# Exponential Stability of Impulsive Delayed Reaction-Diffusion Cellular Neural Networks via Poincaré Integral Inequality

Authors: Xianghong Lai; Tianxiang Yao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This work is devoted to the stability study of impulsive cellular neural networks with time-varying delays and reaction-diffusion terms. By means of new Poincaré integral inequality and Gronwall-Bellman-type impulsive integral inequality, we summarize some novel and concise sufficient conditions ensuring the global exponential stability of equilibrium point. The provided stability criteria are applicable to Dirichlet boundary condition and show that not only the reaction-diffusion coefficients but also the regional features including the boundary and dimension of spatial variable can influence the stability. 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]. As the switching speed of neurons and amplifiers is finite in the implementation of neural networks, time delays are inevitable and therefore a type of more effective models is afterwards introduced, called delayed cellular neural networks (DCNNs). Actually, DCNNs have been found to be helpful in solving some dynamic image processing and pattern recognition problems.As we all know, all the applications of CNNs and DCNNs depend heavily on the dynamic behaviors such as stability, convergence, and oscillatory [8, 9], wherein stability analysis is a major concern in the designs and applications. Correspondingly, 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 often subject to instantaneous perturbations which may be caused by a switching phenomenon, frequency change, or other sudden noise. On this account, neural networks will experience abrupt change at certain instants, exhibiting impulse 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 impulse effects arise naturally in the net. In view of this discovery, many scientists have shown growing interests in the influence that the impulses may have on CNNs or DCNNs with a result that a large number of relevant results have been achieved (see, e.g., [18–24]).Besides impulsive effects, diffusing effects are also nonignorable in reality since the diffusion is unavoidable when the electrons are moving in asymmetric electromagnetic fields. Therefore, the model of impulsive delayed reaction-diffusion neural networks appears as a natural description of the observed evolution phenomena of several real world problems. This one acknowledgement poses a new challenge to the stability research of neural networks.So far, there have been some theoretical achievements [25–33] on the stability of impulsive delayed reaction-diffusion neural networks. Previously, authors of [27–32] studied the stability of impulsive delayed reaction-diffusion neural networks and put forward several stability criteria by impulsive differential inequality and Green formula, wherein the reaction-diffusion term is evaluated to be less than zero by means of Green formula and thereby the presented stability criteria are shown to be wholly independent of diffusion. According to this result, we fail to see the influence of diffusion on stability.Recently, it is encouraging that, for impulsive delayed reaction-diffusion neural network, some new stability criteria involving diffusion are obtained in [25, 26, 33–36]. Meanwhile the estimation of reaction-diffusion term is not merely less than zero, instead a more accurate one is given; that is, the reaction-diffusion term is verified to be less than a negative definite term by using some inequalities together with Green formula. It is thereby testified that the diffusion does contribute to the stability of impulsive neural networks.In [25], the authors quoted the following inequality to deal with the reaction-diffusion terms:
(1)∫Ω*|∂υ(x)∂xj|2dx≥1lj2∫Ω*υ2(x)dx,
where Ω* is a cube |xj|<lj(j=1,2,…,m) and υ(x) is a real-valued function belonging to C01(Ω*). We can easily derive from this inequality that
(2)∫Ω*|∇υ|2dx≥(∫Ω*υ2(x)dx)(∑j=1m1lj2).
For better exploring the influence of diffusion on stability, we wonder if we can get a more accurate estimate of reaction-diffusion term. Fortunately, we find the following new Poincaré integral inequality supporting this idea:
(3)∫𝒮|∇υ(x)|2dx≥4nB2∫𝒮υ2(x)dx.
One can refer to Lemma 3 in Section 2 for the details of this inequality.On the other hand, it is well known that the theory of differential and integral inequalities plays an important role in the qualitative and quantitative study of solution to differential equations. Up till now, there have been many applications of impulsive differential inequalities to impulsive dynamic systems, followed by lots of stability criteria provided. However, these stability criteria appear a bit complicated and we wonder if we can deduce relatively concise stability criteria by using impulsive integral inequalities.Motivated by these, we attempt to, for impulsive delayed neural networks, employ new Poincaré integral inequality to further investigate the influence of diffusion on the stability and combine Gronwall-Bellman-type impulsive integral inequality so as to provide some new and concise stability criteria. The rest of this paper is organized as follows. In Section2, the model of impulsive cellular neural networks with time-varying delays and reaction-diffusion terms as well as Dirichlet boundary condition is outlined; in addition, some facts and lemmas are introduced for later reference. In Section 3, we provide a new estimate on the reaction-diffusion term by the agency of new Poincaré integral inequality and then discuss the global exponential stability of equilibrium point by utilizing Gronwall-Bellman-type impulsive integral inequality with a result of some novel and concise stability criteria presented. To conclude, two illustrative examples are given in Section 4 to verify the effectiveness of our obtained results.


