# Stability of Impulsive Cohen-Grossberg Neural Networks with Time-Varying Delays and Reaction-Diffusion Terms

Authors: Jinhua Huang; Jiqing Liu; Guopeng Zhou
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This work concerns the stability of impulsive Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms as well as Dirichlet boundary condition. By means of Poincaré inequality and Gronwall-Bellman-type impulsive integral inequality, we summarize some new and concise sufficient conditions ensuring the global exponential stability of equilibrium point. The proposed criteria are relevant to the diffusion coefficients and the smallest positive eigenvalue of corresponding Dirichlet Laplacian. In conclusion, two examples are illustrated to demonstrate the effectiveness of our obtained results.


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## Body

## 1. Introduction

Cohen-Grossberg neural networks (CGNNs) were introduced by Cohen and Grossberg in 1983 [1] and have been a hot topic due to their important applications in various fields such as parallel computation, associative memory, image processing, and optimization problems.By reason that time delays are unavoidably encountered for the finite switching speed of neurons and amplifiers in the implementation of neural networks, a more powerful model of delayed Cohen-Grossberg neural networks (DCGNNs) is afterwards proposed. This kind of mathematical models is widely applied in dynamic image processing and pattern recognition problems. It is worth noting that all these applications depend heavily on the dynamical behaviors such as stability, convergence, and oscillatory [2–6]. Meanwhile, stability is an important consideration in the designs and applications of neural networks. The stability of delayed neural networks is a subject of current interest, and therefore considerable theoretical efforts have been put into this topic followed by a large number of stability criteria reported; for example, see [7–12] and the references therein.In real world, however, many evolutionary processes are characterized by abrupt changes at certain instants which may be caused by switching phenomena, frequency changes, or other sudden noises. As such, in the past few years, scientists have become gradually interested in the influence that impulses may have on the CGNNs and DCGNNs, thus obtaining some related results; for example, see [13–18] and the references therein.Actually, besides impulsive effects, we have to recognize that diffusion effects are also nonignorable in reality as diffusion is unavoidable when electrons are moving in asymmetric electromagnetic fields. On this account, the model of neural networks with both impulses and diffusion should be more effective for describing the evolutionary process of practical systems. Based on this consideration, we wonder what the influence of diffusion on the stability of CGNNs and DCGNNs is.So far there have appeared a few theoretical achievements [19–29] on the stability of impulsive reaction-diffusion neural networks with or without delays. Particularly, in [21–26], the main research technique is the impulsive differential inequality whereby the authors discussed the stability of equilibrium point and provided a series of sufficient conditions independent of diffusion. From these results, we fail to see the influence of diffusion on the stability of CGNNs and DCGNNs.Encouragingly, recently there were reported some new results on the stability of CGNNs and DCGNNs in [19, 20, 27]; thereinto, the presented stability criteria derived from the impulsive differential inequality are related to the diffusion terms, and thereby we know the diffusion do contribute to the stability of impulsive neural networks.In this paper, different from [20, 27], we shall consider the case where the boundary condition is Dirichlet boundary condition rather than Neumann boundary condition. Moreover, unlike [19], we shall utilize the new method of Poincarè inequality to deal with the reaction-diffusion terms, and Gronwall-Bellman-type impulsive integral inequality is also introduced for stability analysis. The obtained results show that not only the reaction-diffusion coefficients but also the first eigenvalue of corresponding Dirichlet Laplacian can affect the stability.The rest of this paper is structured as follows. In Section2, the model of impulsive delayed Cohen-Grossberg neural networks with reaction-diffusion terms as well as Dirichlet boundary condition is outlined and some facts and lemmas are introduced for later reference. By the new agencies of Gronwall-Bellman-type impulsive integral inequality and Poincaré inequality, we discuss the global exponential stability of equilibrium point and develop some new and concise algebraic criteria in Section 3. To conclude, two illustrative examples are given in Section 4 to verify the effectiveness of our results.


