# Analysis of Exponential Stability for Neutral Stochastic Cohen-Grossberg Neural Networks with Mixed Delays

Authors: Tianqing Yang; Zuoliang Xiong; Cuiping Yang
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This paper is concerned with the mean-square exponential input-to-state stability problem for a class of stochastic Cohen-Grossberg neural networks. Different from prior works, neutral terms and mixed delays are discussed in our system. By employing the Lyapunov-Krasovskii functional method, Itô formula, Dynkin formula, and stochastic analysis theory, we obtain some novel sufficient conditions to ensure that the addressed system is mean-square exponentially input-to-state stable. Moreover, two numerical examples and their simulations are given to illustrate the correctness of the theoretical results.


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## Body

## 1. Introduction

Over the last few years, Cohen-Grossberg neural networks have been paid more and more attention because of their potential applications in various fields such as signal processing, image processing, pattern recognition, associative memory, programming problems, and combinatorial optimization (see [1–5]). Cohen-Grossberg neural networks model was first introduced by Cohen and Grossberg in 1983, which has become one of the most important neural network models (see [6, 7]). Compared with recurrent neural networks, Hopfield neural networks, and cellular neural networks, it is more challenging and interesting to build Cohen-Grossberg neural networks model. In particular, recurrent neural networks, Hopfield neural networks, and cellular neural networks can be regarded as exceptional cases of Cohen-Grossberg neural networks. Many researchers have further studied the effect of Cohen-Grossberg neural networks models and have obtained some reliable conclusions (see [8–10]). Therefore, it is greatly significant to investigate Cohen-Grossberg neural networks.As we all know, the stability of Cohen-Grossberg neural networks plays an essential role in practical applications since it is a prerequisite to ensure that a real system works properly. Also, the existence of time delays cannot be ignored in real systems for various reasons, such as the finite switching speed of amplifiers, the congestions in the signal processing and transmission between neurons. The time delay may cause instability, oscillation, and divergence to stochastic neural network systems. Given the actual situation, a great number of stability results have been presented on stochastic delayed neural networks studied in the literature (see [11–17]). In addition, neutral-type neural network is a particular time delay system because its main feature that the derivative of system state is connected to delays. In the physical sense, the state trend of a neutral delay system is related to the past state and the current state. Many physical systems can be described as the neutral-type model in the real world. In [18, 19], the stability of neutral-type stochastic neural network has been studied. Meanwhile, the neutral-type time delay is quite different from distributed delays and constant delays. There is no doubt that the distributed delay generally has integral term and acts on probability space. The constant delay is constant and discrete, which means that the effect of the state is delayed by a fixed value on the system. It can be observed that most of the literature works mainly focus on three simple cases containing constant delays, distributed delays, and time-varying delays (see [20–22]). In fact, mixed delays have been considered to be more efficient in modeling neural network systems because these simple delays are often not feasible when neural network systems become more complex. To sum up, it is of great significance to study the stability of stochastic Cohen-Grossberg neural networks with mixed delays.Thankfully, there are a great number of stability results about the stochastic Cohen-Grossberg neural networks with mixed delays (see [23–26]). For example, by applying the LaSalle invariant principle of stochastic differential delay equations and the stochastic analysis theory as well as the adaptive feedback technique, Zhang and Deng [23] introduced several sufficient conditions to ensure the adaptive synchronization of Cohen-Grossberg neural network with mixed time-varying delays and stochastic perturbation. Based on a novel Lyapunov-Krasovskii