## 2. Preliminaries

LetR+=[0,∞) and t0∈R+. Let Rn denote the n-dimensional Euclidean space, and let Ω=∏i=1m[di,ki] be a fixed rectangular region in Rm and M:=max{ki-di:i=1,…,m}. As usual, denote
(4)C01(Ω)={υ∣υandDjυ=∂υ∂xjare  continuous  onΩ,2υ|∂Ω=0,1≤j≤m∂υ∂xj}.Consider the following impulsive cellular neural network with time-varying delays and reaction-diffusion terms:(5)∂ui(t,x)∂t=∑s=1m∂∂xs(Dis∂ui(t,x)∂xs)-aiui(t,x)∂ui(t,x)∂t+∑j=1nbijfj(uj(t,x))+∑j=1ncijfj(uj(t-τj(t),x))∂ui(t,x)∂t222222222222212222222222t≥t0,t≠tk,x∈Ω,22222222222222222222222222i=1,2,…,n,k=1,2,…,(6)ui(tk+0,x)=ui(tk,x)+Pik(ui(tk,x)),1111x∈Ω,i=1,2,…,n,k=1,2,…,
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; Dis=const>0 represents transmission diffusion of the ith unit; activation function fj(uj(t,x)) stands for the output of the jth unit at time t and in space x; bij, cij, and ai are constants: bij indicates the connection strength of the jth unit on the ith unit at time t and in space x, cij denotes the connection weight 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, satisfying 0≤τj(t)≤τ(τ=const) and 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 meeting 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 the impulsive moment tk and in space x.Denote byu(t,x)=u(t,x;t0,φ), u∈Rn, the solution of system (5)-(6), satisfying the initial condition
(7)u(s,x;t0,φ)=φ(s,x),t0-τ≤s≤t0,x∈Ω,
and Dirichlet boundary condition
(8)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].The solutionu(t,x)=u(t,x;t0,φ)=(u1(t,x;t0,φ),…,un(t,x;t0,φ))T of problem (5)–(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:
(9)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 defined by
(10)∥u(t,x;t0,φ)∥Ω2=∑i=1n∫Ωui2(t,x;t0,φ)dx.Before proceeding, we introduce two hypotheses as follows:(H1)
fi(∙):R→R satisfies fi(0)=0, and there exists a constant li>0 such that |fi(y1)-fi(y2)|≤li|y1-y2| for all y1,y2∈R and i=1,2,…,n.(H2)
Pik(∙):R→R is continuous and Pik(0)=0, i=1,2,…,n,  k=1,2,….According to (H1) and (H2), it is easy to see that problem (5)–(8) admits an equilibrium point u=0.Definition 1 (see [25]).
The equilibrium pointu=0 of problem (5)–(8) is said to be globally exponentially stable if there exist constants κ>0 and ϖ≥1 such that
(11)∥u(t,x;t0,φ)∥Ω≤ϖ∥φ∥¯Ωe-κ(t-t0),t≥t0,
where ∥φ∥¯Ω2=supt0-τ≤s≤t0∑i=1n∫Ωφi2(s,x)dx.Lemma 2 (see [37] Gronwall-Bellman-type Impulsive Integral Inequality).
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,…,
(12)q(t)≤c+∫t0tp(s)q(s)ds+∑t0<tk<tηkq(tk),t≥t0,where ηk≥0 and c=const. Then,
(13)q(t)≤c∏t0<tk<t(1+ηk)exp(∫t0tp(s)ds),t≥t0.Lemma 3 (see [38] Poincaré integral inequality).
Let𝒮=∏i=1n[ai,bi] be a fixed rectangular region in Rn and B:=max{bi-ai:i=1,…,n}. For any υ(x)∈C01(𝒮),
(14)∫𝒮υ2(x)dx≤B24n∫𝒮|∇υ(x)|2dx.Remark 4.
