# Asymptotic Stability of Impulsive Reaction-Diffusion Cellular Neural Networks with Time-Varying Delays

Authors: Yutian Zhang
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

This work addresses the asymptotic stability for a class of impulsive cellular neural networks with time-varying delays and reaction-diffusion. By using the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality as well as piecewise continuous Lyapunov functions, we summarize some new and concise sufficient conditions ensuring the global exponential asymptotic stability of the equilibrium point. The provided stability criteria are applicable to Dirichlet boundary condition and showed to be dependent on all of the reaction-diffusion coefficients, the dimension of the space, the delay, and the boundary of the spatial variables. Two examples are finally illustrated to demonstrate the effectiveness of our obtained results.


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## Body

## 1. Introduction

Cellular neural networks (CNNs), proposed by Chua and Yang in 1988 [1, 2], have been the focus of a number of investigations due to their potential applications in various fields such as optimization, linear and nonlinear programming, associative memory, pattern recognition, and computer vision [3–7]. Moreover, on the ground that time delays are unavoidably encountered for the finite switching speed of neurons and amplifiers in implementation of neural networks, it was followed by the introduction of the delayed cellular neural networks (DCNNs) so as to solve some dynamic image processing and pattern recognition problems. Such applications concerning CNNs and DCNNs depend heavily on the dynamical behaviors such as stability, convergence, and oscillatory [8, 9]. Particularly, stability analysis has been a major concern in the designs and applications of the CNNs and DCNNs. The stability of CNNs and DCNNs is a subject of current interest, and considerable theoretical efforts have been put into this topic with many good results reported (see, e.g., [10–13]).With reference to neural networks, however, it is noteworthy that the state of electronic networks is actually subject to instantaneous perturbations more often than not. On this account, the networks experience abrupt change at certain instants which may be caused by a switching phenomenon, frequency change, or other sudden noise; that is, the networks often exhibit impulsive effects [14, 15]. For instance, according to Arbib [16] and Haykin [17], when a stimulus from the body or the external environment is received by receptors, the electrical impulses will be conveyed to the neural net and impulsive effects arise naturally in the net. As a consequence, in the past few years, scientists have become increasingly interested in the influence that impulses may have on the CNNs and DCNNs and a large number of stability criteria have been derived (see, e.g. [18–22]).In reality, besides impulsive effects, diffusion effects are also nonignorable since diffusion is unavoidable when electrons are moving in asymmetric electromagnetic fields. As such, the model of neural networks with both impulses and reaction-diffusion should be more accurate to describe the evolutionary process of the systems in question, and it is necessary to consider the effects of both diffusion and impulses on the stability of CNNs and DCNNs.In the past years, there have been a few theoretical contributions to the stability of CNNs and DCNNs with impulses and diffusion. For instance, Qiu [23] formulated a mathematical model of impulsive neural networks with time-varying delays and reaction-diffusion terms described by impulsive partial differential equations and studied, via delay impulsive differential inequality, the problem of global exponential stability with some stability criteria presented. Remarkably, all of the obtained stability criteria in [23] are independent of the diffusion. In 2008, Li and song [24] investigated a class of impulsive Cohen-Grossberg networks with time-varying delays and reaction-diffusion terms. By establishing a delay inequality with impulsive initial conditions and M-matrix theory, some sufficient conditions ensuring global exponential stability of the equilibrium points are given. Analogous to [23], the proposed stability criteria in [24] are also independent of the diffusion. More recently Pan et al. [25] investigated a class of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion in 2010. By the aid of the delay impulsive differential inequality quoted in [23], several sufficient conditions are exploited ensuring global exponential stability of the equilibrium points. Especially, different from [23, 24], the estimate of the exponential convergence rate depends on reaction-diffusion in [25].In this paper, unlike the methods of impulsive differential inequalities and Poincaré inequality used in [25], we attempt to adopt the new techniques of the impulsive integral inequality of Gronwall-Bellman type and Hardy-Sobolev inequality to investigate the problem of global exponential asymptotic stability for impulsive cellular neural networks with time-varying delays and reaction-diffusion terms. Different from the existing research, we find, besides the reaction-diffusion coefficients, the dimension of the space and the boundary of the spatial variables do influence the stability.The rest of the paper is organized as follows. In Section2, the model of impulsive delayed cellular neural networks with reaction-diffusion terms and Dirichlet boundary condition is outlined, and some facts and lemmas are introduced for later reference. By the new agency of the impulsive integral inequality of Gronwall-Bellman type as well as Hardy-Sobolev inequality, we discuss the global exponential asymptotic stability and develop some new criteria in Section 3. To conclude, two illustrative examples are given to verify the effectiveness of our results in Section 4.