## 2. Preliminaries

LetRn denote the n-dimensional Euclidean space, and let Ω⊂Rm be an open bounded domain with smooth boundary ∂Ω and mesΩ>0. Let R+=[0,∞) and t0∈R+.Consider the following impulsive CGNN with time-varying delays and reaction-diffusion terms:(1)∂ui(t,x)∂t=∑s=1m∂∂xs(Dis∂ui(t,x)∂xs)-ai(ui(t,x))[ωi(ui(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)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,Dis=const>0 represents transmission diffusion of the ith unit, ai(ui(t,x)) represents the amplification function, ωi(ui(t,x)) is the appropriate behavior function, activation function fj(uj(t,x)) stands for the output of the jth unit at time i and in space x and bij and cij are constants: bij indicates the connection strength of the jth unit on the ith unit at time t and in space x, while 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). {tk}(k=1,2,…) is the sequence of 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 impulsive moment tk and in space x.Denote byu(t,x)=u(t,x;t0,φ),u∈Rn, the solution of systems (1)-(2), satisfying the initial condition
(3)u(s,x;t0,φ)=φ(s,x),t0-τ≤s≤t0,x∈Ω,
and Dirichlet boundary condition
(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].The solutionu(t,x)=u(t,x;t0,φ)=(u1(t,x;t0,φ),…,un(t,x;t0,φ))T of problems (1)–(4) 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 left-continuous; that is, the following relations are valid:
(5)ui(tk-0,x)=ui(tk,x),ui(tk+0,x)=ui(tk,x)+Pik(ui(tk,x)).Throughout this paper, we define the norm ofu(t,x;t0,φ) as
(6)∥u(t,x;t0,φ)∥Ω=∑i=1n∫Ωui2(t,x;t0,φ)dx
and make the following assumptions for convenience.(H1)
ai(·):R→R+ is continuous and bounded; that is, there exist constants a_i and a¯i such that
(7)0<a_i≤ai(ζ)≤a¯i<∞,fori=1,…,n.(H2)
ωi(·):R→R is continuous and ωi(0)=0; moreover, there exists constant pi>0 such that
(8)ωi(ζ1)-ωi(ζ2)ζ1-ζ2≥pi>0,forζ1≠ζ2,i=1,…,n.(H3)
fi(·):R→R is continuous and fi(0)=0; furthermore, there exists constant li>0 such that
(9)li=supζ1≠ζ2fi(ζ1)-fi(ζ2)ζ1-ζ2forζ1≠ζ2,i=1,2,…,n.(H4)
Pik(·):R→R is continuous and Pik(0)=0 for i=1,2,…,n and k=1,2,….In the light of (H1)–(H4), it is easy to see that problems (1)-(2) admit an equilibrium point u=0.Definition 1.
The equilibrium pointu=0 of problems (1)-(2) is said to be globally exponentially stable if there exist constants κ>0 and M≥1 such that
(10)∥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 (see [30] (Gronwall-Bellman-type impulsive integral inequality)).
Assume the following.(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,…,
(11)q(t)≤c+∫t0tp(s)q(s)ds+∑t0<tk<tηkq(tk),t≥t0,where ηk≥0 and c=const. Then,
(12)q(t)≤c∏t0<tk<t(1+ηk)exp(∫t0tp(s)ds),t≥t0.Lemma 3 (see [31] (Poincaré inequality)).
Let𝒮 be a bounded region in Rn,v(x)∈C1(𝒮), and v=0 on the boundary of 𝒮; then
(13)λ1∫𝒮v2(x)dx≤∫𝒮|∇v(x)|2dx,
where λ1 is the smallest positive eigenvalue of the following problem:
(14)ΔΨ(x)+λΨ(x)=0,x∈𝒮,Ψ(x)=0,x∈∂𝒮.Lemma 4.
Ifa>0 and b>0, then ab≤(1/ε)a2+εb2 holds for any ε>0.


