# Asymptotic Stability and Exponential Stability of Impulsive Delayed Hopfield Neural Networks

Authors: Jing Chen; Xiaodi Li; Dequan Wang
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A criterion for the uniform asymptotic stability of the equilibrium point of impulsive delayed Hopfield
neural networks is presented by using Lyapunov functions and linear matrix inequality approach. The
criterion is a less restrictive version of a recent result. By means of constructing the extended impulsive Halanay
inequality, we also analyze the exponential stability of impulsive delayed Hopfield neural networks. Some new
sufficient conditions ensuring exponential stability of the equilibrium point of impulsive delayed Hopfield neural
networks are obtained. An example showing the effectiveness of the present criterion is given.


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## Body

## 1. Introduction

In the last several years, Hopfield neural networks (HNN) have received especially considerable attention due to their extensive applications in solving optimization problem, traveling salesman problem, and many other subjects in recent years [1–9]. In hardware implementation of neural networks, time delays are inevitably present due to the finite switching speeds of the amplifiers. Hence, it is vital to investigate the stability of delayed HNN. Recently, various results for the stability of delayed HNN are obtained via different approaches. In [3], Rakkiyappan and Balasubramaniam studied the exponential stability for fuzzy impulsive neural networks by utilizing the Lyapunov-Krasovskii functional and the linear matrix inequality approach. In [8], Li studied the global robust stability for stochastic interval neural networks with continuously distributed delays of neutral type based on the similar methods. In [9], Xia et al. derived some sufficient conditions for the synchronization problem of coupled identical Yang-Yang type fuzzy cellular neural networks with time-varying delays based on using the invariance principle of functional differential equations.On the other hand, impulsive differential equations have attracted a great deal of attention due to its potential applications in biological systems, chemical reactions, and various results are obtained; for instance, see [10–14]. Impulses can make unstable systems stable, and stable systems can become unstable after impulse effects. Hence, the stability properties of impulsive HNN with time delays have become an important topic of theoretical studies and have been investigated by many researchers; see [5, 6, 15–22]. In [5], Zhang and Sun obtained a result for the uniform stability of the equilibrium point of the impulsive HNN systems with time delays by using Lyapunov functions and analysis technique. In [6], global exponential stability of impulsive delay HNN is investigated by applying the piecewise continuous vector Lyapunov function.The purpose of this paper is to present some sufficient conditions for uniform asymptotic stability and global exponential stability of impulsive HNN with time delays by means of constructing the extended impulsive Halanay inequality which is different from that given in [23], Lyapunov functional methods, and linear matrix inequality approach. The results here are also discussed from the point of view of thier comparison with the earlier results. Our results improve and generalize the earlier results. At last, we discuss an example to illustrate the advantage of the results we obtained.


## 2. Systems Description and Preliminaries

Letℝ denote the set of real numbers, let ℝ+ denote the set of nonnegative real numbers, and let ℝn denote the n-dimensional real space equipped with the Euclidean norm ∥·∥.Consider the following impulsive and delayed HNN model:(1)xi′(t)=-cixi(t)+∑j=1n‍aijfj(xj(t))+∑j=1n‍bijgj(xj(t-τj(t)))+Ii,t≠tk,t≥t0,Δxi∣t=tk=xi(tk)-xi(tk-),i∈Λ,k=1,2,…,
