(u(tk,x)),kkkkkkkkkkkkkkkkt=tk,k∈N,u(t,x)=ϕ(t,x),(t,x)∈[-γ,0]×Ω,Ω∈ℝm,
in which
(3)u(t,x)=(u1(t,x),u2(t,x),…,un(t,x))T,u(t-τj(t),x)=(u1(t-τj(t),x),kkkku2(t-τj(t),x),…,kkkun(t-τj(t),x))T(j=1,2,…,p),∫t-r(t)t‍u(s,x)ds=(∫t-r1(t)t‍u1(s,x)ds,∫t-r2(t)t‍u2(s,x)ds,…,∫t-rn(t)t‍un(s,x)ds)T,F=(F1,…,Fn)T,G=(G1,…,Gn)T,Ik=(Ik1,…,Ikn)T(k∈N),ϕ=(ϕ1,…,ϕn)T,D(t,x,u(t,x))=(Dil(t,x,u(t,x)))n×m,∇u=(∇u1,…,∇un)T,∇ui=(∂ui∂x1,…,∂ui∂xm),D∘∇u=(Dil(t,x,u(t,x))∂ui∂xl)n×m,Y=(Y1,…,Yn)T,Yi=(yi1,…,yim)T,∇·Yi=∑l=1m‍∂yil∂xl,∇·Y=(∇·Y1,…,∇·Yn)T.ui(t,x) is the state variable, xi is the space variable, Dil(t,x,u(t,x))≥0 is a diffusion operator, τj(t) and ri(t) are time-varying functions, w(t)=(w1(t),w2(t),…,wn(t))T∈ℝn is a Brownian motion defined on a complete probability space (S,ℱ,𝒫), and Iki is the impulsive function; γ=max1≤j≤p,1≤i≤n{τj,ri}, τj=supt≥0τj(t), and ri=supt≥0ri(t).The organization of this paper is as follows. In Section2, some preliminaries are given. In Section 3, by transforming the solutions of the stochastic reaction-diffusion differential system with delay and impulsive effects into that of the corresponding system without impulses, some stability criteria for the stochastic reaction-diffusion differential system with delay and impulsive effects are derived. In Section 4, the stability criteria are applied to impulsive stochastic reaction-diffusion Cohen-Grossberg neural networks (ISRDCGNNs) with mixed time delays, and sufficient conditions are obtained for the exponential p-stability of the zero solution to the neural networks. In Section 5, a numerical example is provided to illustrate the effectiveness of the theoretical results. A concluding remark is given in Section 6 to end this work.


## 2. Preliminaries

For convenience, we introduce several notations. LetPC([-γ,0] × Ω,ℝn):={u(t,x):[-γ,0]×Ω→ℝn|u(t,x) is continuous at t≠tk,u(tk-,x)=u(tk,x), and u(tk+,x) exists for tk}. For u(t,x)=(u1(t,x),…,un(t,x))T∈ℝn, we define
(4)∥u(t,x)∥=∑i=1n‍∥ui(t,x)∥=∑i=1n‍(∫Ω‍|ui(t,x)|pdx)1/p,
and for ϕ(s,x)=(ϕ1(s,x),…,ϕn(s,x))T∈PC([-γ,0]×Ω,ℝn), we define ∥ϕ(s,x)∥ = sup-γ≤s≤0∑i=1n‍∥ϕi(s,x)∥. Denote PCF0b([-γ,0]×Ω,ℝn): = {u(t,x)∈PC([-γ,0] × Ω,ℝn)|u(t,x) is bounded, F0-measurable, and E∥u(t,x)∥<∞}, PCF0b(δ):={u(t,x)∈PCF0b([-γ,0]×Ω,ℝn)|E∥u(t,x)∥≤δ}. Throughout this paper, we always assume that a product equals to unity if the number of factors is zero.Definition 1.
A functionu(t,x):[-γ,0]×Ω→ℝn is said to be the solution of system (2) if the following conditions are satisfied:(i)
u(t,x) is piecewise continuous with the first kind discontinuity at the points tk, k∈N. Moreover, u(t,x) is left continuous at each point,(ii)
u(t,x) satisfies system (2).Definition 2.
The zero solution of system (2) is said to be as follows.(i)
p-stable if, for any ε>0, there exists a δ>0 such that the initial function ϕ∈PCF0b(δ) implies E∥u(t,x)∥p<ε for (t,x)∈(0,+∞)×Ω. Especially, when p=1, it is said to be stable.(ii)
Exponentiallyp-stable if there is a pair of positive constants λ and K such that, for any initial condition ϕ∈PCF0b([-γ,0]×Ω,ℝn), there holds E∥u(t,x)∥p≤K∥ϕ∥pe-λt, t≥0. Here, λ is called the exponential convergence rate. When p=2 especially, it is said to be exponentially stable in mean square.(iii)
Asymptotically stable if it is stable, and there exists aδ>0 such that the initial function ϕ∈PCF0b(δ) implies limt→∞E∥u(t,x)∥=0.
