# Pth Moment Exponential Stability of Impulsive Stochastic Neural

Authors: Xiaoai Li; Jiezhong Zou; Enwen Zhu
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

This paper investigates the problem ofpth moment exponential stability for a class of stochastic neural networks with time-varying delays and distributed delays under nonlinear impulsive perturbations. By means of Lyapunov functionals, stochastic analysis and differential inequality technique, criteria on pth moment exponential stability of
this model are derived. The results of this paper are completely new and complement
and improve some of the previously known results (Stamova and Ilarionov (2010), Zhang et al. (2005), Li (2010), Ahmed and Stamova (2008), Huang et al. (2008), Huang et al. (2008), and Stamova (2009)). An example is employed to illustrate our feasible results.


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## Body

## 1. Introduction

The dynamics of neural networks have drawn considerable attention in recent years due to their extensive applications in many fields such as image processing, associative memories, classification of patters, and optimization. Since the integration and communication delays are unavoidably encountered in biological and artificial neural systems, it may result in oscillation and instability. The stability analysis of delayed neural networks has been extensively investigated by many researchers, for instance, see [1–30].In real nervous systems, there are many stochastic perturbations that affect the stability of neural networks. The result in Mao [24] suggested that one neural network could be stabilized or destabilized by certain stochastic inputs. It implies that the stability analysis of stochastic neural networks has primary significance in the design and applications of neural networks, such as [7, 12–16, 18, 20, 22–24, 26, 27, 30].On the other hand, it is noteworthy that the state of electronic networks is often subjected to some phenomenon or other sudden noises. On that account, the electronic networks will experience some abrupt changes at certain instants that in turn affect dynamical behaviors of the systems [5, 6, 17–23, 28, 29]. Therefore, it is necessary to take both stochastic effects and impulsive perturbations into account on dynamical behaviors of delayed neural networks [18, 20, 22, 23].Very recently, Li et al. [22] have employed the properties of M-cone and inequality technique to investigate the mean square exponential stability of impulsive stochastic neural networks with bounded delays. Wu et al. [23] studied the exponential stability of the equilibrium point of bounded discrete-time delayed dynamic systems with linear impulsive effects by using Razumikhin theorems. To the best of authors’ knowledge, however, few authors have considered the pth moment exponential stability of impulsive stochastic neural networks with mixed delays.Motivated by the discussions above, our object in this paper is to present the sufficient conditions ensuringpth moment exponential stability for a class of stochastic neural networks with time-varying delays and distributed delays under nonlinear impulsive perturbations by virtue of Lyapunov method, inequality technique and Itô formula. The results obtained in this paper generalize and improve some of the existing results [5, 8, 18, 19, 26–28]. The effectiveness and feasibility of the developed results have been shown by a numerical example.


## 2. Model Description and Preliminaries

Letℝ denote the set of real numbers, ℝn the n-dimensional real space equipped with the Euclidean norm |·|, ℤ+ the set of nonnegative integral numbers. E(·) stands for the mathematical expectation operator. ℒ denotes the well-known ℒ-operator given by the Itô formula. w(t)=(w1(t),…,wm(t)) is m-dimensional Brownian motion defined on a complete probability space (Ω,ℱ,P) with a natural filtration {ℱt}t≥0 generated by {w(s):0≤s≤t}, where we associate Ω with the canonical space generated by w(t) and denote by ℱ the associated σ-algebra generated by w(t) with the probability measure  P.  Let σ(t,x,y)=(σil(t,xi,yi))n×m∈ℝn×m, and σi(t,xi,yi) be ith row vector of σ(t,x,y).In [5, 6], the researchers investigated the following impulsive neural networks with time-varying delays:
(2.1)x˙i(t)=-ai(xi(t))+∑j=1n‍bijfj(xj(t))+∑j=1n‍cijgj(xj(t-τj(t)))+Ii,t≠tk,xi(tk)=pik(x(tk-)),k∈ℤ+,i∈Λ.