According to Lemma  2.1 in [25], we know if 𝒮 is a cube |xj|<lj (j=1,2,…,m) and υ(x) is a real-valued function belonging to C01(𝒮), then
(15)∫𝒮|∂υ(x)∂xj|2dx≥1lj2∫𝒮υ2(x)dx,
which yields
(16)∫𝒮|∇υ|2dx≥(∫𝒮υ2(x)dx)(∑j=1m1lj2).
Through the simple example as follows, we can find that in some cases the estimate ∫𝒮|∇υ(x)|2dx≥(4n/B2)∫𝒮υ2(x)dx shown in Lemma 3 can do better. Let 𝒮=[0,1]×[0,2], we derive from Lemma  2.1 in [25] that
(17)∫𝒮|∇υ|2dx≥(∫𝒮υ2(x)dx)(∑j=1m1lj2)=54∫𝒮υ2(x)dx,
whereas the application of Lemma 3 of this paper will give
(18)∫𝒮|∇υ(x)|2dx≥4nB2∫𝒮υ2(x)dx=2∫𝒮υ2(x)dx,
which is obviously superior to ∫𝒮|∇υ|2dx≥(5/4)(∫𝒮υ2(x)dx).


## 3. Main Results

Theorem 5.
Provided that one has the following:(1)
letD_=min{Dis:i=1,…,n;s=1,…,m}>0 and denote 8mD_/M2=χ;(2)
Pik(ui(tk,x))=-θikui(tk,x), 0≤θik≤2;(3)
there exists a constantγ>0 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 pointu=0 of problem (5)–(8) is globally exponentially stable with convergence rate -(λ+hρeγτ)/2.Proof.
Multiplying both sides of (5) by ui(t,x) and integrating with respect to spatial variable x on Ω, we get
(19)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))dx222222222t≥t0,t≠tk,i=1,…,n,k=1,2,….
Regarding the right-hand part of (19), the first term becomes by using Green formula, Dirichlet boundary condition, Lemma 3, and condition (1) of Theorem 5(20)2∑s=1m∫Ωui(t,x)∂∂xs(Dis∂ui(t,x)∂xs)dx=-2∑s=1m∫ΩDis(∂ui(t,x)∂xs)2dx≤-8mD_M2∫Ωui2(t,x)dx≜-χ∫Ωui2(t,x)dx.
Moreover, From (H1), we have(21)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)+lj2uj2(t,x))dx,(22)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)+lj2uj2(t-τj(t),x))dx.
Consequently, substituting (20)–(22) into (19) produces
(23)d(∫Ωui2(t,x)dx)dt≤-χ∫Ωui2(t,x)dx-2ai∫Ωui2(t,x)dx+∑j=1n∫Ω(bij2ui2(t,x)+lj2uj2(t,x))dx+∑j=1n∫Ω(cij2ui2(t,x)+lj2uj2(t-τj(t),x))dx
for t≥t0, t≠tk, i=1,…,n,  k=1,2,….
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 the first kind discontinuous points tk (k=1,2,…), where it is continuous from the left, that is, Vi(tk-0)=Vi(tk)  (k=1,2,…). In addition, we also see
(24)Vi(tk+0)≤Vi(tk),k=0,1,2,…,
as Vi(t0+0)≤Vi(t0) and the following estimate derived from condition (2) of Theorem 5:
(25)ui2(tk+0,x)=(-θikui(tk,x)+ui(tk,x))2=(1-θik)2ui2(tk,x)≤ui2(tk,x),222222222222222222k=1,2,….
Putt∈(tk,tk+1), k=0,1,2,…. It then results from (23) that
(26)dVi(t)dt≤-χ∫Ωui2(t,x)dx-2ai∫Ωui2(t,x)dx+∑j=1n∫Ω(bij2ui2(t,x)+lj2uj2(t,x))dx+∑j=1n∫Ω(cij2ui2(t,x)+lj2uj2(t-τj(t),x))dx≤(-χ-2ai+∑j=1nbij2+∑j=1ncij2)Vi(t)+maxi=1,…,n(li2)∑j=1nVj(t)+maxi=1,…,n(li2)∑j=1nVj(t-τj(t))222t∈(tk,tk+1),k=0,1,2,⋯.