For system (2), one makes the following assumptions: (H1)
0≤t0<t1<t2<⋯<tk<⋯ are fixed impulsive moments such that tk→∞ as k→∞;(H2)
F,G:R+×Rn×⋯×Rn︸p+2→Rn satisfy F(t,0,…,0)≡0 and G(t,0,…,0)≡0;(H3)
Ik:Rn→Rn satisfies Ik(0)=0,k∈N;(H4)
except for the zero solution of (2), for any solution u(t,x) of (2), u(t,x)≠0 and Ik(u(tk,x))≠-u(tk,x), k∈N.
Denoting(5)Jki:=ui(tk,x)ui(tk,x)+Iki(u1(tk,x),…,un(tk,x)),i=1,2,…,n,k∈N,
we consider the following delayed stochastic reaction-diffusion differential system without impulses:
(6)dv(t,x)=∇·(D~(t,x,v(t,x))∘∇v(t,x))dt+∏0≤tk<t‍JkF~(∫t-r(t)t‍t,v(t,x),v(t-τ1(t),x),…,kkkkkkkkkkkkkkv(t-τp(t),x),kkkkkkkkkkkkkkk∫t-r(t)t‍v(s,x)ds)dt+∏0≤tk<t‍JkG~(∫t-r(t)t‍t,v(t,x),v(t-τ1(t),x),…,v(t-τp(t),x),∫t-r(t)t‍v(s,x)ds)dw(t),v(t,x)=ϕ(t,x),(t,x)∈[-γ,0]×Ω,
where
(7)Jk=diag(Jk1,Jk2,…,Jkn),D~(t,x,v(t,x))=D(t,x,∏0≤tk<t‍Jk-1v(t,x)),F~(∫t-r(t)t‍t,v(t,x),v(t-τ1(t),x),…,v(t-τp(t),x),∫t-r(t)t‍v(s,x)ds)=F(t,∏0≤tk<t‍Jk-1v(t,x),∏0≤tk<t-τ1(t)‍Jk-1v(t-τ1(t),x),…,∏0≤tk<t-τp(t)‍Jk-1v(t-τp(t),x),∫t-r(t)t‍∏0≤tk<s‍Jk-1v(s,x)ds),G~(∫t-r(t)t‍t,v(t,x),v(t-τ1(t),x),…,kkkkv(t-τp(t),x),∫t-r(t)t‍v(s,x)ds)=G(t,∏0≤tk<t‍Jk-1v(t,x),kkkkkkkkkk∏0≤tk<t-τ1(t)‍Jk-1v(t-τ1(t),x),…,kkkkkkkkkk∏0≤tk<t-τp(t)‍Jk-1v(t-τp(t),x),kkkkkkkkkk∫t-r(t)t‍∏0≤tk<s‍Jk-1v(s,x)ds).
A function vectorv(t,x) is a solution of (6) on [-γ,+∞)×Ω if it is absolutely continuous on [-γ,+∞)×Ω and satisfies (6) almost everywhere for t≥0 and x∈Ω.


## 3. Stability Criteria

In this section, we first establish an equivalent relation between the solution of system (2) and that of system (6).Lemma 3.
Assume that (H1)–(H4)  hold. Thenu(t,x)  is a solution of system (2) if and only if v(t,x)  is a solution of system (6), where u(t,x)=∏0≤tk<t‍Jk-1v(t,x) or v(t,x)=∏0≤tk<t‍Jku(t,x).Proof.