where Λ={1,2,…,n}. n≥2 corresponds to the number of units in a neural network; the impulse times tk satisfy 0≤t0<t1<⋯<tk<⋯,limk→+∞tk=+∞; xi corresponds to the membrane potential of the unit i at time t; ci is positive constant; fj, gj denote, respectively, the measures of response or activation to its incoming potentials of the unit j at time t and t-τj(t); constant aij denotes the synaptic connection weight of the unit j on the unit i at time t; constant bij denotes the synaptic connection weight of the unit j on the unit i at time t-τj(t); Ii is the input of the unit i;τj(t) is the transmission delay of the jth neuron such that 0<τj(t)≤τ,t≥t0, and j∈Λ.Assume that system (1) is supplemented with initial conditions of the form
(2)x(s)=ϕ(s),s∈[t0-τ,t0],
where x(s)=(x1(s),x2(s),…,xn(s))T,  ϕ(s)=(ϕ1(s),ϕ2(s),…,ϕn(s))T∈PC([t0-τ,t0],ℝn), and PC(I,ℝn)={ψ:I→ℝn, which is continuous everywhere except at finite number of points tk, at which ψ(tk+) and ψ(tk-) exist and ψ(tk+)=ψ(tk)}. For any given t≥t0, ψ∈PC([t-τ,t],ℝn), the norm of ψ is defined by ∥ψ∥τ=supt-τ≤θ≤t|ψ(θ)|. For anyσ≥0, letPCδ(σ)={ψ∈PC([σ-τ,σ],ℝn):∥ψ∥<δ}.In this paper, we assume that some conditions are satisfied, so that the equilibrium point of (1) without impulse does exist denoted by x*=(x1*,x2*,…,xn*)T; see [2, 5]. Impulsive operator is viewed as perturbation of the equilibrium point of system (1) without impulsive effects. We assume that Δxi∣t=tk=xi(tk)-xi(tk-)=dk(i)(xi(tk-)-x*), dk(i)∈ℝ, and i∈Λ,  k=1,2,….Sincex* is an equilibrium point of (1), one can derive from (1) that the transformation yi=xi-xi*, i∈Λ, transforms system (1) into the following system:
(3)yi′(t)=-ciyi(t)+∑j=1n‍aijΩj(yj(t))+∑j=1n‍bijΓj(yj(t-τj(t))),t≠tk,t≥t0,yi(tk)=(1+dk(i))yi(tk-),i∈Λ,k=1,2,…,
where Ωj(yj(t))=fj(xj*+yj(t))-fj(xj*), Γj(yj(t))=gj(xj*+yj(t-τj(t)))-gj(xj*).Clearly,x* is uniformly asymptotically stable for system (1) if and only if the trivial solution of system (3) is uniformly asymptotically stable. Hence, we only need to prove the stability of the trivial solution of system (3).Remark 1.
Ifxi(tk)=dk(i)xi(tk-), then we cannot get yi(tk)=dk(i)yi(tk-) through the transformation yi=xi-xi*. So some of the results [5] are incorrect.The following notations will be used throughout the paper. The notationAT and A-1 means the transpose of and the inverse of a square matrix A. Let y(t)=(y1(t),y2(t),…,yn(t))T, yτ=(y1(t-τ1(t)), y2(t-τ2(t)),…,yn(t-τn(t)))T; C=diag[-c1,-c2,…,-cn]T, A=(aij)n×n, B=(bij)n×n, Dk=diag[1+dk(1),1+dk(2),…,1+dk(n)]T; Ω(y)=[Ω1(y1),Ω2(y2),…,Ωn(yn)]T, Γ(yτ)=[Γ1(y1(t-τ1(t))), Γ2(y2(t-τ2(t))),…,Γn(yn(t-τn(t)))]T. Then system (3) with initial condition becomes
(4)y′(t)=Cy(t)+AΩ(y(t))+BΓ(yτ),t≠tk,t≥t0,y(tk)=Dky(tk-),k=1,2,…,y(t0+θ)=φ(θ),θ∈[-τ,0],
where φ(θ)=x(t0+θ)-x*.We introduce some definitions as follows.Definition 2 (see [10]).
The functionV(t,s):[0,∞)×ℝn→ℝ+ belongs to class v0 if(A1)
V is continuous on each of the sets [tk-1,tk)×ℝn and lim(t,z)→(tk-,s)V(t,z)=V(tk-,s) exists;(A2)
V(t,s) is locally Lipschitzian in s and V(t,0)≡0.Definition 3 (see [10]).
LetV∈v0, for any (t,s)∈[tk-1,tk)×ℝn; the upper right-hand Dini derivative of V(t,s) along the solution of (4) is defined by
(5)D+V(t,ψ(0))=limsuph→0+{V(t+h,y(t,ψ)(t+h))-V(t,ψ(0))}h.Definition 4 (see [11]).
Assume thaty(t)=y(σ,φ)(t) is the solution of (4) through (σ,φ). Then the zero solution of (4) is said to be(1)
uniformly stable, if for anyσ≥t0 andε>0, there exists a δ=δ(ε)>0 such thatφ∈PCδ(σ) implies that ∥y(σ,φ)(t)∥<ε,t≥σ;(2)
uniformly asymptotically stable, if it is uniformly stable, and there exists aδ>0 such that for anyε>0, σ≥t0, there is a T=T(ε)>0 such that φ∈PCδ(σ) implies that ∥y(σ,φ)(t)∥<ε,t≥σ+T;(3)
globally exponentially stable, if for anyφ∈PC([-τ,0],ℝn), there exist constants M, μ>0 such that
(6)∥y(t)∥<M∥φ∥τe-μ(t-σ),t≥σ.In this paper, we always assume that the following assumption holds:(
H
0
) there exist constants M, N>0 such that
(7)ΩT(y)Ω(y)≤MyTy,ΓT(yτ)Γ(yτ)≤NyτTyτ.In addition, we have the following basic lemmas.Lemma 5 (see [24]).