ChooseV(t) of the form V(t)=∑i=1nVi(t). From (26), one reads
(27)dV(t)dt≤λV(t)+ρ∑j=1nVj(t-τj(t)),2222222t∈(tk,tk+1),k=0,1,2,…,
where λ=maxi=1,…,n(-χ-2ai+∑j=1n(bij2+cij2))+ρ and ρ=nmaxi=1,…,n(li2).
Now constructV*(t)=eγ(t-t0)V(t) again, where γ>0 satisfies γ+λ+hρeγτ>0 and λ+hρeγτ<0. Evidently, V*(t) is also a piecewise continuous function with the first kind discontinuous points tk  (k=1,2,…), where 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 (24)
(28)V*(tk+0)≤V*(tk),k=0,1,2,….
Sett∈(tk,tk+1),  k=0,1,2,…. By virtue of (27), one has
(29)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))222222222222t∈(tk,tk+1),k=0,1,2,….
Choose small enoughε>0. Integrating (29) from tk+ε to t gives
(30)V*(t)≤V*(tk+ε)+(γ+λ)∫tk+εtV*(s)ds+∫tk+εtρeγ(s-t0)∑j=1nVj(s-τj(s))ds2222222t∈(tk,tk+1),k=0,1,2,…,
which yields, after letting ε→0 in (30),
(31)V*(t)≤V*(tk+0)+(γ+λ)∫tktV*(s)ds+∫tktρeγ(s-t0)∑j=1nVj(s-τj(s))ds22222t∈(tk,tk+1),k=0,1,2,….
Next we will estimate the value ofV*(t) at t=tk+1, k=0,1,2,…. For small enough ε>0, we put t=tk+1-ε. An application of (31) leads to, for k=0,1,2,…,
(32)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 (32), there results
(33)V*(tk+1-0)≤V*(tk+0)+(γ+λ)∫tktk+1V*(s)ds+∫tktk+1ρeγ(s-t0)∑j=1nVj(s-τj(s))ds,11111111111111111k=0,1,2,….
Note thatV*(tk+1-0)=V*(tk+1) is applicable for k=0,1,2,…. Thus,
(34)V*(tk+1)≤V*(tk+0)+(γ+λ)∫tktk+1V*(s)ds+∫tktk+1ρeγ(s-t0)∑j=1nVj(s-τj(s))ds
holds for k=0,1,2,…. By synthesizing (31) and (34), we then arrive at
(35)V*(t)≤V*(tk+0)+(γ+λ)∫tktV*(s)ds+∫tktρeγ(s-t0)∑j=1nVj(s-τj(s))ds2222t∈(tk,tk+1],k=0,1,2,….
This, together with (28), results in
(36)V*(t)≤V*(tk)+(γ+λ)∫tktV*(s)ds+∫tktρeγ(s-t0)∑j=1nVj(s-τj(s))ds
for t∈(tk,tk+1],  k=0,1,2,….
Recalling assumptions that0≤τj(t)≤τ and τ•j(t)<(1-(1/h))(h>0), we obtain
(37)∫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,(38)V*(t)≤V*(tk)+(γ+λ)∫tktV*(s)ds+hρeγτ∑j=1n∫tk-τj(tk)t-τj(t)eγ(s-t0)Vj(s)ds111111t∈(tk,tk+1],k=0,1,2,….
By induction argument, we reach(39)V*(tk)≤V*(tk-1)+(γ+λ)∫tk-1tkV*(s)ds1111111+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)ds1111111+hρeγτ∑j=1n∫t1-τj(t1)t2-τj(t2)eγ(s-t0)Vj(s)ds,V*(t1)≤V*(t0)+(γ+λ)∫t0t1V*(s)ds1111111+hρeγτ∑j=1n∫t0-τj(t0)t1-τj(t1)eγ(s-t0)Vj(s)ds.
Therefore,(40)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)ds=V*(t0)+t∈(tk,tk+1],k=0,1,2,….
Since(41)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
(42)V*(t)≤V*(t0)+τhρeγτ∥φ∥¯Ω2V*(t)≤+(γ+λ+hρeγτ)∫t0tV*(s)ds222222222t∈(tk,tk+1],k=0,1,2….