First, we prove the sufficiency. Lettingv(t,x) be a solution of system (6), we derive that, for any t≠tk,
(8)du(t,x)=d(∏0≤tk<t‍Jk-1v(t,x))=∏0≤tk<t‍Jk-1d(v(t,x))=∏0≤tk<t‍Jk-1∇·(D~(t,x,v(t,x))∘∇v(t,x))dt+F~(∫t-r(t)t‍t,v(t,x),v(t-τ1(t),x),…,v(t-τp(t),x),∫t-r(t)t‍v(s,x)ds)dt+G~(∫t-r(t)t‍t,v(t,x),v(t-τ1(t),x),…,v(t-τp(t),x),kkkkkkk∫t-r(t)t‍v(s,x)ds)dw(t)=∏0≤tk<t‍Jk-1∇·(D(t,x,∏0≤tk<t‍Jk-1v(t,x))∘∇(∏0≤tk<t‍Jku(t,x)))dt+F(t,∏0≤tk<t‍Jk-1v(t,x),∏0≤tk<t-τ1(t)‍Jk-1v(t-τ1(t),x),…,∏0≤tk<t-τp(t)‍Jk-1v(t-τp(t),x),∫t-r(t)t‍∏0≤tk<s‍Jk-1v(s,x)ds)dt+G(t,∏0≤tk<t‍Jk-1v(t,x),∏0≤tk<t-τ1(t)‍Jk-1v(t-τ1(t),x),…,∏0≤tk<t-τp(t)‍Jk-1v(t-τp(t),x),∫t-r(t)t‍∏0≤tk<s‍Jk-1v(s,x)ds)dw(t)=∇·(D(t,x,u(t,x))∘∇u(t,x))dt+F(∫t-r(t)t‍t,u(t,x),u(t-τ1(t),x),…,u(t-τp(t),x),∫t-r(t)t‍u(s,x)ds)dt+G(∫t-r(t)t‍t,u(t,x),u(t-τ1(t),x),…,u(t-τp(t),x),kkkkkkkk∫t-r(t)t‍u(s,x)ds)dw(t).
On the other hand, for anyk∈N, we have
(9)u(tk+,x)=limt→tk+∏0≤tj<t‍Jj-1v(t,x)=∏0≤tj≤tk‍Jj-1v(tk+,x)=Jk-1∏0≤tj<tk‍Jj-1v(tk,x)=Jk-1u(tk,x)=u(tk,x)+Ik(u(tk,x)),(10)u(tk-,x)=limt→tk-∏0≤tj<t‍Jj-1v(t,x)=∏0≤tj<tk‍Jj-1v(tk-,x)=∏0≤tj<tk‍Jj-1v(tk,x)=u(tk,x).
Further, ifv(t,x) is a solution of system (6) with initial condition v(t,x)=ϕ(t,x), (t,x)∈[-γ,0]×Ω, then u(t,x)=∏0≤tk<t‍Jk-1v(t,x)=v(t,x)=ϕ(t,x), (t,x)∈[-γ,0]×Ω. Therefore, u(t,x) is a solution of system (2) with initial condition u(t,x)=ϕ(t,x), (t,x)∈[-γ,0]×Ω, and the sufficiency is proved. Similarly, we can prove the necessity.Remark 4.
Lemma3 gives the equivalent relation between the solution of a stochastic reaction-diffusion differential delay system with impulsive effects and the solution of a corresponding system without impulses. Based on the “equivalent method,” the existence and uniqueness of the solution of a stochastic reaction-diffusion differential delay system with impulsive effects can be derived by a new way; that is, any conditions that ensure the existence and uniqueness of the solution of system (6) without impulses will also ensure the existence and uniqueness of the system (2) with impulses.
In what follows, we will reduce the stabilities of system (2) to those of corresponding system (6).Theorem 5.
Under assumptions (H1)–(H4), if there exists a constantM>0 such that, for any t>0,
(11)|∏0≤tk<t‍Jki-1|≤M,i=1,2,…,n,
and the zero solution of (6) is p-stable (exponentially p-stable, asymptotically stable), then the zero solution of (2) is also p-stable (exponentially p-stable, asymptotically stable).Proof.
Letu(t,x) and v(t,x) be the solutions of systems (2) and (6), respectively. Since the zero solution of (6) is p-stable, we have that, for any ε>0, there exists a scalar δ>0 such that the initial condition ϕ∈PCF0b(δ) implies E∥v(t,x)∥p<ε/Mp for (t,x)∈(0,+∞)×Ω. By Lemma 3, u(t,x)=∏0≤tk<t‍Jk-1v(t,x) is a solution of (2) on (t,x)∈[-γ,+∞)×Ω. Furthermore, it is easy to see that
(12)E∥u(t,x)∥p=E∥(u1(t,x),…,un(t,x))T∥p=E∥(∏0≤tk<t‍Jk1-1v1(t,x),…,∏0≤tk<t‍Jkn-1vn(t,x))T∥p=E[∑i=1n‍(∫Ω‍|∏0≤tk<t‍Jki-1vi(t,x)|pdx)1/p]p≤E[M∑i=1n‍(∫Ω‍|vi(t,x)|pdx)1/p]p=MpE∥v(t,x)∥p<Mp·εMp=ε.
Hence, the zero solution of (2) is p-stable. Using similar arguments, we can verify that if the zero solution of (6) is exponentially p-stable (asymptotically stable), then the zero solution of (2) is also exponentially p-stable (asymptotically stable). This completes the proof.In a similar way, we can derive the following results.Theorem 6.