## 3. Main Results

Theorem 5.
Assume the following.(1)
D_=min{Dis:i=1,…,n;s=1,…,m}>0 and denote 2D_λ1=χ.(2)
Pik(ui(tk,x))=-θikui(tk,x),0≤θik≤2.(3)
There exists a constantγ>0 satisfying γ+λ+hρeγτ>0 and λ+hρeγτ<0, where
(15)λ=maxi=1,…,n(-χ-2a_ipi+a¯i∑j=1nbij2+a¯i∑j=1ncij2)+ρ,ρ=maxi=1,…,n(li2)∑i=1na¯i.Then, the equilibrium point u=0 of systems  (1)-(2) is globally exponentially stable with convergence rate -(λ+hρeγτ)/2.Proof.
Multiplying both sides of (1) by ui(t,x), we get
(16)∂ui2(t,x)∂t=2∑s=1mui(t,x)∂∂xs(Dis∂ui(t,x)∂xs)-2ui(t,x)ai(ui(t,x))×[ωi(ui(t,x))-∑j=1nbijfj(uj(t,x))-∑j=1ncijfj(uj(t-τj(t),x))],t≥t0,t≠tk,x∈Ω,k=1,2,…,
which yields, after integrating with respect to spatial variable x on Ω,
(17)d(∫Ωui2(t,x)dx)dt=J1+J2,t≥t0,t≠tk,k=1,2,…,
where J1=2∫Ω∑s=1m(ui(t,x)(∂/∂xs)(Dis(∂ui(t,x))/∂xs))dx,
(18)J2=-2∫Ωui(t,x)ai(ui(t,x))×[ωi(ui(t,x))-∑j=1nbijfj(uj(t,x))-∑j=1ncijfj(uj(t-τj(t),x))]dx.
By combining Green formula, Dirichlet boundary condition, Lemma3, and condition (1) of Theorem 5, we obtain
(19)J1=-2∑s=1m∫ΩDis(∂ui(t,x)∂xs)2dx≤-2D_λ1∫Ωui2(t,x)dx≜-χ∫Ωui2(t,x)dx.
Moreover, it follows from assumptions (H1), (H2), and (H3) that(20)2∫Ωui(t,x)ai(ui(t,x))ωi(ui(t,x))dx≥2a_ipi∫Ω|ui(t,x)|2dx,(21)2∫Ωui(t,x)ai(ui(t,x))∑j=1nbijfj(uj(t,x))dx≤a¯i∑j=1n∫Ω(bij2ui2(t,x)+fj2(uj(t,x)))dx≤a¯i∑j=1n∫Ω2|bij∥ui(t,x)∥fj(uj(t,x))|dx≤a¯i∑j=1n∫Ω(bij2ui2(t,x)+lj2uj2(t,x))dx,(22)2∫Ωui(t,x)ai(ui(t,x))∑j=1ncijfj(uj(t-τj(t),x))dx≤a¯i∑j=1n∫Ω(cij2ui2(t,x)+lj2uj2(t-τj(t),x))dx.
Consequently, substituting (19)–(22) into (17) produces
(23)d(∫Ωui2(t,x)dx)dt≤-χ∫Ωui2(t,x)dx-2a_ipi∫Ωui2(t,x)dx+a¯i∑j=1n∫Ω(bij2ui2(t,x)+lj2uj2(t,x))dx+a¯i∑j=1n∫Ω(cij2ui2(t,x)+lj2uj2(t-τj(t),x))dx
for t≥t0, t≠tk, k=1,2,….
Now define Lyapunov functionVi(t) as Vi(t)=∫Ωuj2(t,x)dx. It is not difficult to see 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, for t=tk (k=0,1,2,…), we know
(24)Vi(tk+0)≤Vi(tk),k=0,1,2,…,
as Vi(t0+0)≤Vi(t0) and ui2(tk+0,x)=(1-θik)2ui2(tk,x)≤ui2(tk,x)(k=1,2,…), supported by condition 2 of Theorem 5.
Putt∈(tk,tk+1), k=0,1,2,…. It is derived from (23) that
(25)dVi(t)dt≤(-χ-2a_ipi+a¯i∑j=1nbij2+a¯i∑j=1ncij2)Vi(t)+a¯imaxi=1,…,n(li2)∑j=1nVj(t)+a¯imaxi=1,…,n(li2)∑j=1nVj(t-τj(t)),t∈(tk,tk+1),k=0,1,2,….
Define functionV(t) of the form V(t)=∑i=1nVi(t) again. From (25), one then reads
(26)dV(t)dt≤λV(t)+ρ∑j=1nVj(t-τj(t)),t∈(tk,tk+1),k=0,1,2,…,
where ρ=maxi=1,…,n(li2)∑i=1na¯i and λ=maxi=1,…,n(-χ-2a_ipi+a¯i∑j=1nbij2+a¯i∑j=1ncij2)+ρ.
ConstructV*(t)=eγ(t-t0)V(t), 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,…), 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 the use of (24)
(27)V*(tk+0)≤V*(tk),k=0,1,2,….
Sett∈(tk,tk+1),k=0,1,2,…. By virtue of (26), one has
(28)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 (28) from tk+ε to t gives
(29)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 letting ε→0 in (29),
(30)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,….
Next, we 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 (30) leads to, for k=0,1,2,…,
(31)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 (31), there results
(32)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,
(33)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 (30) and (33), we then arrive at