## 2. Preliminaries

LetRn denote the n-dimensional Euclidean space, and Ω⊂Rm is a bounded open set containing the origin. The boundary of Ω is smooth and mesΩ>0. Let R+=[0,∞) and t0∈R+.We consider the following impulsive neural networks with time delays and reaction-diffusion terms:(2.1)∂ui(t,x)∂t=∑s=1m∂∂xs(Dis∂ui(t,x)∂xs)-aiui(t,x)+∑j=1nbijfj(uj(t,x))+∑j=1ncijfj(uj(t-τj(t),x))t≥t0,t≠tk,x∈Ω,i=1,2,…,n,k=1,2,…,(2.2)ui(tk+0,x)=ui(tk,x)+Pik(ui(tk,x))x∈Ω,k=1,2,…,i=1,2,…,n,
where n corresponds to the numbers of units in a neural network; x=(x1,…,xm)T∈Ω, ui(t,x), denotes the state of the ith neuron at time t and in space x; smooth functions Dis=Dis(t,x,u)≥0 represent transmission diffusion operators of the ith unit; activation functions fj(uj(t,x)) stand for the output of the jth unit at time t and in space x; bij, cij, ai are constants: bij indicates the strength of the jth unit on the ith unit at time t and in space x, cij denotes the strength of the jth unit on the ith unit at time t-τj(t) and in space x, where τj(t) corresponds to the transmission delay along the axon of the jth unit and satisfies 0≤τj(t)≤τ (τ=const) as well as τ∙j(t)<1-1/h (h>0), and ai>0 represents the rate with which the ith unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs at time t and in space x. The fixed moments tk (k=1,2,…) are called impulsive moments satisfying 0≤t0<t1<t2<⋯ and limk→∞tk=∞; ui(tk+0,x) and ui(tk-0,x) represent the right-hand and left-hand limit of ui(t,x) at time tk and in space x, respectively. Pik(ui(tk,x)) stands for the abrupt change of ui(t,x) at impulsive moment tk and in space x.Denote byu(t,x)=u(t,x;t0,φ), u∈Rn the solution of system (2.1)-(2.2), satisfying the initial condition
(2.3)u(s,x;t0,φ)=φ(s,x),t0-τ≤s≤t0,x∈Ω
and Dirichlet boundary condition
(2.4)u(t,x;t0,φ)=0,t≥t0,x∈∂Ω,
where the vector-valued function φ(s,x)=(φ1(s,x),…,φn(s,x))T is such that ∫Ω∑i=1nφi2(s,x)dx is bounded on [t0-τ,t0] and φi(s,x) (i=1,2,…,n) is first-order continuous differentiable as to s on [t0-τ,t0].The solutionu(t,x)=u(t,x;t0,φ)=(u1(t,x;t0,φ),…,un(t,x;t0,φ))T of problems ((2.5)–(2.8)) is, for the time variable t, a piecewise continuous function with the first kind discontinuity at the points tk (k=1,2,…), where it is continuous from the left, that is the following relations are true:(2.5)ui(tk-0,x)=ui(tk,x),ui(tk+0,x)=ui(tk,x)+Pik(ui(tk,x)).Throughout this paper, the norm ofu(t,x;t0,φ) is governed by (2.6)‖u(t,x;t0,φ)‖Ω=(∑i=1n∫Ωui2(t,x;t0,φ)dx)1/2.Before moving on, we introduce two hypotheses as follows.(H1)
Activation functionfj(uj(t,x)) satisfies fi(0)=0, and there exists constant li>0 such that |fi(y1)-fi(y2)|≤li|y1-y2| holds for all y1,y2∈R and i=1,2,…,n.(H2)
The functionsPik(ui(tk,x)) are continuous on R and Pik(0)=0, i=1,2,…,n, k=1,2,….According to (H1) and (H2), it is easy to see that problems ((2.5)–(2.8)) admits an equilibrium point u=0.Definition 2.1.
The equilibrium pointu=0 of problems ((2.5)–(2.8)) is said to be globally exponentially stable if there exist constants κ>0 and M≥1 such that
(2.7)‖u(t,x;t0,φ)‖Ω≤M‖φ‖¯Ωe-κ(t-t0),t≥t0,
where ∥φ∥¯Ω2=supt0-τ≤s≤t0∑i=1n∫Ωφi2(s,x)dx.Lemma 2.2 (Gronwall-Bellman-type impulsive integral inequality [26]).