Under assumptions (H1)–(H4), if there exists a constantL>0 such that, for any t>0,
(13)|∏0≤tk<t‍Jki|≤L,i=1,2,…,n,
and the zero solution of (2) is p-stable (exponentially p-stable, asymptotically stable), then the zero solution of (6) is also p-stable (exponentially p-stable, asymptotically stable).Combining Theorems5 and 6, one can easily obtain the following results.Theorem 7.
Assume that (H1)–(H4) hold and inequalities (11)  and (13)  are satisfied, then the zero solution of (2) is p-stable (exponentially p-stable, asymptotically stable) if and only if the zero solution of (6) is also p-stable (exponentially p-stable, asymptotically stable).Remark 8.
For impulsive neural networks, many researchers supposed that the impulsive operators are linear (e.g., [7, 12–15]); that is,
(14)xi(tk+)-xi(tk)=-γkixi(tk),0<γki≤2.
From the definition of Jki in Section 2, we have |Jki-1|=|1-γki|<1; then |∏0≤tk<t‍Jki-1|≤1. Obviously, the condition (11) in Theorem 5 is less conservative than (14), in which (11) ensures that the stability of the delayed stochastic reaction-diffusion differential system without impulses can be used to judge the stability of the corresponding system with impulses.Remark 9.
In [2, 8], the authors dealt with the stochastic differential systems with delay and nonlinear impulsive effects, the stability results of which were showed by transforming the system into a corresponding system without impulses. However, the distributed delays and diffusion effects were not taken into account in the previous systems. In this paper, we incorporated stochastic perturbations, reaction-diffusion effects, and mixed time delays into impulsive differential system and derived the stability criteria of the system. It is readily seen that our results are more general than those reported in [2, 8].


## 4. Application to Impulsive Stochastic Reaction-Diffusion Neural Networks

In this section, we apply our previous stability results to analyze the stability of the following ISRDCGNNs with time delays:(15)dui(t,x)=∑l=1m‍∂∂xl(Dil(t,x,ui(t,x))∂ui(t,x)∂xl)dt-αi(ui(t,x))[∑j=1n‍βi(ui(t,x))kkkkkkkkkkkkkkk-∑j=1n‍aijfj(uj(t,x))kkkkkkkkkkkkkkk-∑j=1n‍bijgj(uj(t-τij(t),x))kkkkkkkkkkkkkkk-∑j=1n‍cijhj(∫t-r(t)t‍uj(s,x)ds)∑j=1n‍]dt+∑j=1n‍σij(∫t-r(t)t‍t,uj(t,x),uj(t-τij(t),x),∫t-r(t)t‍uj(s,x)ds)dwi(t),kt≠tk,ui(tk+,x)-ui(tk,x)=Iki(u1(tk,x),…,un(tk,x)),kkkkkkkkkkkkkkt=tk,k∈N,i=1,2,…,n,
where u(t,x)=(u1(t,x),u2(t,x),…,un(t,x))T∈ℝn denotes the state vector associated with the neurons, x∈Ω⊂ℝm, and Ω={x=(x1,x2,…,xm)T||xk|<lk,k=1,2,…,m} is a bounded compact set with smooth boundary ∂Ω and mesΩ>0, Dil(t,x,ui(t,x))≥0 denotes the diffusion function, and let Dil=supt≥0,x∈ΩDil(t,x,ui(t,x)); (aij)n×n, (bij)n×n and (cij)n×n are the interconnection weight matrices, αi(ui(t,x)) represents an amplification function, βi(ui(t,x)) is an appropriately behaved function, fj, gj, and hj denote the activation functions, and (σij)n×n is the diffusion coefficient matrix; σi=(σi1,σi2,…,σin), βi(0)=fi(0)=gi(0)=hi(0)=σij(0)=0, and τij(t)≤τ, r(t)≤r.The boundary condition and the initial value of system (15) are