functional and some new approaches and techniques, Zhu and Cao [26] investigated the problem of exponential stability for a class of Markovian jump impulsive stochastic Cohen-Grossberg neural networks with mixed time delays and known or unknown parameters.Furthermore, it is worth considering that the control input has a great influence on the stochastic Cohen-Grossberg neural networks. Sontag first came up with the input-to-state stability to check robust stability. In [27–29], the mean-square exponential input-to-state stability is more common than the traditional exponential stability. Meanwhile, the traditional exponential stability consists of exponential stability, global asymptotical stability, and almost surely stability. As far as we know, the traditional stability is that the states of stochastic neural networks converge to the equilibrium point as time goes to infinity. However, it does not always happen in our real life. For instance, some systems including the pendulum, finance markets, and stock markets only remain bounded but necessarily converge to the equilibrium point. To date, few researchers have discussed the mean-square exponential input-to-state stability problem of stochastic Cohen-Grossberg neural networks due to the difficulty of mathematics. Hence, this situation encourages our current research.Inspired by the above discussion, we will investigate the problem of the mean-square exponential input-to-state stability analysis for a class of the stochastic neutral-type Cohen-Grossberg neural networks with mixed delays in this paper. Different from the previous stability works, the mean-square exponential input-to-state stability of stochastic neutral-type Cohen-Grossberg neural network has not been investigated. By using the Lyapunov-Krasovskii functional, stochastic analysis theory, Itô formula, and Dynkin formula, the mean-square exponential input-to-state stability is obtained. Moreover, several examples and their simulations are given to illustrate the correctness and effectiveness of obtained results.The remainder of this paper is structured as follows. In Section2, we introduce the mathematical model of neutral stochastic Cohen-Grossberg neural networks, assumptions, and definition of the mean-square exponential stability. The main theoretical result and its proofs are derived in Section 3. In Section 4, two numerical examples and their simulations are given to guarantee the correctness of the proposed results. Finally, the conclusion and further discussion are presented in Section 5.


## 2. Model Description and Preliminaries

In this paper, we mainly focus on the following stochastic neutral-type Cohen-Grossberg neural networks with mixed delays:(1)dxit-∑j=1npijxjt-τ0t=-αixitβixit-∑j=1naijfjxjt-∑j=1nbijgjxjt-τ-∑j=1ncijhjxjt-τ0t-∑j=1ndij∫t-τjttljxjsds+uitdt+∑j=1nσijxjt,xjt-τ,xjt-τ0t,xjt-τ1t,xjt-τ2t,…,xjt-τntdωjt,i=1,2,…,n.With initial conditions xi(t)=ϕi(t),t∈[-τ¯,0], here xi(t) is the state variable of the ith neuron at time t; αi(xi(t)) and βi(xi(t)) denote the amplification functions and behaved functions of the ith unit at time t, respectively. The constants aij,bij,cij, and dij are the connection weight coefficients of the jth unit on the ith unit at time t. fj(xj(t)), gj(xj(t)), hj(xj(t)), and lj(xj(t)) are the neuron activation functions of the jth unit at time t. ui(t) is the external constant input of the ith neuron at time t, and u(t)=(u1(t),u2(t),…,un(t))T belongs to l∞, where l∞ represents the class of essentially bounded functions u from 0,+∞ to Rn with u∞=esssupt≥0ut,t≥0<∞. The noise perturbation σij:Rn+3→R is a Borel measurable function, and ω(t)=ω1t,ω2t,…,ωnt,t≥0 is an n-dimensional standard Brownian motion defined on the complete probability space (Ω,F,P) with a natural filtration Ftt≥0. τ represents the constant delay and τ0(t) denotes the time-varying delay; τi(t)(i=1,2,…,n) are distributed delays. τi(t)(i=0,1,2,…,n) are supposed to be differential and satisfy 0≤τ˙i(t)≤ρ<1, 0≤τi(t)≤τi(i=0,1,2,…,n). Let τ¯=maxt≥0{τ,τ0,τ1,…,τn}, where τ¯ is a positive constant. Let C-τ¯,0;Rn denote the family of continue functions ϕ from [-τ¯,0] to Rn with the norm ϕ=sup-τ¯≤θ≤0ϕθ, where |·| is the Euclidean norm in Rn. LF02([-τ¯,0];Rn) denotes the family of all F0 measurable, C([-τ¯,0];Rn)-valued random variables ϕ={ϕ(s):-τ¯≤s≤0} such that ∫-τ¯0Eϕs2ds<∞, where E[·] stands for the correspondent expectation operator with respect to the given probability measure P.Remark 1.