The authors in [7, 26, 27] studied the stochastic recurrent neural networks with time-varying delays:
(2.2)dxi(t)=(-ai(xi(t))+∑j=1n‍bijfj(xj(t))+∑j=1n‍cijgj(xj(t-τj(t)))+Ii)dt+∑l=1m‍σil(t,xi(t),xi(t-τi(t)))dwl(t).In this paper, we will study the generalized stochastically perturbed neural network model with time-varying delays and distributed delays under nonlinear impulses defined by the state equations:(2.3)dxi(t)=[-ai(xi(t))+∑j=1n‍bijfj(xj(t))+∑j=1n‍cijgj(xj(t-τj(t)))+∑j=1n‍dij∫-∞t‍Kij(t-s)·hj(xj(s))ds+Ii]dt+∑l=1m‍σil(t,xi(t),xi(t-τi(t)))dwl(t),t≠tk,xi(tk)=pik(x(tk-)),k∈ℤ+,i∈Λ,
where Λ={1,2,…,n}, the time sequence {tk} satisfies 0=t0<t1<t2<⋯<tk<tk+1⋯, limk→∞tk=∞; x(t)=(x1(t),x2(t),…,xn(t))T and xi(t) corresponds to the state of the ith unit at time t; bij, cij,  and dij denote the constant connection weight; τj(t) is the time-varying transmission delay and satisfies 0≤τj(t)≤τj,0<ηj=inft∈R+{1-τ˙j(t)}, for j∈Λ.  fj(·),  gj(·),  hj(·) denote the activation functions of the jth neuron; the delay kernel Kij(·) is the real-valued nonnegative piecewise continuous functions defined on [0,∞);  n corresponds to the numbers of units in a neural network; Ii denotes the external bias on the ith unit; pik(x(tk)) represents the abrupt change of the state xi(t) at the impulsive moment tk.System (2.3) is supplemented with initial condition given by
(2.4)xi(s)=φi(s),s∈(-∞,0],i∈Λ,
where φ(s)=(φ1(s),φ2(s),…,φn(s))T∈PCBℱ0b((-∞,0],ℝn)=˙BPCBℱ0b. Denote by PCBℱ0b the family of all bounded ℱ0-measurable, PC((-∞,0],ℝn)-value random variables φ, satisfying supθ∈(-∞,0]E|φ(θ)|p<∞, where PC((-∞,0],ℝn)={φ:[-∞,0]→ℝn} is continuous everywhere except at finite number of points tk,  at which φ(tk+) and φ(tk-) exist and φ(tk+)=φ(tk).The norms are defined by the following norms, respectively:(2.5)∥φ∥p=sups∈(-∞,0](∑i=1n‍|φi(s)|p)1/p,∥x∥p=(∑i=1n‍|xi|p)1/p.Throughout this paper, the following standard hypothesis are needed.(H1) Functionsai(·):ℝ→ℝ are continuous and monotone increasing, that is, there exist real numbers ai>0 such that
(2.6)ai(u)-ai(v)u-v≥ai,
for all u,v∈ℝ,  u≠v,  i∈Λ.(H2)  Functionsfj, gj,andhj are Lipschitz-continuous on ℝ with Lipschitz constants Ljf>0,Ljg>0,  and Ljh>0, respectively. That is,
(2.7)|fj(u)-fj(v)|≤Ljf|u-v|,|gj(u)-gj(v)|≤Ljg|u-v|,|hj(u)-hj(v)|≤Ljh|u-v|,
for all u,v∈ℝ,  i∈Λ.(H3)  The delay kernelsKij:[0,∞)→ℝ+ satisfy
(2.8)Kij(s)≤𝒦(s)∀i,j∈Λ,s∈[0,∞),∫0∞‍𝒦(s)eμ0sds<∞,
where 𝒦(s):[0,∞)→ℝ+ is continuous and integrable, and the constant μ0 denotes some positive number.(H4)  There exist nonnegative constantsei, li such that
(2.9)[σi(t,u′,v′)-σi(t,u,v)][σi(t,u′,v′)-σi(t,u,v)]T≤ei|u′-u|2+li|v′-v|2,
for all u,v,u′,v′∈ℝ,  i∈Λ.(H5)  There exist nonnegative matrixesPk=(pijk)n×n such that
(2.10)|pik(u1,u2,…,un)-pik(v1,v2,…,vn)|p≤∑j=1n‍pijk|uj-vj|p,
for any (u1,u2,…,un)T,(v1,v2,…,vn)T∈ℝn, where p≥1 is an integer.We end this section by introducing three definitions.Definition 2.1.
A constant vectorx*=(x1*,x2*,…,xn*)T∈ℝn is said to be an equilibrium point of system (2.3) if x* is governed by the algebraic system
(2.11)ai(xi*)=∑j=1n‍bijfj(xj*)+∑j=1n‍cijgj(xj*)+∑j=1n‍dij∫-∞t‍Kij(t-s)hj(xj*)ds+Ii,
where it is assumed that impulse functions pik(·) satisfy pik(x1*,x2*,…,xn*)=xi* for all i∈Λ and σ(t,xi*,xi*)=0.Definition 2.2.
The equilibriumx*=(x1*,x2*,…,xn*)T∈ℝn of system (2.3) is said to be pth moment exponentially stable if there exist λ>0 and M≥1 such that
(2.12)E∥x(t)-x*∥p≤E∥φ-x*∥pMe-λtfort≥0,
where x(t) is an any solution of system (2.3) with initial value φ∈PCBF0b((-∞,0],ℝn).Definition 2.3 (Forti and Tesi, 1995 [25]).