(16)ui(t,x)∣∂Ω=0,(t,x)∈[-δ,+∞)×∂Ω,i=1,2,…,n,ui(s,x)=ϕi(s,x),(s,x)∈[-δ,0]×Ω,i=1,2,…,n,
where δ=max{τ,r}.Equivalently, we consider the following stochastic reaction-diffusion Cohen-Grossberg neura1 networks without impulses:(17)dvi(t,x)=∑l=1m‍∂∂xl(Dil(t,x,∏0≤tk<t‍Jki-1vi(t,x))∂vi(t,x)∂xl)dt-∏0≤tk<t‍Jki·αi(∏0≤tk<t‍Jki-1vi(t,x))×[∑j=1n‍βi(∏0≤tk<t‍Jki-1vi(t,x))kkkkk-∑j=1n‍aijfj(∏0≤tk<t‍Jkj-1vj(t,x))kkkkk-∑j=1n‍bijgj(∏0≤tk<t-τij(t)‍Jkj-1vj(t-τij(t),x))kkkkk-∑j=1n‍cijhj(∫t-r(t)t‍∏0≤tk<s‍Jks-1vj(s,x)ds)]dt+∏0≤tk<t‍Jki·∑j=1n‍σij(t,∏0≤tk<t‍Jkj-1vj(t,x),kkkkkkkkkkkkkkkkk∏0≤tk<t-τij(t)‍Jkj-1vj(t-τij(t),x),kkkkkkkkkkkkkkkkkk∫t-r(t)t‍∏0≤tk<s‍Jkj-1vj(s,x)ds)dwi(t),kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkki=1,2,…,n.Throughout this section, we make the following assumptions:(H5)
αi(u) is a continuous function, and 0<α_i≤αi(u)≤α¯i for all u∈ℝ, i=1,2,…,n;(H6)
there exists a constantβi>0 such that
(18)βi(u)-βi(v)u-v≥βi,
for all u,v∈ℝ(u≠v), i=1,2,…,n;(H7)
there exist positive constantsFi, Gi, and Hi such that
(19)Fi=supu≠v|fi(u)-fi(v)u-v|,Gi=supu≠v|gi(u)-gi(v)u-v|,Hi=supu≠v|hi(u)-hi(v)u-v|,
for all u,v∈ℝ(u≠v), i=1,2,…,n;(H8)
there exist positive constantssij(1), sij(2), and sij(3)(i,j=1,2,…,n) such that
(20)(σi(t,u(1),u(2),u(3))-σi(t,v(1),v(2),v(3)))×(σi(t,u(1),u(2),u(3))-σi(t,v(1),v(2),v(3)))T≤∑j=1n‍(sij(1)|uj(1)-vj(1)|2+sij(2)|uj(2)-vj(2)|2+sij(3)|uj(3)-vj(3)|2),
for all u(k)=(u1(k),u2(k),…,un(k)), v(k)=(v1(k),v2(k),…,vn(k)), k=1,2,3.The following lemmas are useful in proving our main results.Lemma 10 (see [16]).
LetQ be an n×n matrix with nonpositive off-diagonal elements; then Q is an M-matrix if and only if there exists a vector ξ>0 such that Qξ>0.Lemma 11 (see [10]).
Letp≥2 be a positive integer, lk(k=1,2,…,m) positive constants, and cube Ω={x=(x1,x2,…,xm)T∈ℝm||xk|<lk,k=1,2,…,m}.  Let h(x) be a real-valued function belonging to C1(Ω) which vanishes on the boundary ∂Ω of Ω, that is, h(x)|∂Ω=0. Then
(21)∫Ω‍|h(x)|pdx≤p2lk24∫Ω‍|h(x)|p-2|∂h(x)∂xk|2dx,kkkkkkkkkkkkkk=1,2,…,m.Lemma 12 (see [17]).
Leta,b≥0, p≥i>0; then
(22)ap-ibi≤p-ipap+ipbp.Theorem 13.
Under assumptions (H5)–(H8),  if inequalities (11)  and (13)  are satisfied and Q+T is an M-matrix, where
(23)Q=(qij)n×n,qij=-LMα-i(|aij|Fj+|bij|Gj+r|cij|Hj),T=(tij)n×n,tij=-(p-1)L2M2(sij(1)+sij(2)+r2sij(3))(i≠j),tii=4(p-1)p∑l=1m‍Dilli2+pα_iβi-(p-1)LMα-i×∑j=1n‍[(p-2)LM2α-i|aij|Fj+|bij|Gj+r|cij|Hjkkkkkkkkk+(p-2)LM2α-i(sij(1)+sij(2)+r2sij(3))]-(p-1)L2M2(sii(1)+sii(2)+r2sii(3)),
then the zero solution of system (15) is exponentially p-stable.Proof.
SinceQ+T is an M-matrix, by Lemma 10, there exists ξ=(ξ1,ξ2,…,ξn)T such that (Q+T)ξ>0; that is,
(24)[4(p-1)p∑l=1m‍Dilli2+pα_iβi-(p-1)LMα-i×∑j=1n‍((p-2)LM2α-i|aij|Fj+|bij|Gj+r|cij|Hjkkkkkkkkkkk+(p-2)LM2α-i(sij(1)+sij(2)+r2sij(3)))∑l=1m‍]ξi-LMα-i∑j=1n‍[(p-1)LMα-i|aij|Fj+|bij|Gj+r|cij|Hjkkkkkkkkkkkkkk+(p-1)LMα-i(sij(1)+sij(2)+r2sij(3))]ξj>0,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkki=1,2,…,n.