Assume that
(A1) the sequence{tk} satisfies 0≤t0<t1<t2<⋯, with limk→∞tk=∞,
(A2)q∈PC1[R+,R] and q(t) is left-continuous at tk, k=1,2,…,
(A3)p∈C[R+,R+] and for k=1,2,…(2.8)q(t)≤c+∫t0tp(s)q(s)ds+∑t0<tk<tηkq(tk),t≥t0,
where ηk≥0 and c=const. Then,
(2.9)q(t)≤c∏t0<tk<t(1+ηk)exp⁡(∫t0tp(s)ds),t≥t0.Lemma 2.3 (Hardy-Sobolev inequality [27]).
LetΩ⊂Rm(m≥3) be a bounded open set containing the origin and u∈H1(Ω)={ω∣ω∈L2(Ω),Diω=∂ω/∂xi∈L2(Ω),1≤i≤m}. Then there exists a positive constant Cm=Cm(Ω) such that
(2.10)(m-2)24∫Ωu2|x|2dx≤∫Ω|∇u|2dx+Cm∫∂Ωu2dσ.Lemma 2.4.
Ifa>0 and b>0, then ab≤(1/ε)a2+εb2 holds for any ε>0.


## 3. Main Results

Theorem 3.1.
Provided that(1)
forx=(x1,…,xm)T∈Ω(m≥3), there exists a constant β such that |x|2=∑s=1mxs2<β. In addition, there exists a constant D̲>0 such that Dis=Dis(t,x,u)≥D̲>0. Denote D̲(m-2)2/2β=χ,(2)
Pik(ui(tk,x))=-θikui(tk,x), 0≤θik≤2,(3)
there exists a constantγ satisfying γ+λ+hρeγτ>0 as well as λ+hρeγτ<0, where λ=maxi=1,…,n(-χ-2ai+∑j=1n(bij2+cij2))+ρ, ρ=nmaxi=1,…,n(li2),then, the equilibrium point u=0 of problems ((2.5)–(2.8)) is globally exponentially stable with convergence rate – (λ+hρeγτ)/2.Proof.
Multiplying both sides of (2.1) by ui(t,x) and integrating with respect to spatial variable x on Ω, we get
(3.1)d(∫Ωui2(t,x)dx)dt=2∑s=1m∫Ωui(t,x)∂∂xs(Dis∂ui(t,x)∂xs)dx-2ai∫Ωui2(t,x)dx+2∑j=1nbij∫Ωui(t,x)fj(uj(t,x))dx+2∑j=1ncij∫Ωui(t,x)fj(uj(t-τj(t),x))dxt≥t0,t≠tk,k=1,2,….
Regarding the right-hand part of (3.1), the first term becomes by using Green formula, Dirichlet boundary condition, Lemma 2.3, and condition 1 of Theorem 3.1(3.2)2∑s=1m∫Ωui(t,x)∂∂xs(Dis∂ui(t,x)∂xs)dx=-2∑s=1m∫ΩDis(∂ui(t,x)∂xs)2dx≤-D̲(m-2)22∫Ωui2(t,x)|x|2dx≤-D̲(m-2)22β∫Ωui2(t,x)dx≜-χ∫Ωui2(t,x)dx.
Moreover, we derive from (H1) that(3.3)2∑j=1nbij∫Ωui(t,x)fj(uj(t,x))dx≤2∑j=1n|bij|∫Ω|ui(t,x)||fj(uj(t,x))|dx≤2∑j=1n∫Ωlj|bij||ui(t,x)||uj(t,x)|dx≤∑j=1n∫Ω(bij2ui2(t,x)+lj2ui2(t,x))dx,2∑j=1ncij∫Ωui(t,x)fj(uj(t-τj(t),x))dx≤2∑j=1n|cij|∫Ω|ui(t,x)||fj(uj(t-τj(t),x))|dx≤2∑j=1n∫Ωlj|cij||ui(t,x)||uj(t-τj(t),x)|dx≤∑j=1n∫Ω(cij2ui2(t,x)+lj2ui2(t-τj(t),x))dx.
Consequently, substituting ((2.10)–(3.14)) into (3.1) produces
(3.4)d(∫Ωui2(t,x)dx)dt≤-χ∫Ωui2(t,x)dx-2ai∫Ωui2(t,x)dx+∑j=1n∫Ω(bij2ui2(t,x)+lj2ui2(t,x))dx+∑j=1n∫Ω(cij2ui2(t,x)+lj2ui2(t-τj(t),x))dxfort≥t0,t≠tk,k=1,2,….