For any vectorsa, b∈ℝn, the inequality
(8)±2aTb≤aTXa+bTX-1bholds, in whichX is any n×n matrix with X>0.Lemma 6 (see [25]).
Assume that there exist constantsP, Q>0 and m(t)∈PC([t0-τ,∞),ℝ+) such that(i)
fort=tk, m(tk)≤γkm(tk-), γk>0 are constants and satisfy maxk∈ℤ+{1/γk,1}<P/Q;(ii)
fort≥t0, t≠tk,
(9)D+m(t)≤-Pm(t)+Qm~(t),where m~(t)=supt-τ≤s≤tm(s).Then fort≥t0,
(10)m(t)≤m~(t0)(∏t0<tk≤t‍γk)e-λ(t-t0),
where λ satisfies the following inequality:
(11)0<λ≤P-Qmaxk∈ℤ+{1γk,1}·eλτ.


## 3. Main Results

In this section, we will establish some theorems which provide sufficient conditions for uniformly asymptotically stable and global exponential stability of system (1).Theorem 7.
The equilibrium point the system (1) is uniformly asymptotically stable, if there exists n×n symmetric, and positive definite matrix P satisfies the following conditions:(H1)
η¯≐∏k=1∞‍max{ηk,1}<∞, where ηk is the largest eigenvalue of P-1DkPDk;(H2)
λ3<(-M-N)/λ1, where λ1 is the smallest eigenvalue of P and λ3 is the largest eigenvalue of P-1(CP+PC+PAATP+PBBTP).Proof.
First, we will prove that the zero solution of system (4) is uniformly stable. For any ε>0, we may choose a δ>0 such that δ≤(λ1/η¯λ2)ε, where λ2 is the largest eigenvalue of P. For any σ≥t0,φ∈PCδ(σ), let y(t)=y(σ,φ)(t) be a solution of (4) through (σ,φ),σ≥t0 (for convenience, that we assume σ=t0); then we can prove that ∥y(t)∥<ε,t≥t0.
Consider the following Lyapunov function:V(t,y(t))=yT(t)Py(t); then we have
(12)λ1∥y(t)∥2≤V(t,y(t))≤λ2∥y(t)∥2.
By virtue of Lemma5, we obtain for t∈[tk,tk+1), k=1,2,…,
(13)D+V(t,y(t))|(4)=(yT(t))′Py(t)+yT(t)Py′(t)=(Cy(t)+AΩ(y(t))+BΓ(yτ))TPy(t)+yT(t)P(Cy(t)+AΩ(y(t))+BΓ(yτ))=yT(t)CPy(t)+ΩT(y(t))ATPy(t)+ΓT(yτ)BTPy(t)+yT(t)PCy(t)+yT(t)PAΩ(y(t))+yT(t)PBΓ(yτ)=yT(t)(CP+PC)y(t)+2ΩT(y(t))ATPy(t)+2ΓT(yτ)BTPy(t)≤yT(t)(CP+PC)y(t)+ΩT(y(t))Ω(y(t))+yT(t)PAATPy(t)+ΓT(yτ)Γ(yτ)+yT(t)PBBTPy(t)≤yT(t)(CP+PC+PAATP+PBBTP)y(t)+ΩT(y(t))Ω(y(t))+ΓT(yτ)Γ(yτ)≤λ3yT(t)Py(t)+MyT(t)y(t)+NyτTyτ≤λ3yT(t)Py(t)+Mλ1-1yT(t)Py(t)+Nλ1-1yτTPyτ≤[λ3+Mλ1-1]yT(t)Py(t)+Nλ1-1yτTPyτ.
First, it is obvious that fort0-τ≤t≤t0,
(14)λ1∥y(t)∥2≤V(t,y(t))≤λ2δ2≤η¯-1λ1ε2.
Then we can prove that fort∈[t0,t1),
(15)V(t,y(t))≤η¯-1λ1ε2.