(34)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,….
This, together with (27), results in
(35)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 the assumptions that0≤τj(t)≤τ and τ•j(t)<1-(1/h)(h>0), we therefore obtain
(36)∫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,
(37)V*(t)≤V*(tk)+(γ+λ)∫tktV*(s)ds+hρeγτ∑j=1n∫tk-τj(tk)t-τj(t)eγ(s-t0)Vj(s)ds,t∈(tk,tk+1],k=0,1,2,….
By induction argument, we reach(38)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.
Thus,
(39)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,t∈(tk,tk+1],k=0,1,2,….
Since(40)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
(41)V*(t)≤V*(t0)+τhρeγτ∥φ∥¯Ω2+(γ+λ+hρeγτ)∫t0tV*(s)ds,t∈(tk,tk+1],k=0,1,2….
According to Lemma2, we assert that
(42)V*(t)≤(V*(t0)+τhρeγτ∥φ∥¯Ω2)×exp{(γ+λ+hρeγτ)(t-t0)},t≥t0,
which reduces to
(43)∥u(t,x;t0,φ)∥Ω≤1+τhρeγτ∥φ∥¯Ω×exp{(λ+hρeγτ2)(t-t0)},t≥t0.
This completes the proof.Remark 6.
According to the conditions of Theorem5, we see that the reaction-diffusion terms can influence the stability of equilibrium point u=0. Specifically, the acting factors include the reaction-diffusion coefficients and the first eigenvalue of corresponding Dirichlet Laplacian.Remark 7.
It is not difficult to see that there must exist constantγ>0 satisfying condition 3 of Theorem 5 if λ<-hρ.Theorem 8.
Assume the following.(1)
D_=min{Dis:i=1,…,n;s=1,…,m}>0 and denote 2D_λ1=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 satisfying γ+λ+hρeγτ>0 and λ+hρeγτ+ln(1+α)/μ<0, where λ=maxi=1,…,n(-χ-2a_ipi+a¯i∑j=1nbij2+a¯i∑j=1ncij2)+ρ and ρ=maxi=1,…,n(li2)∑i=1na¯i.Then, the equilibrium pointu=0 of systems  (1)-(2) 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, for t=tk(k=0,1,2,…), we derive from condition 2 of Theorem 8 that
(44)ui2(tk+0,x)-ui2(tk,x)=(1-θik)2ui2(tk,x)-ui2(tk,x)≤αui2(tk,x).
Thereby,
(45)V(tk+0)≤αV(tk)+V(tk),k=0,1,2,….
Construct functionV*(t)=eγ(t-t0)V(t) again, 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; that is, V*(tk-0)=V*(tk)(k=1,2,…). And for t=tk(k=0,1,2,…), it follows from (45) that
(46)V*(tk+0)≤αV*(tk)+V*(tk),k=0,1,2,…
Sett∈(tk,tk+1],k=0,1,2,…. Following the same procedure as shown in the proof of Theorem 5, we get
(47)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,t∈(tk,tk+1],k=0,1,2,….
The relations (46) and (47) yield
(48)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)dθ,t∈(tk,tk+1],k=0,1,2,….
By induction argument, we obtain
(49)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,
(50)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),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 5 into (50), (50) becomes, for t∈(tk,tk+1],k=0,1,2,…,
(51)V*(t)-V*(t0)≤αV*(t0)+τhρeγτ∥φ∥¯Ω2+(γ+λ+hρeγτ)∫t0tV*(s)ds+α∑t0<tk<tV(tk).
It then results from Lemma 2 that, for t≥t0,
(52)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)).
On the other hand, sinceinfk=1,2,…(tk-tk-1)≥μ, one has k≤(tk-t0)/μ. Thereby,
(53)(1+α)k≤exp{ln(1+α)μ(tk-t0)}≤exp{ln(1+α)μ(t-t0)}
and (52) can be rewritten as
(54)V*(t)≤((α+1)V*(t0)+τhρeγτ∥φ∥¯Ω2)×exp((γ+λ+hρeγτ+ln(1+α)μ)(t-t0))
which implies
(55)∥u(t,x;t0,φ)∥Ω≤(α+1+τhρeγτ)∥φ∥¯Ω×exp(12(λ+hρeγτ+ln(1+α)μ)(t-t0)),t≥t0.