We define a Lyapunov functionVi(t) as Vi(t)=∫Ωui2(t,x)dx. It is easy to find that Vi(t) is a piecewise continuous function with points of discontinuity of the first kind tk (k=1,2,…), where it is continuous from the left, that is, Vi(tk-0)=Vi(tk) (k=1,2,…). In addition, due to Vi(t0+0)≤Vi(t0) and the following estimate derived from condition 2 of Theorem 3.1(3.5)ui2(tk+0,x)=(-θikui(tk,x)+ui(tk,x))2=(1-θik)2ui2(tk,x)≤ui2(tk,x)(k=1,2,…),
we have(3.6)Vi(tk+0)≤Vi(tk),k=0,1,2,….
holds for t=tk (k=0,1,2,…). Put t∈(tk,tk+1), k=0,1,2,…. Then for the derivative dVi(t)/dt of Vi with respect to problems ((2.5)–(2.8)), it results from (3.4) that
(3.7)dVi(t)dt≤-χ∫Ωui2(t,x)dx-2ai∫Ωui2(t,x)dx+∑j=1n∫Ω(bij2ui2(t,x)+lj2ui2(t,x))dx+∑j=1n∫Ω(cij2ui2(t,x)+lj2uj2(t-τj(t),x))dx≤(-χ-2ai+∑j=1nbij2+∑j=1ncij2)Vi(t)+max⁡i=1,…,n(li2)∑j=1nVj(t)+max⁡i=1,…,n(li2)∑j=1nVj(t-τj(t))t∈(tk,tk+1),k=0,1,2,….
Choose V(t) of the form V(t)=∑i=1nVi(t). From (3.7), one then reads
(3.8)dV(t)dt≤(max⁡i=1,…,n(-χ-2ai+∑j=1n(bij2+cij2))+nmax⁡i=1,…,n(li2))V(t)+nmax⁡i=1,…,n(li2)∑j=1nVj(t-τj)=λV(t)+ρ∑j=1nVj(t-τj(t))t∈(tk,tk+1),k=0,1,2,….
ConstructV*(t)=eγ(t-t0)V(t), where γ satisfies γ+λ+hρeγτ>0 and λ+hρeγτ<0. Evidently, V*(t) is also a piecewise continuous function with points of discontinuity of the first kind tk (k=1,2,…), in which it is continuous from the left, that is V*(tk-0)=V*(tk) (k=1,2,…). Moreover, at t=tk (k=0,1,2,…), we find by use of (3.6)
(3.9)V*(tk+0)≤V*(tk),k=0,1,2,….
Sett∈(tk,tk+1), k=0,1,2,…. By virtue of (3.8), one has
(3.10)dV*(t)dt=γeγ(t-t0)V(t)+eγ(t-t0)dV(t)dt≤γeγ(t-t0)V(t)+(λV(t)+ρ∑j=1nVj(t-τj(t)))eγ(t-t0)=(γ+λ)V*(t)+ρeγ(t-t0)∑j=1nVj(t-τj(t))t∈(tk,tk+1),k=0,1,2,….
Choose small enoughε>0. Integrating (3.10) from tk+ε to t gives
(3.11)V*(t)≤V*(tk+ε)+(γ+λ)∫tk+εtV*(s)ds+∫tk+εtρeγ(s-t0)∑j=1nVj(s-τj(s))ds,t∈(tk,tk+1),k=0,1,2,…
which yields after lettingε→0 in (3.11)
(3.12)V*(t)≤V*(tk+0)+(γ+λ)∫tktV*(s)ds+∫tktρeγ(s-t0)∑j=1nVj(s-τj(s))ds,t∈(tk,tk+1),k=0,1,2,….
We now proceed to estimate the value ofV*(t) at t=tk+1, k=0,1,2,…. For small enough ε>0, we put t=tk+1-ε. Now an application of (3.12) leads to, for k=0,1,2,…,(3.13)V*(tk+1-ε)≤V*(tk+0)+(γ+λ)∫tktk+1-εV*(s)ds+∫tktk+1-ερeγ(s-t0)∑j=1nVj(s-τj(s))ds.
If we letε→0 in (3.13), there results
(3.14)V*(tk+1-0)≤V*(tk+0)+(γ+λ)∫tktk+1V*(s)ds+∫tktk+1ρeγ(s-t0)∑j=1nVj(s-τj(s))ds,k=0,1,2,….
Note thatV*(tk+1-0)=V*(tk+1) is applicable for k=0,1,2,…. Thus,
(3.15)V*(tk+1)≤V*(tk+0)+(γ+λ)∫tktk+1V*(s)ds+∫tktk+1ρeγ(s-t0)∑j=1nVj(s-τj(s))ds
holds fork=0,1,2,…. By synthesizing (3.12) and (3.15), we then arrive at
(3.16)V*(t)≤V*(tk+0)+(γ+λ)∫tktV*(s)ds+∫tktρeγ(s-t0)∑j=1nVj(s-τj(s))dst∈(tk,tk+1],k=0,1,2,….