Suppose that this is not true; then there existst^∈[t0,t1) such thatV(t^,y(t^))>η¯-1λ1ε2.
Set(16)tˇ=sup{t∣s∈[t0,t),V(s,y(s))≤η¯-1λ1ε2}.
It is obvious thattˇ<t^. Then it follows that(Ia)
V(t,y(t))≤η¯-1λ1ε2,t∈[t0,tˇ);(IIa)
V(tˇ,y(tˇ))=η¯-1λ1ε2;(IIIa)
for anyδ>0, there exists tδ∈(tˇ,tˇ+δ) such that V(tδ,y(tδ))>η¯-1λ1ε2.
So(17)V(tˇ,y(tˇ))=η¯-1λ1ε2≥V(t,yτ),tˇ-τ≤t≤tˇ.
In view of condition(H2), from (13), we obtain
(18)D+V(tˇ,y(tˇ))≤[λ3+Mλ1-1]yT(tˇ)Py(tˇ)+Nλ1-1yτTPyτ≤[λ3+Mλ1-1+Nλ1-1]V(tˇ,y(tˇ))<0,
which is a contradiction with (IIIa). Hence, (15) holds.
Considering(19)V(t1,y(t1))=yT(t1)Py(t1)=yT(t1-)D1PD1y(t1-)≤η1yT(t1-)Py(t1-)=η1V(t1-,y(t1-))≤η1η¯-1λ1ε2≤max{η1,1}η¯-1λ1ε2,
we will prove that fort∈[t1,t2),
(20)V(t,y(t))≤max{η1,1}η¯-1λ1ε2.
Suppose that this is false; then we can define(21)t→=sup{t∣s∈[t1,t),V(s,y(s))≤max{η1,1}η¯-1λ1ε2}.
Similarly, we can obtain(Ib)
V(t,y(t))≤max{η1,1}η¯-1λ1ε2,t∈[t1,t→);(IIb)
V(t→,y(t→))=max{η1,1}η¯-1λ1ε2;(IIIb)
for anyδ>0, there exists tδ∈(t→,t→+δ) such that V(tδ,y(tδ))>max{η1,1}η¯-1λ1ε2.
So(22)V(t→,y(t→))=max{η1,1}η¯-1λ1ε2≥V(t,yτ),t→-τ≤t≤t→.
In fact, ift→-τ≥t1, then it is obvious that inequality (22) holds. If t→-τ<t1, then V(t,yτ)≤η¯-1λ1ε2≤max{η1,1}η¯-1λ1ε2=V(t→,y(t→)). So, inequality (22) still holds.
Considering condition(H2), from (13), we obtain
(23)D+V(t→,y(t→))≤[λ3+Mλ1-1]yT(t→)Py(t→)+Nλ1-1yτTPyτ≤[λ3+Mλ1-1+Nλ1-1]V(t→,y(t→))<0,
which contradicts (IIIb). Hence, (20) holds.
By induction hypothesis, we may prove, in general, that fort∈[tm,tm+1),
(24)V(t,y(t))≤∏k=1m‍max{ηk,1}η¯-1λ1ε2;
that is,(25)V(t,y(t))≤∏t0<tk≤t‍max{ηk,1}η¯-1λ1ε2.
Finally, we arrive at(26)λ1∥y(t)∥2≤V(t,y(t))≤λ1ε2,t≥t0.
Therefore, we obtain∥y(t)∥<ε,t≥t0. In view of the choice of δ, the zero solution of (4) is uniformly stable; that is, the equilibrium point of (1) is uniformly stable.
Next we show the uniformly asymptotical stability. For any givenG>0, we find a corresponding δ>0 such that for any φ∈PCδ(σ) implies that ∥y(t)∥=∥y(t0,φ)(t)∥≤G, t≥σ=t0; that is, V(t,y(t))≤λ2G2.
For any smallε∈(0,G), we choose N→=N→(ε)∈ℤ+ such that
(27)N→λ12λ2ε<G≤(N→+1)λ12λ2ε,(28)N→>-Nλ3λ1+M+N.
In fact, it is feasible to choose small enoughε∈(0,G) such that N→ in (27) is large enough to satisfy (28).
Sinceη¯≐∏k=1∞‍max{ηk,1}<∞ implies that ∑k=1∞‍(max{ηk,1}-1)<∞, there exists sufficient large N⋆∈ℤ+ such that
(29)∑i=N⋆∞(max{ηi,1}-1)≤λ1ε26λ2G2,max{ηk,1}<16N→+1,k≥N⋆.