A mapH:ℝn→ℝn is a homeomorphism of ℝn onto itself if H is continuous and one-to-one, and its inverse map H-1 is also continuous.


## 3. Main Result

For convenience, we denote that(3.1)ϕi=pai-∑j=1n‍∑l=1p-1‍(|bij|pαl,ijLjfpβl,ij+|cij|pγl,ijLjgpδl,ij+|dij|pξl,ijLjhpηl,ij∫0∞‍𝒦(s)ds)-∑j=1n‍μjμi|bji|pαp,jiLifpβp,ji,ψi=∑j=1n‍μjμi|cji|pγp,jiLigpδp,ji,Φi=ϕi-p(p-1)2ei-(p-1)(p-2)2li,Ψi=ψi+(p-1)li,ζij=μiμj|dij|pξp,ijLjhpηp,ij,μ-=maxi{μi},μ_=mini{μi},
where μi are positive constant, αl,ij, βl,ij, γl,ij, δl,ij, ξl,ij, and ηl,ij are real numbers and satisfy
(3.2)∑l=1p‍αl,ij=1,∑l=1p‍βl,ij=1,∑l=1p‍γl,ij=1,∑l=1p‍δl,ij=1,∑l=1p‍ξl,ij=1,∑l=1p‍ηl,ij=1.Lemma 3.1.
Ifai(i=1,2,…,p) denote p nonnegative real numbers, then
(3.3)a1a2⋯ap≤a1p+a2p+⋯+appp,
where p≥1 denotes an integer.
A particular form of (3.3), namely,
(3.4)a1p-1a2≤(p-1)a1pp+a2pp,forp=1,2,3,….Lemma 3.2.
IfH:ℝn→ℝn is a continuous function and satisfies the following conditions. (1)
H(x) is injective on ℝn, that is, H(x)≠H(y) for all x≠y.(2)
∥H(x)∥→∞ as ∥x∥→∞.
Then,H(x) is homeomorphism of ℝn.Theorem 3.3.
System (2.3) exists a unique equilibrium x* under the assumptions (H1)–(H3) if the following condition is also satisfied:
(H6)ϕi>ψi+∑j=1nζji∫0∞‍𝒦(s)ds.Proof.
Defining a mapH(x)=(h1(x),h2(x),…,hn(x))T∈C0(ℝn,ℝn), where
(3.5)hi(x)=-ai(xi)+∑j=1n‍bijfj(xj)+∑j=1n‍cijgj(xj)+∑j=1n‍dij∫-∞t‍Kij(t-s)hj(xj)ds+Ii,
the map H is a homeomorphism on ℝn if it is injective on ℝn and satisfies ∥H(x)∥→∞ as ∥x∥→∞.
In the following, we will prove thatH(x) is a homeomorphism.
Firstly, we claim thatH(x) is an injective map on ℝn. Otherwise, there exist xT,yT∈ℝn, and xT≠yT such that H(x)=H(y), then
(3.6)ai(xi)-ai(yi)=∑j=1n‍bij[fj(xj)-fj(yj)]+∑j=1n‍cij[gj(xj)-gj(yj)]+∑j=1n‍dij∫-∞t‍Kij(t-s)[hj(xj)-hj(yj)]ds.
It follows from (H1)–(H3) that
(3.7)ai|xi-yi|≤∑j=1n‍|bij|Ljf|xj-yj|+∑j=1n‍|cij|Ljg|xj-yj|+∑j=1n‍|dij|Ljh|xj-yj|∫-∞t‍𝒦(t-s)ds.
Therefore,
(3.8)∑i=1n‍pμiai|xi-yi|p≤∑i=1n‍∑j=1n‍pμi|bij|Ljf|xi-yi|p-1|xj-yj|+∑i=1n‍∑j=1n‍pμi|cij|Ljg|xi-yi|p-1|xj-yj|+∑i=1n‍∑j=1n‍pμi|dij|Ljh|xi-yi|p-1|xj-yj|∫-∞t‍𝒦(t-s)ds≤∑i=1n‍∑j=1n‍μi(∑l=1p-1‍|bij|pαl,ijLjfpβl,ij|xi-yi|p+|bij|pαp,ijLjfpβp,ij|xj-yj|p)+∑i=1n‍∑j=1n‍μi(∑l=1p-1‍|cij|pγl,ijLjgpδl,ij|xi-yi|p+|cij|pγp,ijLjgpδp,ij|xj-yj|p)+∑i=1n‍∑j=1n‍μi∫-∞t‍𝒦(t-s)ds∑l=1p-1‍|dij|pξl,ijLjhpηl,ij|xi-yi|p+∑i=1n‍∑j=1n‍μi∫-∞t‍𝒦(t-s)ds|dij|pξp,ijLjhpηp,ij|xj-yj|p=∑i=1n‍μi(pai-ϕi+ψi+∑j=1n‍ζji∫0∞‍𝒦(s)ds)|xi-yi|p.