We can choose a sufficiently small constantϵ>0 such that
(25)[4(p-1)p∑l=1m‍Dilli2+pα_iβi-ϵ-(p-1)LMα-i×∑j=1n‍((p-2)LM2α-i|aij|Fj+|bij|Gj+r|cij|Hjkkkkkkkk+(p-2)LM2α-i(sij(1)+sij(2)+r2sij(3)))∑l=1m‍]ξi-LMα-i∑j=1n‍[(|aij|Fj+(p-1)LMα-isij(1))kkkkkkkkkkkk+eϵτ(|bij|Gj+(p-1)LMα-isij(2))kkkkkkkkkkkk+eϵr-1ϵ(|cij|Hj+(p-1)LMrα-isij(3))]ξj>0.
Let wi(t,x)=eϵt∫Ω‍|vi(t,x)|pdx, p≥2, i=1,2,…,n. By the Ito^ differential formula, the stochastic derivative of wi(t,x) along (17) can be derived as follows:
(26)Lwi(t,x)=ϵeϵt∫Ω‍|vi(t,x)|pdxkkkkkkkkkkkk+peϵtsgn(vi(t,x))∫Ω‍|vi(t,x)|p-1×[-∑j=1n‍cijhj(∫t-r(t)t‍∏0≤tk<s‍Jks-1vj(s,x)ds)]∑l=1m‍∂∂xl(Dil(t,x,∏0≤tk<t‍Jki-1vi(t,x))∂vi(t,x)∂xl)kkkkkkkkkkk-∏0≤tk<t‍Jki·αi(∏0≤tk<t‍Jki-1vi(t,x))kkkkkkkkkkk×[βi(∏0≤tk<t‍Jki-1vi(t,x))kkkkkkkkkkk-∑j=1n‍aijfj(∏0≤tk<t‍Jkj-1vj(t,x))kkkkkkkkkkk-∑j=1n‍bijgj(∏0≤tk<t-τij(t)‍Jkj-1vj(t-τij(t),x))kkkkkkkkkkk-∑j=1n‍cijhj(∫t-r(t)t‍∏0≤tk<s‍Jks-1vj(s,x)ds)]]dx+p(p-1)2eϵt(∏0≤tk<t‍Jki)2∫Ω‍|vi(t,x)|p-2σiσiTdx,kkkkkkkkkkkkkkkkkkkkkkkkkkkkkkkki=1,2,…,n.
By Lemma11, we derive from (16) that
(27)psgn(vi(t,x))∫Ω‍|vi(t,x)|p-1×∑l=1m‍∂∂xl(Dil(t,x,∏0≤tk<t‍Jki-1vi(t,x))kkkkkkkkkkkkkkkkk×∂vi(t,x)∂xl∏0≤tk<t‍)dx=p∫Ω‍|vi(t,x)|p-2vi(t,x)∇·(Dil(t,x,∏0≤tk<t‍Jki-1vi(t,x))∂vi(t,x)∂xl)l=1mdx≤-p(p-1)∫Ω‍∑l=1m‍Dil|vi(t,x)|p-2(∂vi(t,x)∂xl)2dx≤-4(p-1)p∑l=1m‍Dilli2∫Ω‍|vi(t,x)|pdx.
Applying assumptions (H5)–(H8) and Lemma 12, we can deduce from (11), (13), and (27) that
(28)Lwi(t,x)≤eϵt∫Ω‍[(ϵ-4(p-1)p∑l=1m‍Dilli2-pα_iβi)×|vi(t,x)|p+pLMα-i|vi(t,x)|p-1×∑j=1n‍(∫t-r(t)t‍|aij|Fj|vj(t,x)|kkkkkkkkkkkkkkkk+|bij|Gj|vj(t-τij(t),x)|kkkkkkkkkkkkkkkk+|cij|Hj∫t-r(t)t‍|vj(s,x)|ds)+p(p-1)2L2M2|vi(t,x)|p-2×∑j=1n‍(∫t-r(t)t‍sij(1)vj2(t,x)kkkkkkkkkkkkkkkk+sij(2)vj2(t-τij(t),x)kkkkkkkkkkkkkkkk+rsij(3)∫t-r(t)t‍vj2(s,x)ds)4(p-1)p∑l=1m‍Dilli2]dx≤eϵt∫Ω‍[(ϵ-4(p-1)p∑l=1m‍Dilli2-pα_iβi)kkkkkkkk×|vi(t,x)|p+(p-1)LMα-i|vi(t,x)|p×∑j=1n‍(|aij|Fj+|bij|Gj+r|cij|Hj)+LMα-i∑j=1n‍(∫t-r(t)t‍|aij|Fj|vj(t,x)|p+|bij|kkkkkkkkkkkkkkkkkkkk×Gj|vj(t-τij(t),x)|pkkkkkkkkkkkkkkkkkkkk+|cij|Hjkkkkkkkkkkkkkkkkkkk×∫t-r(t)t‍|vj(s,x)|pds∫t-r(t)t‍)+(p-1)(p-2)2L2M2|vi(t,x)|p×∑j=1n‍(sij(1)+sij(2)+r2sij(3))+(p-1)L2M2×∑j=1n‍(∫t-r(t)t‍sij(1)|vj(t,x)|pkkkkkkkkkkkkkkkk+sij(2)|vj(t-τij(t),x)|pkkkkkkkkkkkkkkkk+rsij(3)kkkkkkkkkkkkkkk×∫t-r(t)t‍|vj(s,x)|p∫t-r(t)t‍)ds4(p-1)p∑l=1m‍Dilli2]dx≤[ϵ-4(p-1)p∑l=1m‍Dilli2-pα_iβi+(p-1)LMα-i×∑j=1n‍((p-2)LM2α-i|aij|Fj+|bij|Gj+r|cij|Hjkkkkkkkkkkkkk+(p-2)LM2α-ikkkkkkkkkkkk×(sij(1)+sij(2)+r2sij(3))(p-2)LM2α-i)∑l=1m‍]wi(t,x)+LMα-i∑j=1n‍[((p-1)LMα-i|aij|Fj+(p-1)LMα-isij(1))kkkkkkkkkkkkkk×wj(t,x)kkkkkkkkkkkkkk+eϵτ((p-1)LMα-i|bij|Gjkkkkkkkkkkkkkkkkkkkk+(p-1)LMα-isij(2))kkkkkkkkkkkkkk×wj(t-τij(t),x)kkkkkkkkkkkkk+(|cij|Hj+(p-1)LMrα-isij(3))kkkkkkkkkkkkk×∫t-r(t)t‍eϵ(t-s)wj(s,x)ds(p-1)LMrα-i].
Further, we can get
(29)D+(Ewi(t,x))≤[ϵ-4(p-1)p∑l=1m‍Dilli2kkkkk-pα_iβi+(p-1)LMα-ikkkkk×∑j=1n‍((p-2)LM2α-i|aij|Fj+|bij|Gj+r|cij|Hjkkkkkkkkkk+(p-2)LM2α-i(sij(1)+sij(2)+r2sij(3)))]Ewi(t,x)kkkkk+LMα-i∑j=1n‍[(|aij|Fj+(p-1)LMα-isij(1))Ewj(t,x)k+eϵτ((p-1)LMα-i|bij|Gj+(p-1)LMα-isij(2))kkkkkkkkkkkkkkkkkkkkk×Ewj(t-τij(t),x)k+(|cij|Hj+(p-1)LMrα-isij(3))kkkkk×∫t-r(t)t‍eϵ(t-s)Ewj(s,x)ds(p-1)LMα-i].
Denotingk0=∥ϕ∥p/min1≤i≤n{ξi}, we have
(30)Ewi(t,x)=eϵtE∥vi(t,x)∥p≤E∥vi(t,x)∥p≤k0ξi,t∈[-δ,0],i=1,2,…,n.
In what follows, we prove that
(31)Ewi(t,x)≤k0ξi,t≥0,i=1,2,…,n.
In fact, if (31) is not true, then there exist i0∈{1,2,…,n} and t*∈[0,+∞) such that
(32)Ewi0(t*,x)≤k0ξi0,D+Ewi0(t*,x)>0,Ewj(t,x)≤k0ξj,t∈[-δ,t*],j=1,2,…,n.
However, (25), (29), and (32) imply that
(33)D+(Ewi0(t*,x))≤k0{[ϵ-4(p-1)p∑l=1m‍Di0lli02-pα_i0βi0+(p-1)LMα-i0kkkkkkkk×∑j=1n‍((p-2)LM2α-i0|ai0j|Fjkkkkkkkkkkkkkkk+|bi0j|Gj+r|ci0j|Hjkkkkkkkkkkkkkkk+(p-2)LM2α-i0(si0j(1)+si0j(2)+r2si0j(3)))∑l=1m‍Di0lli02]ξi0+LMα-i0kkkkkkkk×∑j=1n‍[(|ai0j|Fj+(p-1)LMα-i0si0j(1))kkkkkkkkkkkkkkkk+eϵτ(|bi0j|Gj+(p-1)LMα-i0si0j(2))kkkkkkkkkkkkkkkk+eϵr-1ϵ((p-1)LMrα-i0|ci0j|Hjkkkkkkkkkkkkkkkkkkkkkkkkkk+(p-1)LMrα-i0si0j(3))]ξj∑l=1m‍}<0,
which is a contradiction. Hence, (31) holds, which leads to
(34)E∥vi(t,x)∥p≤k0ξie-ϵt,t≥0,i=1,2,…,n.