Let(30)w1=[1+∑k=1∞(max{ηk,1}-1)]λ2G2|[λ1λ3+M+N]N→+N|·2ε2+1.
Next we show that there existsT1∈[N⋆,N⋆+w1] such that
(31)V(T1,y(T1))<λ1ε22N→.
Or else, for allt∈[N⋆,N⋆+w1],
(32)V(t,y(t))≥λ1ε22N→.
Thus, we get(33)V(t,y(t))+λ1ε22≥λ1ε22(N→+1)≥λ2G2>V(ξ,yτ),t-τ≤ξ≤t.
From (13), we have
(34)D+V(t,y(t))|(4)≤[λ3+Mλ1-1]yT(t)Py(t)+Nλ1-1yτTPyτ≤[λ3+Mλ1-1]V(t,y(t))+Nλ1-1(V(t,y(t))+λ1ε22)≤[λ3+Mλ1-1+Nλ1-1]V(t,y(t))+ε22N≤[λ3+Mλ1-1+Nλ1-1]λ1ε22N→+ε22N≤{[λ3λ1+M+N]N→+N}ε22.
Integrating the above inequality fromN⋆ to N⋆+w1, we have
(35)V(N⋆+w1,y(N⋆+w1))≤V(N⋆,y(N⋆))+{[λ3λ1+M+N]N→+N}ε22w1+∑N⋆<tk<N⋆+w1[V(tk)-V(tk-)]≤V(N⋆,y(N⋆))+{[λ3λ1+M+N]N→+N}ε22w1+∑N⋆<tk<N⋆+w1[max{ηk,1}-1]V(tk-)≤λ2G2+{[λ3λ1+M+N]N→+N}ε22w1+∑N⋆<tk<N⋆+w1[max{ηk,1}-1]λ2G2≤{[λ3λ1+M+N]N→+N}ε22w1+{1+∑N⋆<tk<N⋆+w1[max{ηk,1}-1]}λ2G2<0,
which is a contradiction. So (31) holds. We may choose T1=N⋆+w1.
We next claim that for allt>T1,
(36)V(t,y(t))<(N→2+14)λ1ε2.
Suppose that this is not true; then there exists aτ2>T1 such that
(37)V(τ2,y(τ2))≥(N→2+14)λ1ε2,
and for T1<t<τ2,
(38)V(t,y(t))<(N→2+14)λ1ε2.
Suppose thatT1∈[tm,tm+1), m∈ℤ+. We claim that τ2≥tm+1. Otherwise, τ2∈[T1,tm+1). Since (31) holds, it is clear that there exists a τ1∈[T1,τ2) such that
(39)V(τ1,y(τ1))=N→2λ1ε2,V(τ1,y(τ1))≤V(t,x(t))≤V(τ2,x(τ2)),τ1≤t≤τ2.
Furthermore, we note that(40)V(t,y(t))+λ1ε22≥λ1ε22(N→+1)≥λ2G2>V(ξ,yτ),t-τ≤ξ≤t.
From (13), we have
(41)D+V(t,y(t))≤{[λ3λ1+M+N]N→+N}ε22<0,τ1≤t≤τ2,
which implies that
(42)V(τ2,y(τ2))≤V(τ1,y(τ1)).
This is a contradiction.
Hence, we obtainτ2≥tm+1; without loss of generality, we may suppose that τ2∈[tm+q,tm+q+1),q≥1. Next we first claim that there exists τ1>0 satisfying τ2>τ1>T1 such that
(43)V(τ1,y(τ1))≥N→2λ1ε2.
Suppose that this is false; then for allt∈(T1,τ2),
(44)V(t,y(t))<N→2λ1ε2,
which implies that τ2=tm+q in view of (37). Consequently, we have
(45)V(tm+q,y(tm+q))≥(N→2+14)λ1ε2,V(tm+q-,y(tm+q-))<N→2λ1ε2,
which implies that
(46)(max{ηm+q,1}-1)λ2G2≥(max{ηm+q,1}-1)V(tm+q-,y(tm+q-))>14λ1ε2.
Hence, we getmax{ηm+q,1}>1+(λ1ε2/4λ2G2), which contradicts (29). So (43) holds.
Therefore, there are two situationsτ2>tm+q and τ2=tm+q. Next we discuss them, respectively.(b1)
Ifτ2>tm+q, let
(47)τ1~=infτ1≤t<τ2{N←2t∣s∈[t,τ2],N→2λ1ε2≤V(s,y(s))<V(τ2,y(τ2))N←2}.