According to Lemma2, we know
(43)V*(t)≤(V*(t0)+τhρeγτ∥φ∥¯Ω2)×exp{(γ+λ+hρeγτ)(t-t0)},t≥t0
which reduces to
(44)∥u(t,x;t0,φ)∥Ω≤1+τhρeγτ∥φ∥¯Ωexp{(λ+hρeγτ2)(t-t0)},2222222222222222222222222222222222222222t≥t0.
This completes the proof.Remark 6.
According to Theorem5, we see that the diffusion can really influence the stability of equilibrium point u=0 of problem (5)–(8), wherein the factors embrace not only the reaction-diffusion coefficients but also the regional features including the dimension and boundary of spatial variable. Owing to the employ of new Poincaré integral inequality, in this paper, the estimation of reaction-diffusion terms is superior to that in [25] in some cases, and this will be helpful to further know the influence of diffusion on stability. What is more, from condition (1) of Theorem 5, we also see that the dimension of spatial variable has an impact on the stability while this is not mentioned in [25].Remark 7.
Among the three conditions of Theorem5, condition (3) is critical and therefore we must ensure the existence of constant γ>0. Fortunately, it is not difficult to find that there must exist a constant γ>0 satisfying condition (3) if λ<-hρ which is easily checked.Theorem 8.
Providing that one has the following:(1)
letD_=min{Dis:i=1,…,n;s=1,…,m}>0 and denote 8mD_/M2=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 a constantγ>0 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 (5)–(8) is globally exponentially stable with convergence rate -(1/2)(λ+hρeγτ+(ln(1+α)/μ)).Proof.
Define 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 the first kind discontinuous points tk, k=1,2,…, where it is continuous from the left, that is, V(tk-0)=V(tk)  (k=1,2,…). Furthermore, when t=tk  (k=0,1,2,…), it follows from condition (2) of Theorem 8 that
(45)ui2(tk+0,x)-ui2(tk,x)=(1-θik)2ui2(tk,x)-ui2(tk,x)≤αui2(tk,x).
Thereby,(46)V(tk+0)≤αV(tk)+V(tk),k=0,1,2,….
Construct another Lyapunov functionV*(t)=eγ(t-t0)V(t), where γ>0 satisfies γ+λ+hρeγτ>0 and λ+hρeγτ+(ln(1+α)/μ)<0. Then, V*(t) is also a piecewise continuous function with the first kind discontinuous points tk, k=1,2,…, where it is continuous from the left, and for t=tk (k=0,1,2,…), it results from (46) that
(47)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 5, we get
(48)V*(t)≤V*(tk+0)+(γ+λ)∫tktV*(s)ds+hρeγτ∑j=1n∫tk-τj(tk)t-τj(t)eγ(θ-t0)Vj(θ)dθ+hρeγτt∈(tk,tk+1],k=0,1,2,….
The relations (47) and (48) yield
(49)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)dθ2222222222t∈(tk,tk+1],k=0,1,2,….
By induction argument, we reach(50)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,
(51)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)t0eγ(θ-t0)Vj(θ)dθ+α∑t0<tk<tV(tk)22222222222222222222222t∈(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 5 into (51), (51) becomes
(52)V*(t)-V*(t0)≤αV*(t0)+τhρeγτ∥φ∥¯Ω2+(γ+λ+hρeγτ)∫t0tV*(s)ds+α∑t0<tk<tV(tk)2222222222222222222t∈(tk,tk+1],k=0,1,2….
It then results from Lemma 2 that, for t≥t0,
(53)V*(t)≤((α+1)V*(t0)+τhρeγτ∥φ∥¯Ω2)V*(t)≤×∏t0<tk<t(1+α)exp((γ+λ+hρeγτ)(t-t0))V*(t)=((α+1)V*(t0)+τhρeγτ∥φ∥¯Ω2)V*(t)×(1+α)kexp((γ+λ+hρeγτ)(t-t0)).
On the other hand, sinceinfk=1,2,…(tk-tk-1)≥μ, one has k≤(tk-t0)/μ. Thereby,
(54)(1+α)k≤exp{ln(1+α)μ(tk-t0)}≤exp{ln(1+α)μ(t-t0)}
and (53) can be rewritten as
(55)V*(t)≤((α+1)V*(t0)+τhρeγτ∥φ∥¯Ω2)×exp((γ+λ+hρeγτ+ln(1+α)μ)(t-t0))
which implies
(56)∥u(t,x;t0,φ)∥Ω≤(α+1+τhρeγτ)∥φ∥¯Ω×exp(12(λ+hρeγτ+ln(1+α)μ)(t-t0)),2222222222222222222222222222222222222t≥t0.