In fact, letτ=0 and τj(t)=0(j=2,…,n) and replace σij(xj(t),xj(t-τ),xj(t-τ0(t)),xj(t-τ1(t)),xj(t-τ2(t)),…,xj(t-τn(t))) in system (1) with σij(xj(t),xj(t-τ0(t)),xj(t-τ1(t))); our model reduces to the model in [15]. In other words, the model discussed in [15] is a special case of the system (1). Since the delays are limited to time-varying delays τ0(t) and τ1(t), our model is more general and useful in real life.In order to obtain our main results, we need the following assumptions and definition.Assumption 2.
The functionsβi(·), fi(·), gi(·), hi(·), and li(·) satisfy Lipschitz condition; i.e., there exist positive constants Bi, Li, Mi, Ni, and Qi such that (2)βix-βiy≤Bix-y,fix-fiy≤Łix-y,gix-giy≤Mix-y,hix-hiy≤Nix-y,lix-liy≤Qix-y,for all x,y∈R,i=1,2,…,n.Assumption 3.
There exist nonnegative constantsμij, νij, δij, and ξij(1), ξij(2), ⋯, ξij(n) such that (3)σijx,y,z,k1,k2,…,kn-σijx′,y′,z′,k1′,k2′,…,kn′2(4)≤μijx-x′2+νijy-y′2+δijz-z′2+∑i=1nξijiki-ki′2,for all x,x′,y,y′,z,z′,k1,k1′,k2,k2′,…,kn,kn′∈R,i,j=1,2,…,n.Assumption 4.
α j ( 0 ) = 0 or βj(0)=fj(0)=gj(0)=hj(0)=lj(0)=uj(0)≡0,σij(0,0,…,0)≡0,i,j=1,2,…,n.Assumption 5 (see [1]).
α i ( x i ( t ) ) are bounded functions, and there exist positive constants α_i and α¯i such that 0<α_i≤αi(xi(t))≤α¯i, i=1,2,…,n.Assumption 6 (see [26]).
There exist positive constantsηi, such that xi(t)βi(xi(t))≥ηixi2(t),i=1,2,…,n.Assumption 7.
P = ( p i j ) n × n is a matrix, · denotes the matrix norm which satisfies ρ0=P∈(0,1).Under Assumptions2–4, it is not difficult to check that the Lipschitz and linear growth conditions hold, and thus there exists a unique solution of system (1). Let x(t;ϕ) denote the solution which satisfies x(s)=ϕ(s) on -τ¯≤s≤0 in LF02([-τ¯,0];Rn). It is clear that under Assumption 4, system (1) admits a trivial solution x(t;0)≡0 corresponding to the initial data ϕ=0.Definition 8 (Zhu and Cao [11]).
The trivial solution of system (1) is said to be mean-square exponentially input-to-state stable if for every ϕ∈LF02([-τ¯,0];Rn) and u(t)∈l∞, there exist scalars α0>0, λ0>0 and γ0>0 such that the following inequality holds: (5)Ext,ψ2≤α0e-λ0tEϕ2+γ0u∞2.


## 3. Main Results

In this section, the mean-square exponential input-to-state stability of the trivial solution for system (1) is discussed under Assumptions 2–7.Theorem 9.
Under Assumptions2–7, the trivial solution of system (1) is mean-square exponentially input-to-state stable, if there exist positive constants qi, ri, vi, wij(i,j=1,2,…,n), such that the following conditions hold:(6)R1i=-2qiα_iηi+qiα¯i+ri+vi+∑j=1nwji+∑j=1nqiα¯iBipij+Ljaij+Mjbij+Njcij+τ¯Qjdij+qjα¯jLiaji+μji+eλτ¯τ¯α¯jQidji+∑j=1n∑k=1nqjα¯jpjkLiaji+eλτ¯τ¯Qidji<0R2i=-1-ρe-λτ¯vi+∑j=1nqjα¯jBjpji+α¯jNicji+α¯jpji+δji+∑j=1n∑k=1nqjα¯jpjiLkajk+Mkbjk+Nkcjk+τ¯Qkdjk+Nipjkcji<0R3i=-e-λτ¯ri+∑j=1nqjα¯jMibji+νji+∑j=1n∑k=1nqjα¯jMipjkbji<0Aki=∑j=1nqjξjik-1-ρe-λτ¯wki<0.Proof.