We first show thatτ1~<tm+q. Suppose on the contrary that τ1~∈[tm+q,τ2); then
(48)V(t,y(t))+λ1ε22≥λ1ε22(N→+1)V(t,y(t))+λ1ε22≥λ2G2>V(ξ,yτ),t-τ≤ξ≤t,t∈[τ1~,τ2].
From (13), we have
(49)D+V(t,y(t))≤{[λ3λ1+M+N]N→+N}ε22<0,τ1~≤t≤τ2,
which implies  that
(50)V(τ2,y(τ2))≤V(τ1~,y(τ1~));
which is a contradiction with the definition of τ1~. Thus we obtain that τ1~<tm+q. Suppose that τ1~∈[tm+k,tm+k+1),1≤k<q.
We also have two cases.(b1a)
Ifτ1~ is not impulsive point, that is to say τ1~>tm+k, then considering the definition of τ1~, we have
(51)V(τ1~,y(τ1~))=N→2λ1ε2.
By the same argument as the above mentioned, we obtain that (48) still holds.
Hence, from (13), we get
(52)(N→2+14)λ1ε2≤V(τ2,y(τ2))≤V(τ1~,y(τ1~))+∑i=m+k+1m+q‍[V(ti)-V(ti-)]≤N→2λ1ε2+∑i=m+k+1m+q(max{ηi,1}-1)V(ti-)≤N→2λ1ε2+∑i=m+k+1m+q(max{ηi,1}-1)λ2G2,
which implies that
(53)14λ1ε2≤∑i=m+k+1m+q(max{ηi,1}-1)λ2G2,
which is a contradiction with (29). So τ1~ is some impulsive point. (b1b)
Ifτ1~ is some impulsive point, that is to see τ1~=tm+k, then from the definition of τ1~, it is clear that
(54)V(τ1~-,y(τ1~-))=V(tm+k-,y(tm+k-))<N→2λ1ε2,which implies that
(55)V(τ1~,y(τ1~))=V(tm+k,y(tm+k))≤ηm+kV(tm+k-,y(tm+k-))<ηm+kN→2λ1ε2≤max{ηm+k,1}N→2λ1ε2.
On the other hand, note that inequality (48) still holds; from (13) and (29), we have
(56)(N→2+14)λ1ε2≤V(τ2,y(τ2))≤V(τ1~,y(τ1~))+∑i=m+k+1m+q[V(ti)-V(ti-)]<max{ηm+k,1}N→2λ1ε2+∑i=m+k+1m+q(max{ηi,1}-1)V(ti-),
which implies that
(57)14λ1ε2≤(max{ηm+k,1}-1)N→2λ1ε2+∑i=m+k+1m+q(max{ηi,1}-1)λ2G2≤16N→N→2λ1ε2+∑i=m+k+1m+q(max{ηi,1}-1)λ2G2.
That means(58)16λ1ε2<∑i=m+k+1m+q(max{ηi,1}-1)λ2G2,
which contradicts (29).
Hence, the first situationτ2>tm+q is impossible. (b2)
Ifτ2=tm+q, then by the same arguments as in the proof in (b1) and (43), we have
(59)V(τ2-,y(τ2-))>N→2λ1ε2.
Then let(60)τ1~=infτ1≤t<τ2{t∣s∈[t,τ2],N→2λ1ε2≤V(s,y(s))<V(τ2,y(τ2))N←2}.
The rest of the arguments are omitted. Finally we can find our desirable contradiction. Hence, (36) holds.
With above mentioned, the same arguments as before, if we replaceT1 with N⋆, then there exists a T2=T1+w1=N⋆+2w1 such that for t>T2,
(61)V(t,y(t))<N→2λ1ε2.
LetT2 replace T1; then there exists a T3=N⋆+3w1 such that for t>T3,
(62)V(t,y(t))<(N→2-14)λ1ε2.
By induction hypothesis, we may prove, in general, that there exists aT2N→=N⋆+2N→w1 such that t>T2N←,
(63)V(t,y(t))<(N→2-N→-22)λ1ε2=λ1ε2.