Therefore,
(35)E∥v(t,x)∥p≤K∥ϕ∥pe-ϵt,t≥0,
where K=(max1≤i≤n{ξi}/min1≤i≤n{ξi})np. This means that the zero solution of system (17) is exponentially p-stable and the exponential convergence rate equals ϵ. By Theorem 5, we can obtain that the zero solution of system (15) is exponentially p-stable. The proof is complete.Remark 14.
The stability of impulsive Cohen-Grossberg neural networks without spacial diffusion or distributed delays or stochastic disturbance, which are special cases of system (15), have been studied in [5, 13, 18–20]. It should be noted that the main result in [13] is a special case of Theorem 13. Further, the stability criteria derived in [5, 18–20] are dependent on the intervals of adjoining impulsive moments, while our results are independent of that. Thus, our results are new, and they effectually complement or improve the previously published results.Remark 15.
In [19, 21–26], the reaction-diffusion neural networks have been investigated. Nevertheless, the diffusion terms were eliminated by inequality analysis techniques, and the derived conditions for the stability of neural networks are the same as those obtained in the cases when there are no reaction-diffusion terms in the systems. Thus, our results including reaction-diffusion terms are less conservative than those in [19, 21–26].Remark 16.
As far as we know, almost all the existing results concerning the stability of neural networks are based on 2-norm (e.g., [5, 11, 19, 21, 22, 26–28]). In this paper, we derived the stability criteria of ISRDCGNNs with mixed time delays in terms of p-norm. Hence, our results generalize and improve the existing results reported in the previous literature.


## 5. Numerical Example

In this section, we give an example to illustrate the main theoretical results in Sections3 and 4.In system (15), let
(36)n=m=2,α1(x)=2.5+0.5cosx,α2(x)=3-sinx,β1(x)=8x,β2(x)=10x,fj(x)=gj(x)=hj(x)=12(|x+1|+|x-1|),lj=1(j=1,2),r(t)=1+sint,(Dil(x))=(6688),(aij)=(0.1-0.2-0.3-0.5),(bij)=(0.80.80.40.2),(cij)=(0.3-0.20.1-0.2),(τij(t))=(0.02sin2(t)0.01|cost|0.03cos2(t)0.02|sin2t|),(σij(x,y,z))=(0.1x-0.2y-0.1z0.2x+0.3y+0.1z0.5x+0.4y-0.3z0.3x+0.1y+0.2z),Ik1(u1(tk,x),u2(tk,x))=12k+2u1(tk,x)+1(2k)2u1(tk,x)sin2(u2(tk,x)),Ik2(u1(tk,x),u2(tk,x))=1223ku2(tk,x)+1(2k)2u2(tk,x)cos2(u1(tk,x)).By direct calculation, we obtain that(37)α_1=2,α¯1=3,α_2=2,α¯2=4,β1=8,β2=10,Fi=Gi=Hi=1,τ=0.03,r=2,(sij(1))=(0.040.120.60.18),(sij(2))=(0.080.180.480.06),(sij(3))=(0.040.060.360.12),Jk1-1=1+12k+2+1(2k)2sin2(u2(tk,x)),Jk2-1=1+1223k+1(2k)2cos2(u1(tk,x)),(38)L=1,M=2.
Hence, assumptions (H5)–(H8) and inequalities (11) and (13) are satisfied. Taking p=4, it is not difficult to compute that
(39)Q=(-9.0-8.4-7.2-8.8),T=(34.60-6.48-30.2432.48),Q+T=(25.60-14.88-37.4423.68),
in which Q+T is an M-matrix, and all the conditions of Theorem 13 are satisfied. From Theorem 13, we know that the zero solution of system (15) with the parameters and functions above is exponentially 4-stable (see Figure 1).Transient behaviors of the state variablesu1(t,x) and u2(t,x) in the example.
(a)
(b)


## 6. Concluding Remark

In this paper, we incorporated stochastic perturbations, reaction-diffusion effects, and mixed time delays into impulsive differential systems. First, an equivalent relation between the solution of a stochastic reaction-diffusion differential system with time delays and impulsive effects and that of corresponding system without impulses was established. Second, some stability criteria for the stochastic reaction-diffusion differential system with time delays and impulsive effects were derived by transforming the solutions of the system to those of corresponding one without impulses. Third, the stability criteria were applied to ISRDCGNNs with mixed time delays, and sufficient conditions were obtained for the exponentialp-stability of the zero solution to the neural networks. Lastly, a numerical example was provided to illustrate the effectiveness of our theoretical results. Our stability results provide a new, convenient, and efficient approach to study the stability of stochastic reaction-diffusion differential systems with time delays and impulsive effects, and some previously published results are generalized and improved.



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