The proof is completed.As(57)2∑j=1nbij∫Ωui(t,x)f(uj(t,x))dx≤∑j=1n∫Ω(ε1bij2ui2(t,x)+lj2ε1uj2(t,x))dx,2∑j=1ncij∫Ωui(t,x)f(uj(t-τj(t),x))dx≤∑j=1n∫Ω(ε2cij2ui2(t,x)+lj2ε2uj2(t-τj(t),x))dx
hold for any ε1,ε2>0. In the sequel, analogous to the proofs of Theorems 5 and 8 we arrive at the following.Theorem 9.
Provided that one has the following:(1)
letD_=min{Dis:i=1,…,n;s=1,…,m}>0 and denote 8mD_/M2=χ;(2)
Pik(ui(tk,x))=-θikui(tk,x), 0≤θik≤2;(3)
there exist constantsγ>0 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 (5)–(8) is globally exponentially stable with convergence rate -(λ+hρeγτ)/2.Remark 10.
According to Theorem5, we know that there must exist constant γ>0 satisfying condition (3) of Theorem 9 if there are constants ε1,ε2>0 such that λ<-hρ.Theorem 11.
Assume that one has the following:(1)
letD_=min{Dis:i=1,…,n;s=1,…,m}>0 and denote 8mD_/M2=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>0 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 point u=0 of problem (5)–(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,…),
(58)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 Theorem8, we reach the following.Theorem 12.
Assume that one has the following:(1)
letD_=min{Dis:i=1,…,n;s=1,…,m}>0 and denote 8mD_/M2=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 constantsγ>0 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 point u=0 of problem (5)–(8) is globally exponentially stable with convergence rate -(1/2)(λ+hρeγτ+(ln(1+α)/μ)).Remark 13.
Different from Theorems5–11, the impulsive part in Theorem 12 could be nonlinear and this will be of more applicability. Actually, Theorems 5–11 can be regarded as the special cases of Theorem 12.


## 4. Examples

Example 14.
Consider system (5)–(8) equipped with Pik(ui(tk,x))=1.343ui(tk,x). Let n=2, m=2, Ω=[0,1.5]×[0,2], τj(t)=(3/4)arctan(t), a1=a2=6.5, (Dis)2×2=(1.22.32.21.5), (bij)2×2=(-0.231.3-0.143.2), (cij)2×2=(-0.1-0.20.25-0.13), and fj(uj)=(2/4)(|uj+1|-|uj-1|).ForM=2 and D_=1.2, we compute χ=4.8. This, together with li=2/2, yields
(59)ρ=nmaxi=1,…,n(li2)=1,(60)λ=maxi=1,…,n(-χ-2ai+∑j=1n(bij2+cij2))+ρ=-6.461.Leth=4. Since λ=-6.461<-4=-hρ, we conclude from Theorem 5 that the equilibrium point u=0 of this system is globally exponentially stable.Example 15.
Consider system (5)–(8) equipped with Pik(ui(tk,x))=arctan(0.5ui(tk,x)). Let n=2, m=2, τj(t)=(1/π)arctan(t), Ω=[0,1.5]×[0,2], ai=6.5, (Dis)2×2=(1.22.32.23.5), (bij)2×2=(-0.231.3-0.143.2), (cij)2×2=(-0.1-0.20.25-0.13), fj(uj)=(2/4)(|uj+1|-|uj-1|), and tk=tk-1+2k.ForM=2 and D_=1.2, we compute χ=4.8. This, together with li=2/2 and ε1=ε2=1, yields
(61)ρ=nε2maxi=1,…,n(li2)=1,λ=maxi=1,…,n(-χ-2ai+∑j=1n(ε1bij2+ε2cij2))+nε1maxi=1,…,n(li2)=-6.461.Lettingτ=0.5, h=4, μ=2, and α=1.5, we can find γ=0.78 satisfying
(62)γ+λ+hρeγτ=0.2269>0,λ+hρeγτ+ln(1+α)μ=-0.0949<0.It is then concluded from Theorem12 that this system is globally exponentially stable.



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*Source: 131836-2013-03-31.xml*