Therefore, we obtain that∥y(t)∥<ε,t>N⋆+2Nw1. In view of the choice of N⋆, N→, and w1, the zero solution of (4) is uniformly asymptotically stable; that is, the equilibrium point of (1) is uniformly asymptotically stable. The proof of Theorem 7 is therefore complete.Let|1+dk(i)|≠|1+dk(j)|, ci≠cj, i≠j, i,j∈Λ, k=1,2,…, in Theorem 7; then we can have the following result.Corollary 8.
The equilibrium point of system (1) is uniformly asymptotically stable, if there exists n×n symmetric and positive definite matrix P satisfying(I1)
η¯≐∏k=1∞‍max{maxi∈Λ(1+dk(i))2,1}<∞;(I2)
λ3<(-M-N)/λ1, where λ1 is the smallest eigenvalue of P and λ3 is the largest eigenvalue of 2C+AATP+BBTP.Remark 9.
For using the less conservative conditions in Theorem7, our results obviously improve some results established in the earlier references. In [5], condition y(tk)=Dy(tk-) holds for all k∈ℤ+; here note in our Theorem 7 that we only require that the solutions satisfy the hypothesis (H1) at impulsive points. In addition, our conditions are without requirement of the range of the largest eigenvalues of P-1DTPD on (0,1), which are milder than the restrictions in [5].By utilizing Lemma6, we will give some sufficient conditions for globally exponential stability of the equilibrium point of system (1).Theorem 10.
Assume that there existsn×n symmetric and positive definite matrix P such that λ1>0 is the smallest eigenvalue of P, λ3 is the largest eigenvalue of P-1(CP+PC+PAATP+PBBTP), ηk is the largest eigenvalue of P-1DkPDk, and λ1, λ3 and ηk satisfy the following conditions: (H3)
one has(64)λ3+Mλ1+Nλ1·maxk∈ℤ+{1ηk,1}<0,(H4)
there exist constants𝕌(>0), δ(≥0) such that δ<μ and the following inequality
(65)∑k=1mlnηk-δ(tm-t0)<𝕌∀m∈ℤ+holds,where
μ
satisfies the following inequality:
(66)
0
<
μ
≤
-
λ
3
-
M
λ
1
-
N
λ
1
max
k
∈
ℤ
+
{
1
η
k
,
1
}
·
e
μ
τ
.Then the equilibrium point of the system (1) is globally exponentially stable, and the exponential convergence rate is equal to (
μ
-
δ
)/2.Proof.
Lety(t)=y(σ,φ)(t) be a solution of (4) through (σ,φ), σ≥t0 (for convenience, we assume that σ=t0). We next will prove that the zero solution of (4) is globally exponentially stable. Construct the Lyapunov functional as in Theorem 7; that is, V(t,y(t))=yT(t)Py(t); then we obtain the following:(1)
λ1∥y(t)∥2≤V(t,y(t))≤λ2∥y(t)∥2, where λ2 is the largest eigenvalue of P;(2)
fort∈[tk,tk+1), k=1,2,…,
(67)D+V(t,y(t))|(4)≤[λ3+Mλ1-1]yT(t)Py(t)+Nλ1-1yτTPyτ;(3)
V(tk,y(tk))≤ηkV(tk-,y(tk-)).
From(2), we have for t∈[tk,tk+1), k=1,2,…,
(68)D+V(t,y(t))|(4)≤-PV(t,y(t))+QV~(t,y(t)),
where V~(t)=supt-τ≤s≤tV(s),   P=-λ3-Mλ1-1, and Q=Nλ1-1.
For anyt≥t0, suppose that t∈[tm,tm+1), m≥0. By Lemma 6 and condition (H4), we obtain
(69)V(t)≤V~(t0)(∏k=1m‍ηk)e-μ(t-t0)≤V~(t0)e𝕌·eδ(tm-t0)·e-μ(t-t0)≤V~(t0)e𝕌·eδ(t-t0)·e-μ(t-t0)≤λ2e𝕌∥φ∥τ2e-(μ-δ)(t-t0),
where μ satisfies inequality (66).
Hence, we obtain for anyt≥t0,
(70)∥y(t)∥≤e(1/2)𝕌λ2λ1∥φ∥τe-((μ-δ)/2)(t-t0),t≥t0,
where μ satisfies inequality (66).
Therefore, the zero solution of system (4) is globally exponentially stable; that is, the equilibrium point of system (1) is globally exponentially stable. The proof of Theorem 10 is complete.Remark 11.
In Theorem10, if supn∈ℤ+(∏k=1n‍ηk)<∞, then we can choose δ=0 in condition (H3).