From (H6), it leads to a contradiction with our assumption. Therefore, H(x) is an injective map on ℝn.
To demonstrate the property∥H(x)∥→∞ as ∥x∥→∞, we have
(3.9)∑i=1n‍sgn(xi)pμi(hi(x)-hi(0))|xi|p-1=∑i=1n‍sgn(xi)pμi|xi|p-1(-ai(xi)+∑j=1n‍bijfj(xj)+∑j=1n‍cijgj(xj))+∑i=1n‍sgn(xi)pμi|xi|p-1∑j=1n‍dij∫-∞t‍Kij(t-s)hj(xj)ds≤-∑i=1n‍paiμi|xi|p+∑i=1n‍μi(pai-ϕi+ψi+∑j=1n‍ζji∫0∞‍𝒦(s)ds)|xi|p=-∑i=0n‍μi(ϕi-ψi-∑j=1n‍ζji∫0∞‍𝒦(s)ds)|xi|p≤-ω∥x∥pp,
where ω=min1≤i≤n{μi(ϕi-ψi-∑j=1nζji∫0∞‍𝒦(s)ds)}. Then, we have
(3.10)ω∥x∥pp≤pμ-∑i=1n‍|hi(x)-hi(0)||xi|p-1.
Using the Hölder inequality, we obtain
(3.11)∥x∥pp≤pμ-ω(∑i=1n‍|xi|p)1-1/p(∑i=1n‍|hi(x)-hi(0)|p)1/p,
which leads to
(3.12)∥x∥p≤pμ-ω(∥H(x)∥p+∥H(0)∥p).
From (3.12), we see that ∥H(x)∥→∞ as ∥x∥→∞. Thus, the map H(x) is a homeomorphism on ℝn under the sufficient condition (H6), and hence it has a unique fixed point x*. This fixed point is the unique solution of the system (2.3). The proof is now complete.To establish some sufficient conditions ensuring thepth moment exponential stability of the equilibrium point of x* of system (2.3), we transform x* to the origin by using the transformation yi(t)=xi(t)-x* for i∈Λ. Then system (2.3) can be rewritten as the following form:
(3.13)dyi(t)=[-a~i(yi(t))+∑j=1n‍bijf~j(yj(t))+∑j=1n‍cijg~j(yj(t-τj(t))+∑j=1n‍dij∫-∞t‍Kij(t-s)h~j(yj(s))ds]dt+∑l=1m‍σ~il(t,yi(t),yi(t-τi(t))dwl(t),t≠tk,yi(tk)=p~ik(y(tk-)),k∈Z+,i∈Λ,
where
(3.14)a~i(yi(t))=ai(yi(t)+xi*)-ai(xi*),f~j(yj(t))=fj(yj(t)+xj*)-fj(xj*),g~j(yj(t-τj(t)))=gj(yj(t-τj(t))+xj*)-gj(xj*),h~j(yj(s))=hj(yj(s)+xj*)-hj(xj*),σ~ij(t,yj(t),yj(t-τj(t)))=σij(t,yj(t)+xj*,yj(t-τj(t))+xj*)-σij(t,xj*,xj*),p~ik(y(tk))=pik(y(tk)+x*)-pik(x*).
In order to obtain our results, the following assumptions are necessary.(H7)  Whenηi≥1,  Φi>Ψi+∑j=1nζji∫0∞‍𝒦(s)ds for any i∈Λ.When0<ηi≤1,  Φi>(Ψi/ηi)+∑j=1nζji∫0∞‍𝒦(s)ds for any i∈Λ.(H8)  There exist constantλ∈(0,μ0) and α∈[0,λ), ρk=maxi∈Λ{1,∑j=1n(μj/μi)pjik}≤eα(tk-tk-1) such that
(3.15)λ<Φi-Ψiηieλτ-∑j=1n‍ζji∫0∞‍𝒦(s)eλsds∫0∞‍𝒦(s)eλssds≤∫0∞‍𝒦(s)eμ0sds.Theorem 3.4.
Assume that (H1)–(H5) and (H7)-(H8) hold, then system (2.3) exists a unique equilibrium x*, and the equilibrium point is pth moment exponentially stable.Proof.
From (H7), if0<ηi≤1, then
(3.16)ϕi>Φi>Ψiηi+∑j=1n‍ζji∫0∞‍𝒦(s)ds>ψiηi+∑j=1n‍ζji∫0∞‍𝒦(s)ds>ψi+∑j=1n‍ζji∫0∞‍𝒦(s)ds,
if ηi≥1,
(3.17)ϕi>Φi>Ψi+∑j=1n‍ζji∫0∞‍𝒦(s)ds>ψi+∑j=1n‍ζji∫0∞‍𝒦(s)ds,
for any i∈Λ.