This, together with (3.9), results in
(3.17)V*(t)≤V*(tk)+(γ+λ)∫tktV*(s)ds+∫tktρeγ(s-t0)∑j=1nVj(s-τj(s))dsfort∈(tk,tk+1],k=0,1,2,….
Recalling assumptions that0≤τj(t)≤τ and τ•j(t)<1-1/h (h>0), we have
(3.18)∫tktρeγ(s-t0)∑j=1nVj(s-τj((s)))ds=∑j=1n∫tk-τj(tk)t-τj(t)ρeγ(θ+τj(s)-t0)Vj(θ)11-τ•j(s)dθ≤hρeγτ∑j=1n∫tk-τj(tk)t-τj(t)eγ(θ-t0)Vj(θ)dθ.
Hence,
(3.19)V*(t)≤V*(tk)+(γ+λ)∫tktV*(s)ds+hρeγτ∑j=1n∫tk-τj(tk)t-τj(t)eγ(s-t0)Vj(s)dst∈(tk,tk+1],k=0,1,2,….
By induction argument, we reach(3.20)V*(tk)≤V*(tk-1)+(γ+λ)∫tk-1tkV*(s)ds+hρeγτ∑j=1n∫tk-1-τj(tk-1)tk-τj(tk)eγ(s-t0)Vj(s)ds,⋮V*(t2)≤V*(t1)+(γ+λ)∫t1t2V*(s)ds+hρeγτ∑j=1n∫t1-τj(t1)t2-τj(t2)eγ(s-t0)Vj(s)ds,V*(t1)≤V*(t0)+(γ+λ)∫t0t1V*(s)ds+hρeγτ∑j=1n∫t0-τj(t0)t1-τj(t1)eγ(s-t0)Vj(s)ds.
Therefore,(3.21)V*(t)≤V*(t0)+(γ+λ)∫t0tV*(s)ds+hρeγτ∑j=1n∫t0-τj(t0)t-τj(t)eγ(s-t0)Vj(s)ds≤V*(t0)+(γ+λ)∫t0tV*(s)ds+hρeγτ∑j=1n∫t0-τj(t0)teγ(s-t0)Vj(s)ds=V*(t0)+(γ+λ+hρeγτ)∫t0tV*(s)ds+hρeγτ∑j=1n∫t0-τj(t0)t0eγ(s-t0)Vj(s)dst∈(tk,tk+1],k=0,1,2,….
Since
(3.22)hρeγτ∑j=1n∫t0-τj(t0)t0eγ(s-t0)Vj(s)ds≤hρeγτ∑j=1n∫t0-τt0Vj(s)ds=hρeγτ∫t0-τt0(∑j=1n∫Ωφj2(s,x)dx)ds≤τhρeγτ‖φ‖¯Ω2,
we claim
(3.23)V*(t)≤V*(t0)+τhρeγτ‖φ‖¯Ω2+(γ+λ+hρeγτ)∫t0tV*(s)dst∈(tk,tk+1],k=0,1,2….
According to Lemma2.2, we claim
(3.24)V*(t)≤(V*(t0)+τhρeγτ‖φ‖¯Ω2)exp⁡{(γ+λ+hρeγτ)(t-t0)},t≥t0
which reduces to(3.25)‖u(t,x;t0,φ)‖Ω≤1+τhρeγτ‖φ‖¯Ωexp⁡{(λ+hρeγτ2)(t-t0)},t≥t0.
This completes the proof.Remark 3.2.
According to the conditions of Theorem3.1, we see that the reaction-diffusion term do influence the stability of problem ((2.5)–(2.8)). Moreover, besides the reaction-diffusion coefficients, the dimension of the space and the boundary of spatial variables have also an effect on the stability of the equilibrium point u=0.Theorem 3.3 ..
Providing that(1)
forx=(x1,…,xm)T∈Ω(m≥3), there exist constants β such that |x|2=∑s=1mxs2<β, in addition, there exists constant D̲>0 such that Dis=Dis(t,x,u)≥D̲>0; denote D̲(m-2)2/2β=χ,(2)
Pik(ui(tk,x))=-θikui(tk,x), 1-1+α≤θik≤1+1+α, α≥0,(3)
infk=1,2…(tk-tk-1)>μ,(4)
there exists constantγ which satisfies γ+λ+hρeγτ>0 and λ+hρeγτ+ln(1+α)/μ<0, where λ=maxi=1,…,n(-χ-2ai+∑j=1n(bij2+cij2))+ρ and ρ=nmaxi=1,…,n(li2), then, the equilibrium point u=0 of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate –(1/2)(λ+hρeγτ+ln(1+α)/μ).Proof.