Letηk∈(0,1],k=1,2,…, in Theorem 7; then we can have the following result.Corollary 12.
Assume that there existsn×n symmetric and positive definite matrix P such that λ1 is the smallest eigenvalue of P, λ3 is the largest eigenvalue of P-1(CP+PC+PAATP+PBBTP), ηk is the largest eigenvalue of P-1DkPDk, and condition
(71)λ1λ3+M+N<0holds.Then the equilibrium point of the system (1) is globally exponentially stable, and the exponential convergence rate is equal to
μ
/
2
, where
μ
satisfies the following inequality:
(72)
0
<
μ
≤
-
λ
3
-
M
λ
1
-
N
λ
1
max
k
∈
ℤ
+
η
k
·
e
μ
τ
.


## 4. Example

In this section, we present a numerical example to illustrate our results.Example 1.
We consider Example  1 in [2] as follows:
(73)u1′(t)=-2.5u1(t)-0.5f(u1(t))+0.1f(u2(t))-0.1f(u1(t-τ))+0.2f(u2(t-τ))-1,u2′(t)=-2u1(t)+0.2f(u1(t))-0.1f(u2(t))+0.2f(u1(t-τ))+0.1f(u2(t-τ))+4,
with impulses
(74)dk(1)=1+12k2-1,dk(2)=1+1k2-1,tk=k,k∈ℤ+.
The delayed feedback matrixesA, B, and C are
(75)C=(-2.500-2),A=(-0.50.10.2-0.1),B=(-0.10.20.20.1).
Since the activation function in [2] is described by f=fi=0.5(|x+1|-|x-1|), i=1,2, then we have M=N=1. We may choose P=E (unit matrix); then λ1=λ2=1. Note that
(76)P-1DkPDk=Dk2=(1+12k2001+1k2),P-1(CP+PC+PAATP+PBBTP)=2C+AAT+BBT=(-4.69-0.11-0.11-3.9)=Δ.
then we get ηk=1+1/k2, η¯=∏k=1∞‍(1+1/k2)<∞. Matrix Δ,s characteristic equation is
(77)λ2+8.59λ+18.2789=0.By a straightforward calculation, we obtain that the largest characteristic rootλ3≈-3.885<-2. By Theorem 7, the equilibrium point of system (73) is uniformly asymptotically stable with impulses (74) for any τ>0, which is shown in Figure 1(a).(a) State trajectories of system (73) with impulses (74) and τ=3. (b) State trajectories of system (73) with impulses (79) and τ=0.17.
(a)
(b)However, the criteria in [5] are invalid here. In fact, condition y(tk)=Dy(tk-) is not satisfied here. Moreover, because of the impulsive effect, the criteria in [2] are also invalid here. Therefore, our results are less conservative than those given result in [2, 5].Furthermore, letτ=0.17. Note that ∏k=1∞‍(1+1/k2)<∞, λ3=-3.885<-2; then one can choose δ=0 in Theorem 10. So all conditions of Theorem 10 are satisfied. Therefore, the equilibrium point of system (73) is globally exponentially stable, and the exponential convergence rate is equal to μ, where μ>0 satisfies
(78)μ+e0.17μ-2.885<0.Ifdk(1), dk(2) in above example are given as follows:
(79)dk(1)={2.2-1,k=2n-1,0.23-1,k=2n,n∈ℤ+,dk(2){1.7-1,k=2n-1,0.4-1,k=2n,n∈ℤ+,
then we finally get λ3=-3.885<-3.500, δ=0. So all conditions of Theorem 10 are still satisfied. Therefore, the equilibrium point of system (73) is globally exponentially stable with impulses (79), and the exponential convergence rate is equal to μ, where μ>0 satisfies
(80)μ+2.5e0.17μ-2.885<0.The simulation is shown in Figure1(b). However, it is easy to check that the impulsive delayed Halanay inequality in [23, 26] is not feasible here. Thus our results can be applied to the case not covered in [23, 26].


## 5. Conclusion

The uniform asymptotic stability and global exponential stability of impulsive HNN with time delays are considered in this paper. Some new stability conditions are obtained by means of constructing the extended impulsive Halanay inequality, Lyapunov functional methods, and linear matrix inequality approach. Moreover, our results can be applied to the case not covered in some other existing criteria. Hence, the results extend and improve the earlier publications. An example is given to illustrate the feasibility of the results and the effects of impulses.



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*Source: 638496-2013-09-25.xml*