Then regardless of cases, (H6) and(3.18)Φi>Ψiηi+∑j=1n‍ζji∫0∞‍𝒦(s)ds
are satisfied. Therefore, system (2.3) exists a unique equilibrium point x*.
Now, we define(3.19)ui(t,y(t))=eλtμi|yi(t)|p=ui(t),U(t,y(t))=∑i=1n‍ui(t).
For t≠tk, k∈ℤ+, we obtain
(3.20)ℒU=Ut+12trace[σ~T(t,yi(t),yi(t-τi(t)))Uyiyi(t,y)σ~(t,yi(t),yi(t-τi(t)))]+∑i=1n‍Uyi(-a~i(yi(t))+∑j=1n‍bijf~j(yj(t))+∑j=1n‍cijg~j(yj(t-τj(t))))+∑i=1n‍∑j=1n‍Uyidij∫-∞t‍Kij(t-s)h~j(yj(s))ds≤∑i=1n‍(λ-pai+p(p-1)2ei)ui(t)+peλt{∑i=1n‍∑j=1n‍μi|bij|Ljf|yi(t)|p-1|yj(t)|+∑i=1n‍∑j=1n‍μi[(p-1)2|cij|Ljg|yi(t)|p-1|yj(t-τj(t))|+∑i=1n‍∑j=1n‍μi+(p-1)2li|yi(t)|p-2|yi(t-τi(t))|2]+∑i=1n‍∑j=1n‍μi|dij|Ljh|yi(t)|p-1∫0∞‍𝒦(s)|yj(t-s)|ds}≤∑i=1n‍(λ-pai+p(p-1)2ei)ui(t)+p(p-1)2eλt∑i=1n‍μili|yi(t)|p-2|yi(t-τi(t))|2+eλt∑i=1n‍∑j=1n‍μi(∑l=1p-1‍|bij|pαl,ijLjfpβl,ij|yi(t)|p+|bij|pαp,ijLjfpβp,ij|yj(t)|p)+eλt∑i=1n‍∑j=1n‍μi(∑l=1p-1‍|cij|pγl,ijLjgpδl,ij|yi(t)|p+|cij|pγp,ijLjgpδp,ij|yj(t)|p)+eλt∑i=1n‍∑j=1n‍μi∫0∞‍𝒦(s)(∑l=1p-1‍|dij|pξl,ijLjhpηl,ij|yi(t)|p+|dij|pξl,ijLjhpηp,ij|yj(t-s)|p)ds≤∑i=1n‍[∑j=1n‍∑l=1p-1‍(|bij|pαl,ijLjfpβl,ij+|cij|pγl,ijLjgpδl,ij+|dij|pξl,ijLjhpηl,ij∫0∞‍𝒦(s)ds)]ui(t)+∑i=1n‍(λ-pai+p(p-1)2ei+(p-1)(p-2)2li+∑j=1n‍μjμi|bji|pαp,jiLjfpβp,ji)ui(t)+∑i=1n‍∑j=1n‍μiμj|dij|pξp,ijLjhpηp,ij∫0∞‍𝒦(s)eλsuj(t-s)ds+∑i=1n‍eλτi(t)(∑j=1n‍μjμi|cji|pγp,jiLigpδp,ji+(p-1)li)ui(t-τi(t))=∑i=1n‍((λ-Φi)ui(t)+Ψieλτi(t)ui(t-τi(t)))+∑i=1n‍∑j=1n‍ζij∫0∞‍𝒦(s)eλsuj(t-s)ds.
while Lemma 3.1 is used in the second inequality. Let
(3.21)V(t,y(t))=U(t,y(t))+∑i=1n‍Ψieλτi∫t-τi(t)t‍ui(s)1-τ˙i(ψi-1(s))ds+∑i=1n‍∑j=1n‍ζij∫0∞‍𝒦(s)eλs∫t-st‍uj(r)drds,
for t≥0, where ψi(s)=s-τi(s), applying Ito^ formula to (3.21), we can get
(3.22)ℒV=ℒU+∑i=1n‍Ψieλτi(ui(t)1-τ˙i(ψi-1(t))-ui(t-τi(t))1-τ˙i(t)(t-τi(t))′)+∑i=1n‍∑j=1n‍ζij∫0∞‍𝒦(s)eλs(uj(t)-uj(t-s))ds≤-∑i=1n‍{Φi-λ-Ψiηieλτ-∑j=1n‍ζji∫0∞‍𝒦(s)eλsds}ui(t).
From (H8), we have
(3.23)ℒV<0.