Define a Lyapunov functionV of the form V(t)=∑i=1nVi(t), where Vi(t)=∫Ωui2(t,x)dx. Obviously, V(t) is a piecewise continuous function with points of discontinuity of the first kind tk, k=1,2,…, where it is continuous from the left, that is, V1(tk-0)=V1(tk) (k=1,2,…). Furthermore, for t=tk (k=0,1,2,…), it follows from condition 2 of Theorem 3.3 that
(3.26)ui2(tk+0,x)-ui2(tk,x)=(1-θik)2ui2(tk,x)-ui2(tk,x)≤αui2(tk,x).
Thereby,
(3.27)V(tk+0)≤αV(tk)+V(tk),k=0,1,2,….
Construct another Lyapunov function defined byV*(t)=eγ(t-t0)V(t), where γ satisfies γ+λ+hρeγτ>0 and λ+hρeγτ+(ln(1+α)/μ)<0. Then, V*(t) is also a piecewise continuous function with points of discontinuity of the first kind tk, k=1,2,…, where it is continuous from the left, that is V*(tk-0)=V*(tk) (k=1,2,…). And for t=tk (k=0,1,2,…), it results from (3.27) that
(3.28)V*(tk+0)≤αV*(tk)+V*(tk),k=0,1,2,….
Sett∈(tk,tk+1], k=0,1,2,…. Following the same procedure as in Theorem 3.1, we get
(3.29)V*(t)≤V*(tk+0)+(γ+λ)∫tktV*(s)ds+hρeγτ×∑j=1n∫tk-τj(tk)t-τj(t)eγ(θ-t0)Vj(θ)dθt∈(tk,tk+1],k=0,1,2,….
The relations (3.28) and (3.29) yield
(3.30)V*(t)-V*(tk)≤αV*(tk)+(γ+λ)∫tktV*(s)ds+hρeγτ×∑j=1n∫tk-τj(tk)t-τj(t)eγ(θ-t0)Vj(θ)dθt∈(tk,tk+1],k=0,1,2,….
By induction argument, we arrive at(3.31)V*(tk)-V*(tk-1)≤αV*(tk-1)+(γ+λ)∫tk-1tkV*(s)ds+hρeγτ∑j=1n∫tk-1-τj(tk-1)tk-τj(tk)eγ(θ-t0)Vj(θ)dθ,⋮V*(t2)-V*(t1)≤αV*(t1)+(γ+λ)∫t1t2V*(s)ds+hρeγτ∑j=1n∫t1-τj(t1)t2-τj(t2)eγ(θ-t0)Vj(θ)dθ,V*(t1)-V*(t0)≤αV*(t0)+(γ+λ)∫t0t1V*(s)ds+hρeγτ∑j=1n∫t0-τj(t0)t1-τj(t1)eγ(θ-t0)Vj(θ)dθ.
Hence,
(3.32)V*(t)-V*(t0)≤αV*(t0)+(γ+λ)∫t0tV*(s)ds+hρeγτ∑j=1n∫t0-τj(t0)t-τj(t)eγ(θ-t0)Vj(θ)dθ+α∑t0<tk<tV(tk)≤αV*(t0)+(γ+λ+hρeγτ)∫t0tV*(s)ds+hρeγτ∑j=1n∫t0-τj(t0)t0eγ(θ-t0)Vj(θ)dθ+α∑t0<tk<tV(tk)t∈(tk,tk+1],k=0,1,2,….
Introducinghρeγτ∑j=1n∫t0-τj(t0)t0eγ(θ-t0)Vj(θ)dθ≤τhρeγτ∥φ∥¯Ω2 as shown in the proof of Theorem 3.1 into (3.32), the expression becomes
(3.33)V*(t)-V*(t0)≤αV*(t0)+τhρeγτ‖φ‖¯Ω2+(γ+λ+hρeγτ)∫t0tV*(s)ds+α∑t0<tk<tV(tk)t∈(tk,tk+1],k=0,1,2….
It then results from Lemma2.2 that
(3.34)V*(t)≤((α+1)V*(t0)+τhρeγτ‖φ‖¯Ω2)∏t0<tk<t(1+α)exp⁡((γ+λ+hρeγτ)(t-t0))=((α+1)V*(t0)+τhρeγτ‖φ‖¯Ω2)(1+α)kexp⁡((γ+λ+hρeγτ)(t-t0)),t≥t0.