The proof is completed.Due to Lemma4, we know that the following inequalities:
(56)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,x))dx≤∑j=1n∫Ω(ε2cij2ui2(t,x)+lj2ε2ui2(t-τj,x))dx
hold for any ε1,ε2>0. Thus, in a similar way to the proofs of Theorems 5–8, we can prove the following theorems.Theorem 9.
Assume the following.(1)
D_=min{Dis:i=1,…,n;s=1,…,m}>0 and denote 2D_λ1=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-2a_ipi+a¯i∑j=1n(ε1bij2+ε2cij2))+(maxi=1,…,n(li2)/ε1)∑i=1na¯i, and ρ=(maxi=1,…,n(lj2)/ε2)∑i=1na¯i.
Then, the equilibrium pointu=0 of systems  (1)-(2) is globally exponentially stable with convergence rate -(λ+hρeγτ)/2.Remark 10.
There must exist constantγ>0 satisfying condition 3 of Theorem 9 if there are constants ε1,ε2>0 such that λ<-hρ.Theorem 11.
Assume the following.(1)
D_=min{Dis:i=1,…,n;s=1,…,m}>0 and denote 2D_λ1=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 satisfying γ+λ+hρeγτ>0 and λ+hρeγτ+ln(1+α)/μ<0, where
(57)λ=maxi=1,…,n(-χ-2a_ipi+a¯i∑j=1n(ε1bij2+ε2cij2))+maxi=1,…,n(li2)ε1∑i=1na¯i,ρ=maxi=1,…,n(li2)ε2∑i=1na¯i.
Then, the equilibrium pointu=0 of systems (1)-(2) 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
(58)ui2(tk+0,x)-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)
for t=tk(k=1,2,…). In an identical way with the proof of Theorem 8, we can present the following.Theorem 12.
Assume the following.(1)
LetD_=min{Dis:i=1,…,n;s=1,…,m}>0 and denote 2D_λ1=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
(59)λ=maxi=1,…,n(-χ-2a_ipi+a¯i∑j=1n(ε1bij2+ε2cij2))+maxi=1,…,n(li2)ε1∑i=1na¯i,ρ=maxi=1,…,n(li2)ε2∑i=1na¯i.
Then, the equilibrium pointu=0 of systems   (1)-(2) 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 problems (1)–(4) with Pik(ui(tk,x))=1.343ui(tk,x); moreover, n=2, m=2, Ω={(x1,x2)T∣x12+x22<1}, ai(u1(t,x))=1, ω1(u1(t,x))=6.5u1(t,x), ω2(u2(t,x))=8.5u2(t,x), (Dis)=(1.22.32.21.5), (bij)=(-0.231.3-0.143.2), (cij)=(-0.1-0.20.25-0.13), fj(uj)=(2/4)(|uj+1|-|uj-1|), and τj(t)=(3/4)arctan(t).
Asλ1=5.783 and D_=1.2, we know χ=13.8792. Further, for li=2/2, a_i=ai¯=1, p1=6.5, and p2=8.5, we compute
(60)ρ=maxi=1,…,n(li2)∑i=1na¯i=1,λ=maxi=1,…,n(-χ-2a_ipi+a¯i∑j=1nbij2+a¯i∑j=1ncij2)+ρ=-15.5402.
Leth=4. Since λ=-15.5402<-4=-hρ, we therefore conclude from Theorem 5 that the zero solution of this system is globally exponential stable.Example 15.
Consider problems (1)–(4) with Pik(ui(tk,x))=arctan(0.5ui(tk,x)); moreover, n=2, m=2, Ω={(x1,x2)T∣x12+x12<1}, ai(u1(t,x))=1, ω1(u1(t,x))=6.5u1(t,x), ω2(u2(t,x))=8.5u2(t,x), (Dis)=(1.22.32.21.5), (bij)=(-0.231.3-0.143.2), (cij)=(-0.1-0.20.25-0.13), fj(uj)=(2/4)(|uj+1|-|uj-1|), τj(t)=(1/π)arctan(t), and tk=tk-1+k.
Asλ1=5.783 and D_=1.2, we know χ=13.8792. Further, for li=2/2,a_i=ai¯=1, p1=6.5,p2=8.5, and εi=1, we compute
(61)ρ=maxi=1,…,nε2∑i=1na¯i=1,λ=maxi=1,…,n(-χ-2a_ipi+a¯i∑j=1n(ε1bij2+ε2cij2))+maxi=1,…,n(li2)ε1∑i=1na¯i=-15.5402.
Letτ=0.5,h=4,μ=1,θik=0.5,  and α=1.5; we can find γ=2.4 such that
(62)γ+λ+hρeγτ=0.1403>0,λ+hρeγτ+ln(1+α)μ=-1.3434<0.
Therefore it is concluded from Theorem12 that the zero solution of this system is globally exponential stable.



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*Source: 409758-2013-03-28.xml*