On the other hand, we have
(3.24)EU(tk,y(tk))=Eeλtk∑i=1n‍μi|yi(tk)|p≤Eeλtk∑i=1n‍μi∑j=1n‍pijk|yj(tk-)|p≤ρkEU(tk-,y(tk-)),(3.25)EV(tk,y(tk))=EU(tk,y(tk))+E(∑i=1n‍Ψieλτi∫tk-τi(tk)tk‍ui(s)1-τ˙i(ψi-1(s))ds)+E(∑i=1n‍∑j=1n‍ζij∫0∞‍𝒦(s)eλs∫tk-stk‍uj(r)drds)≤ρkEV(tk-,y(tk-)),
for t∈[tk-1,tk),  k∈Z+, by (3.23), (3.25), and (H8), we have
(3.26)EU(t,y(t))≤EV(t,y(t))≤EV(tk-1,y(tk-1))≤ρk-1EV(tk-1-,y(tk-1-))≤ρ0ρ1⋯ρk-1EV(t0,y(t0))≤eαt1⋯eα(tk-1-tk-2)EV(t0,y(t0))≤eαtEV(0,y(0)),
where ρ0=1.
On the other hand, we observe that(3.27)V(0,y(0))≤∑i=1n‍μi|yi(0)|p+∑i=1n‍Ψieλτi∫-τ0‍ui(s)1-τ˙i(ψi-1(s))ds+∑i=1n‍∑j=1n‍ζij∫0∞‍𝒦(s)eλs∫-s0‍uj(r)drds≤∑i=1n‍μi(1+∑j=1n‍ζji∫0∞‍𝒦(s)eλssds+Ψiτηieλτi)sup-∞<s≤0|yi(s)|p.
It follows that
(3.28)E(∑i=1n‍|yi(t)|p)≤Me-(λ-α)tsup-∞<s≤0∑i=1n‍|yi(s)|p,
for t≥0, where
(3.29)1≤M=μ-μ_(1+maxi∈Λ{∑j=1n‍ζji∫0∞‍𝒦(s)eλssds+Ψiτηieλτi})<∞,
which means that
(3.30)E∥x(t)-x*∥p≤E∥φ-x*∥pMe-(λ-α)tfort≥t0.
Therefore, the equilibrium point of system (2.3) is pth moment exponentially stable.Corollary 3.5.
Ifp≥2, under the assumptions (H1)–(H5), system (2.3) exists a unique equilibrium point x*, and x* is pth moment exponentially stable if the following two conditions are satisfied:
(H9)(3.31)pai-(p-1)∑j=1n(|bij|Ljf+|cij|Ljg+|dij|Ljh∫0∞‍𝒦(s)ds)-∑j=1n|bji|Lif>∑j=1n‍(|cji|Lig+|dji|Lih∫0∞‍𝒦(s)ds)+p(p-1)2(li+ei).
(H10)  There exist constantλ∈(0,μ0) and α∈[0,λ),  ρk=maxi∈Λ{1,∑j=1n(μj/μi)pjik}≤eα(tk-tk-1) such that
(3.32)λ<pai-(p-1)∑j=1n‍(|bij|Ljf+|cij|Ljg+|dij|Ljh∫0∞‍𝒦(s)ds)-∑j=1n‍|bji|Lif-p(p-1)2ei-(p-2)(p-1)2li-(∑j=1n‍|cji|Lig+(p-1)li)eλτ-∑j=1n‍|dji|Lih∫0∞‍𝒦(s)eλsds.Proof.
In Theorem3.4, let αl,ij=βl,ij=γl,ij=δl,ij=ξl,ij=ηl,ij=1/p,  μi=1 for all i,j,l∈Λ.  The result is obtained directly.Remark 3.6.
In Corollary3.5, if p=2, the conditions (H9) and (H10) are less weak than the following conditions: 
(H9)(3.33)2mini∈Λai-maxi∈Λ(∑j=1n‍|bij|Ljf)-maxi∈Λ(∑j=1n‍|cij|Ljg)-maxi∈Λ(∑j=1n‍|dij|Ljh∫0∞‍𝒦(s)ds)>∑j=1n‍maxi∈Λ(|bji|Lif)+∑j=1n‍maxi∈Λ(|cji|Lig)+∑j=1n‍maxi∈Λ(|dji|Lih∫0∞‍𝒦(s)ds)+maxi∈Λli+∑j=1n‍maxi∈Λei.
(H10) There exist constant λ∈(0,μ0) and α∈[0,λ), ρk=maxi∈Λ{1,∑j=1n(μj/μi)pjik}≤eα(tk-tk-1) such that
(3.34)λ<2mini∈Λai-maxi∈Λ(∑j=1n‍|bij|Ljf)-maxi∈Λ(∑j=1n‍|cij|Ljg)-maxi∈Λ(∑j=1n‍|dij|Ljh∫0∞‍𝒦(s)ds)-∑j=1n‍maxi∈Λ|bji|Lif-maxi∈Λei-(∑j=1n‍maxi∈Λ|cji|Lig+maxi∈Λli)eλτ-∑j=1n‍maxi∈Λ|dji|Lih∫0∞‍𝒦(s)eλsds,
while (H9) and (H10) were asked in Theorem  3.2 in [18].Corollary 3.7.