On the other hand, sinceinfk=1,2,…(tk-tk-1)>μ, one has k<(tk-t0)/μ. Thereby,
(3.35)(1+α)k<exp{ln⁡(1+α)μ(tk-t0)}<exp{ln⁡(1+α)μ(t-t0)}.
And (3.34) can be rewritten as
(3.36)V*(t)≤((α+1)V*(t0)+τhρeγτ‖φ‖¯Ω2)exp⁡((γ+λ+hρeγτ+ln⁡(1+α)μ)(t-t0))
which implies
(3.37)‖u(t,x;t0,φ)‖Ω≤(α+1+τhρeγτ)‖φ‖¯Ωexp⁡(12(λ+hρeγτ+ln⁡(1+α)μ)(t-t0)),t≥t0.
The proof is completed.Remark 3.4.
Theorem3.1 is in fact the special case of Theorem 3.3 by choosing α=0. Due to Lemma 2.4, we know
(3.38)2∑j=1nbij∫Ωui(t,x)f(uj(t,x))dx≤∑j=1n∫Ω(ε1bij2ui2(t,x)+lj2ε1ui2(t,x))dx,2∑j=1ncij∫Ωui(t,x)f(uj(t-τj,x))dx≤∑j=1n∫Ω(ε2cij2ui2(t,x)+lj2ε2ui2(t-τj,x))dx
hold for any ε1,ε2>0.
In the sequel, we follow the same procedures as in Theorems3.1 and 3.3 to find the following theorems.Theorem 3.5.
Provided that(1)
forx=(x1,…,xm)T∈Ω(m≥3), there exists a constant β such that |x|2=∑s=1mxs2<β. in addition, there exists a constant D̲>0 such that Dis=Dis(t,x,u)≥D̲>0; denote D̲(m-2)2/2β=χ,(2)
Pik(ui(tk,x))=-θikui(tk,x), 0≤θik≤2,(3)
there exist constantsγ and ε1,ε2>0 such that γ+λ+hρeγτ>0 and λ+hρeγτ<0, where λ=maxi=1,…,n(-χ-2ai+∑j=1n(ε1bij2+ε2cij2))+(n/ε1)maxi=1,…,n(li2) and ρ=(n/ε2)maxi=1,…,n(li2), then, the equilibrium point u=0 of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate –(λ+hρeγτ)/2.Theorem 3.6.
Assume that(1)
forx=(x1,…,xm)T∈Ω(m≥3), there exists a constant β such that |x|2=∑s=1mxs2<β; In addition, there exists a constant D̲>0 such that Dis=Dis(t,x,u)≥D̲>0; denote D̲(m-2)2/2β=χ,(2)
Pik(ui(tk,x))=-θikui(tk,x), 1-1+α≤θik≤1+1+α, α≥0,(3)
infk=1,2,…(tk-tk-1)>μ,(4)
there exist constantsγ and ε1,ε2>0 such that γ+λ+hρeγτ>0 and λ+hρeγτ+ln(1+α)/μ<0, where λ=maxi=1,…,n(-χ-2ai+∑j=1n(ε1bij2+ε2cij2))+(n/ε1)maxi=1,…,n(li2) and ρ=(n/ε2)maxi=1,…,n(li2).
Then, the equilibrium pointu=0 of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate -(1/2)(λ+hρeγτ+ln(1+α)/μ).
Further, on the condition that|Pik(ui(tk,x))|≤θik|ui(tk,x)|, where θik2<(α-1)/2 and α≥1, we obtain, for t=tk (k=1,2,…),
(3.39)ui2(tk+0,x)-ui2(tk,x)=(Pik(ui(tk,x))+ui(tk,x))2-ui2(tk,x)≤2(ui(tk,x))2+2(Pik(ui(tk,x)))2-ui2(tk,x)≤(2+2θik2)(ui(tk,x))2-ui2(tk,x)≤αui2(tk,x).
Identical with the proof of Theorem3.3, we present the theorem as follows.Theorem 3.7.
Assume that(1)
forx=(x1,…,xm)T∈Ω(m≥3), there exists a constant β such that |x|2=∑s=1mxs2<β. in addition, there exists a constant D̲>0 such that Dis=Dis(t,x,u)≥D̲>0. Denote D̲(m-2)2/2β=χ,(2)
|Pik(ui(tk,x))|≤θik|ui(tk,x)|, where θik2≤(α-1)/2 and α≥1,(3)
infk=1,2,…(tk-tk-1)>μ,(4)
there exist constantsγandε1,ε2>0such thatγ+λ+hρeγτ>0andλ+hρeγτ+ln(1+α)/μ<0,whereλ=maxi=1,…,n(-χ-2ai+∑j=1n(ε1b2ij+ε2cij2))+(n/ε1)maxi=1,…,n(li2)andρ=(n/ε2)maxi=1,…,n(li2).
Then, the equilibrium pointu=0 of problem ((2.5)–(2.8)) is globally exponentially stable with convergence rate -1/2(λ+hρeγτ+ln(1+α)/μ).Remark 3.8.
Different from Theorems3.1–3.6, the impulsive part in Theorem 3.7 could be nonlinear and this will be of more applicability. Actually, Theorems 3.1–3.6 can be regarded as the special cases of Theorem 3.7.


## 4. Examples

Example 4.1.
Consider the following impulsive reaction-diffusion delayed neural network:(4.1)∂ui(t,x)∂t=∑s=1m∂∂xs(Dis∂ui(t,x)∂xs)-aiui(t,x)+∑j=1nbijfj(uj(t,x))+∑j=1ncijfj(uj(t-τj(t),x))t≥0,t≠tk,x∈Ω,k=1,2,…,i=1,…,n
with the impulsive effects characterized by
(4.2)u1(tk+0,x)=u1(tk,x)+1.343u1(tk,x),u2(tk+0,x)=u2(tk,x)+1.343u2(tk,x)k=1,2,…,x∈Ω
and initial condition (2.3) and Dirichlet condition (2.4), where n=2, m=4, Ω={(x1,…,x4)T|∑i=14xi2<4}, a1=a2=6.5, (Dis)2×4=(1.22.32.53.11.83.22.73.4), (bij)2×2=(-0.231.3-0.13), (cij)2×2=(-0.1-0.20.1-0.3), fj(uj)=(1/4)(|uj+1|-|uj-1|), 0≤τj(t)≤0.5, and τ•j(t)<0. For β=4 and D̲=1.2, we compute χ=0.6. This, together with the chosen li=1/2, yields
(4.3)λ=max⁡i=1,…,n(-χ-2ai+∑j=1n(bij2+cij2))+nmax⁡i=1,…,n(li2)=-4,ρ=nmax⁡i=1,…,n(li2)=12.
By selectingγ=2.6, τ=0.5 and h=1, we estimate
(4.4)γ+λ+hρeγτ=2.6-4+12e1.3>0,λ+hρeγτ=-4+12e1.3<0.
According to Theorem3.1, we therefore conclude that the system in Example 4.1 is globally exponential stable.Example 4.2.
Consider the following impulsive reaction-diffusion delayed neural network:(4.5)∂ui(t,x)∂t=∑s=1m∂∂xs(Dis∂ui(t,x)∂xs)-aiui(t,x)+∑j=1nbijfj(uj(t,x))+∑j=1ncijfj(uj(t-τj(t),x))t≥0,t≠tk,x∈Ω,k=1,2,…,i=1,…,n
with the impulsive effects featured by
(4.6)u1(tk+0,x)=u1(tk,x)+arctan(0.5u1(tk,x)),u2(tk+0,x)=u2(tk,x)+arctan(0.5u2(tk,x))k=1,2,…,x∈Ω
and initial condition (2.3) and Dirichlet condition (2.4), where n=2, m=4, Ω={(x1,…,x4)T|∑i=14xi2<4}, a1=a2=6.5, (Dis)2×4=(1.22.32.53.11.83.22.73.4), (bij)2×2=(-0.231.3-0.13), (cij)2×2=(-0.1-0.20.1-0.3), fj(uj)=1/4(|uj+1|-|uj-1|), 0≤τj(t)≤0.5, τ•j(t)<0, and infk=1,2,…(tk-tk-1)>1. For β=4 and D̲=1.2, we compute χ=0.6. This, together with li=1/2 and ε1=ε2=1, yields
(4.7)ρ=nε2max⁡i=1,…,n(li2)=12,λ=max⁡i=1,…,n(-χ-2ai+∑j=1n(ε1bij2+ε2cij2))+nε1max⁡i=1,…,n(li2)=-4.
Selectα=1.5 by setting θik=0.5. Hence, we compute by letting μ=1, γ=3, τ=0.5, and h=1 that
(4.8)γ+λ+hρeγτ=3-4+12e1.5>0,λ+hρeγτ+ln⁡(1+α)μ=-4+12e1.5+ln⁡2.5<0.
It is then concluded from Theorem3.7 that the system in Example 4.2 is globally exponentially stable.



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*Source: 501891-2011-11-22.xml*