Under the assumptions(H2) and (H8), system (2.1) exists a unique equilibrium point x*, and x* is globally exponentially stable if 0<ηi≤1, and the following condition is also satisfied:
(3.35)pai-(p-1)∑j=1n‍(Ljf|bij|pαij/(p-1)+Ljg|cij|pγij/(p-1))-∑j=1n‍μjμi|bji|p(1-αji)Lif>1ηi∑j=1n‍μjμi|cji|p(1-γji)Lig.Proof.
In Theorem3.4, when p>1, choosing αl,ij=αij/(p-1),  γl,ij=γij/(p-1) for l=1,2,…,p-1, and dij=0,  βl,ij=δl,ij=1/p for all l∈Λ, then the result is obtained.Remark 3.8.
If0<ηi≤1,  Lif=Lig=Li,  it follows from (3.35) that
(3.36)pai-(p-1)∑j=1n‍(Lj|bij|pαij/(p-1)+Lj|cij|pγij/(p-1))-∑j=1n‍μjμi|bji|p(1-αji)Li>∑j=1n‍μjμi|cji|p(1-γji)Li,
this is less conservative than the following inequality:
(3.37)mini∈Λ{pai-(p-1)∑j=1n‍(Lj|bij|pαij/(p-1)+Lj|cij|pγij/(p-1))-∑j=1n‍μjμi|bji|p(1-αji)Li}>maxi∈Λ{∑j=1n‍μjμi|cji|p(1-γji)Li},
while (3.37) was required in Theorem  3.1 in [5] and in the only theorem in [8].
Whenp=2,  αij=γij=1/2,  μi=1,  (3.37) is equal to
(3.38)mini∈Λ{2ai-∑j=1n‍(Lj|bij|+Lj|cij|)-∑j=1n‍|bji|Li}>maxi∈Λ{∑j=1n‍|cji|Li},
while (3.38) was required in Theorem  3.2 in [19].Similar to Corollaries3.5 and 3.7, the following results are directly gained from Theorem 3.4.Corollary 3.9.
Under the assumptions (H2), (H4), and (H8), system (2.2) exists a unique equilibrium point x*, and x* is pth moment exponentially stable if 0<ηi≤1, and the following condition is satisfied:
(3.39)pai-(p-1)∑j=1n‍(|bij|Ljf+|cij|Ljg)-∑j=1n‍μjμi|bji|Lif-p(p-1)2ei-(p-1)(p-2)2li>1ηi(μjμi∑j=1n‍|cji|Lig+(p-1)li).
Especially, if p=2, then the equilibrium x* is exponentially stable in mean square if the following condition is also satisfied:
(3.40)2ai-∑j=1n‍(|bij|2αijLjf2βij+|cij|2γijLjg2δij)-∑j=1n‍μjμi|bji|2-2αjiLif2-2βji-ei>1ηi(μjμi∑j=1n‍|cji|2-2γjiLig2-2δji+li).Remark 3.10.
If0<ηi≤1, it follows from (3.39) that
(3.41)pai-(p-1)∑j=1n‍(|bij|Ljf+|cij|Ljg)-∑j=1n‍μjμi|bji|Lif-p(p-1)2ei-(p-1)(p-2)2li>μjμi∑j=1n‍|cji|Lig+(p-1)li.
This is less conservative than the following inequality:
(3.42)min1≤i≤n{pai-(p-1)∑j=1n‍(|bij|Ljf+|cij|Ljg)-∑j=1n‍μjμi|bji|Lif-∑j=1n‍(p-1)(p-2)2(ei+li)min1≤i≤n-∑j=1n‍μjμi(p-1)ei}>max1≤i≤n{μjμi∑j=1n‍|cji|Lig+μjμi(p-1)li},
while (3.42) was required in Theorem 3.3 in [26].Remark 3.11.
If0<ηi≤1, it follows from (3.40) that
(3.43)2ai-∑j=1n(|bij|2αijLjf2βij+|cij|2γijLjg2δij)-∑j=1nμjμi|bji|2-2αjiLif2-2βji-ei>μjμi∑j=1n|cji|2-2γjiLig2-2δji+li,
this is less conservative than the following inequality:
(3.44)min1≤i≤n{2ai-∑j=1n(|bij|2αijLjf2βij+|cij|2γijLjg2δij)-∑j=1nμjμi|bji|2-2αjiLif2-2βji-max1≤i≤n{μi}μiei}>max1≤i≤n{μjμi∑j=1n|cji|2-2γjiLig2-2δji+max1≤i≤n{μi}μili},
while (3.44) was required in Theoerm 3.3 in [27].


## 4. Illustrative Example

In this section, we will give an example to show that the conditions given in the previous sections are less weak than those given in some of the earlier literatures, such as [18].Consider the following stochastic neural networks with mixed time delays(4.1)dxi(t)=(-aixi(t)+∑j=12‍bijfj(xj(t))+∑j=12‍cijgj(xj(t-τj(t)))+∑j=12‍dij·∫-∞t‍Kij(t-s)hj(xj(s))ds+Ii∑j=12‍)dt+σi(t,xi(t),xi(t-τi(t)))dwi(t),
where t≠tk,  i∈Λ={1,2},  tk=k,  fj(s)=gj(s)=hj(s)=(1/2)[|x+1|-|x-1|], and  Kij(s)=e-s.
(4.2)(ai)2×1=(1.73),(bij)2×2=(0.10.20.40.3),(cij)2×2=(0.20.30.10.4),(dij)2×2=(0.10.20.20.3),(Ii)2×1=(0.942.5),(σi)2×1=(0.1(x1(t-τ1(t))-1.2)0.1(x2(t-τ2(t))-1.4)).In the following, we introduce the following nonlinear impulsive controllers:(4.3)x1(tk)=0.05sin(x1(tk-)-1.2)-0.02x2(tk)+1.48,k∈ℤ+,x2(tk)=-0.03x1(tk)+0.04cos(x2(tk-)-1.4)+1.72,k∈ℤ+.
In this case, we have Ljf=Ljg=Ljh=1,  𝒦(s)=e-s,  ej=0,  lj=0.01 for j=1,2, and  μ0=0.9.Forp=2, we can compute that
(4.4)2a1-∑j=12‍[|b1j|Ljf+|c1j|Ljg+|d1j|Ljh∫0∞‍𝒦(s)ds]-∑j=12‍|bj1|L1f=1.8>∑j=12‍[|cj1|L1g+|dj1|L1h∫0∞‍𝒦(s)ds]+l1+e1=0.61,2a2-∑j=12‍[|b2j|Ljf+|c2j|Ljg+|d2j|Ljh∫0∞‍𝒦(s)ds]-∑j=12‍|bj2|L2f=3.8>∑j=12‍[|cj2|L1g+|dj2|L1h∫0∞‍𝒦(s)ds]+l2+e2=1.21,Pk=(0.20.080.120.16).
Choosing μi=1 for i=1,2,  we have ρk=maxi∈Λ{1,∑j=12pjik}=1  and λ=0.6andα=0.5,  then
(4.5)0.6=λ<2a1-∑j=12‍[|b1j|Ljf+|c1j|Ljg+|d1j|Ljh∫0∞‍𝒦(s)ds]-∑j=12‍|bj1|L1f-e1-[∑j=12‍|cj1|L1g+l1]eλτ-∑j=12‍|dj1|L1h∫0∞‍𝒦(s)eλsds=1.05-0.31e0.12=0.7,0.6=λ<2a2-∑j=12‍[|b2j|Ljf+|c2j|Ljg+|d2j|Ljh∫0∞‍𝒦(s)ds]-∑j=12‍|bj2|L2f-e2-[∑j=12‍|cj2|L2g+l2]eλτ-∑j=12‍|dj2|L2h∫0∞‍𝒦(s)eλsds=2.55-0.71e0.12=1.749,∫0∞𝒦(s)eλssds=6.25≤∫0∞𝒦(s)eμ0sds=10.
Thus, all conditions of Theorem 3.4 in this paper are satisfied; the equilibrium solution is exponentially stable in mean square. From above discussion, it is easy to see that
(4.6)2mini∈Λai-maxi∈Λ(∑j=12‍|bij|Ljf)-maxi∈Λ(∑j=12‍|cij|Ljg)-maxi∈Λ(∑j=12‍|dij|Ljh∫0∞‍𝒦(s)ds)-∑j=1n‍maxi∈Λ(|bji|Lif)=1.0<∑j=12‍maxi∈Λ(|cji|Lig)+∑j=12‍maxi∈Λ(|dji|Lih∫0∞‍𝒦(s)ds)+maxi∈Λli+∑j=12‍maxi∈Λei=1.11,
which implies that the condition (H9) in [18] do not hold for this example. So our results are less weaker than some previous results.


## 5. Conclusion

In this paper, we investigate thepth moment exponential stability for stochastic neural networks with mixed delays under nonlinear impulsive effects. By means of Lyapunov functionals, stochastic analysis, and differential inequality technique, some sufficient conditions for the pth moment exponential stability of this system are derived. The results of this paper are new, and they supplement and improve some of the previously known results [5, 8, 18, 19, 26, 27]. Moreover, examples are given to illustrate the effectiveness of our results. Furthermore, the method given in this paper may be extended to study other neural networks, such as the model in [29] and stochastic Cohen-Grossberg neural networks in [30], and we can get improved results too.



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*Source: 175934-2012-12-11.xml*