Letx(t;ϕ)=x(t). To obtain the mean-square exponential input-to-state stability, we can construct the following Lyapunov-Krasovskii functional:(7)Vt,xt=∑i=15Vit,xtwhere(8)V1t,xt=eλt∑i=1nqixit-∑j=1npijxjt-τ0t2V2t,xt=∫t-τteλs∑i=1nrixis2dsV3t,xt=∫t-τ0tteλs∑i=1nvixis2dsV4t,xt=∑i=1n∫t-τitteλs∑j=1nwjixis2dsV5t,xt=∑i=1n∑j=1nqiα¯iQjdij1+∑k=1npik∫-τ¯0∫t+θtxjs2eλs+τ¯dsdθBy employing Itô formula, we get the following stochastic differential equation:(9)dVt,xt=LVt,xtdt+Vxt,xtσtdωt,where Vx(t,x(t))=(∂V(t,x(t)/∂x1),…,∂V(t,x(t))/∂xn), σ(t)=(σij(t))n×n, σij(t)=σij(xj(t),xj(t-τ),xj(t-τ0(t)),xj(t-τ1(t)),xj(t-τ2(t)),…,xj(t-τn(t))), and L is the weak infinitesimal operator which satisfies (10)LV1t,xt=λeλt∑i=1nqixit-∑j=1npijxjt-τ0t2+2eλt∑i=1nqixit-∑j=1npijxjt-τ0t×-αixitβixit-∑j=1naijfjxjt-∑j=1nbijgjxjt-τ-∑j=1ncijhjxjt-τ0t-∑j=1ndij∫t-τjttljxjsds+uit+eλt∑i=1nqi∑j=1nσij2xjt,xjt-τ,xjt-τ0t,xjt-τ1t,xjt-τ2t,…,xjt-τnt≤2λeλtmax1≤i≤nqi∑i=1nxi2t+ρ02xi2t-τ0t-2eλtαixitβixit∑i=1nqixit+2eλtαixitβixit∑i=1nqi∑j=1npijxjt-τ0t+2eλtαixit∑i=1nqixit∑j=1naijfjxjt-2eλtαixit∑i=1nqi∑j=1npijxjt-τ0t∑k=1naikfkxkt+2eλtαixit∑i=1nqixit∑j=1nbijgjxjt-τ-2eλtαixit∑i=1nqi∑j=1npijxjt-τ0t∑k=1nbikgkxkt-τ+2eλtαixit∑i=1nqixit∑j=1ncijhjxjt-τ0t-2eλtαixit∑i=1nqi∑j=1npijxjt-τ0t∑k=1ncikhkxkt-τ0t+2eλtαixit∑i=1nqixit∑j=1ndij∫t-τjttljxjsds-2eλtαixit∑i=1nqi∑j=1npijxjt-τ0t∑k=1ndik∫t-τkttlkxksds-2eλtαixit∑i=1nqixituit+2eλtαixit∑i=1nqi∑j=1npijxjt-τ0tuit+eλt∑i=1nqi∑j=1nσij2xjt,xjt-τ,xjt-τ0t,xjt-τ1t,xjt-τ2t,…,xjt-τnt,
whereP=pijn×n is a matrix and Px≤Px=ρ0x; under Assumptions 2 and 3, we can obtain (11)LV1t,xt≤2λeλtmax1≤i≤nqi∑i=1nxi2t+ρ02xi2t-τ0t-2eλtαixitβixit∑i=1nqixit+2eλtαixitβixit∑i=1nqi∑j=1npijxjt-τ0t+2eλtαixitLj∑i=1nqi∑j=1naijxjtxit+2eλtαixitLk∑i=1nqi∑j=1npij∑k=1naikxktxjt-τ0t+2eλtαixitMj∑i=1nqi∑j=1nbijxjt-τxit+2eλtαixitMk∑i=1nqi∑j=1npij∑k=1nbikxkt-τxjt-τ0t+2eλtαixitNj∑i=1nqi∑j=1ncijxjt-τ0txit+2eλtαixitNk∑i=1nqi∑j=1npij∑k=1ncikxkt-τ0txjt-τ0t+2eλtαixitQj∑i=1nqi∑j=1ndijxit∫t-τjttxjsds+2eλtαixitQk∑i=1nqi∑j=1npij∑k=1ndikxjt-τ0t∫t-τkttxksds+2eλtαixit∑i=1nqixituit+2eλtαixit∑i=1nqi∑j=1npijxjt-τ0tuit+eλt∑i=1n∑j=1nqjμjixit2+eλt∑i=1n∑j=1nqjνjixit-τ2+eλt∑i=1n∑j=1nqjδjixit-τ0t2+eλt∑i=1n∑j=1nqjξji1xit-τ1t2+eλt∑i=1n∑j=1nqjξji2xit-τ2t2+⋯+eλt∑i=1n∑j=1nqjξjinxit-τnt2.By using the elementary inequality 2ab≤a2+b2 and under Assumptions 5 and 6, we have (12)LV1t,xt≤2λeλtmax1≤i≤nqi∑i=1nxi2t+ρ02xi2t-τ0t-2eλtα_iηi∑i=1nqixit2+eλtα¯iBi∑i=1nqi∑j=1npijxjt-τ0t2+xit2+eλtα¯iLj∑i=1nqi∑j=1naijxjt2+xit2+eλtα¯iLk∑i=1nqi∑j=1npij∑k=1naikxkt2+xjt-τ0t2+eλtα¯iMj∑i=1nqi∑j=1nbijxjt-τ2+xit2+eλtα¯iMk∑i=1nqi∑j=1npij∑k=1nbikxkt-τ2+xjt-τ0t2+eλtα¯iNj∑i=1nqi∑j=1ncijxjt-τ0t2+xit2+eλtα¯iNk∑i=1nqi∑j=1npij∑k=1ncikxkt-τ0t2+xjt-τ0t2+eλtα¯iQj∑i=1nqi∑j=1ndijτjtxit2+∫t-τjttxjs2ds+eλtα¯iQk∑i=1nqi∑j=1npij∑k=1ndikτktxjt-τ0t2+∫t-τkttxks2ds+eλtα¯i∑i=1nqixit2+uit2+eλtα¯i∑i=1nqi∑j=1npijxjt-τ0t2+uit2+eλt∑i=1n∑j=1nqjμjixit2+eλt∑i=1n∑j=1nqjνjixit-τ2+eλt∑i=1n∑j=1nqjδjixit-τ0t2+eλt∑i=1n∑j=1nqjξji1xit-τ1t2+eλt∑i=1n∑j=1nqjξji2xit-τ2t2+⋯+eλt∑i=1n∑j=1nqjξjinxit-τnt2≤eλt∑i=1n2λmax1≤i≤nqi-2qiα_iηi+qiα¯i+∑j=1nqiα¯iBipij+Ljaij+Mjbij+Njcij+τ¯Qjdij+qjα¯jajiLi+μji+∑j=1n∑k=1nqjα¯jLipjkajixit2+eλt∑i=1n2λρ02max1≤i≤nqi+∑j=1nqjα¯jBjpji+α¯jNicji+α¯jpji+δji+∑j=1n∑k=1nqjα¯jpjiLkajk+Mkbjk+Nkcjk+τ¯Qkdjk∑j=1n∑k=1nqj+Nipjkcjixit-τ0t2+eλt∑i=1n∑j=1nqjα¯jMibji+νji+∑k=1nα¯jMipjkbjixit-τ2+eλt∑i=1nqiα¯i1+∑j=1npijuit2+eλt∑i=1n∑j=1nqiα¯iQjdij∫t-τjttxjs2ds+∑k=1nQkpijdik∫t-τkttxks2ds+eλt∑i=1n∑j=1nqjξji1xit-τ1t2+eλt∑i=1n∑j=1nqjξji2xit-τ2t2+⋯+eλt∑i=1n∑j=1nqjξjinxit-τnt2.
By Itô formula, we get(13)LV2t,xt=eλt∑i=1nrixit2-eλt-τ∑i=1nrixit-τ2≤eλt∑i=1nrixit2-eλt-τ¯∑i=1nrixit-τ2LV3t,xt=eλt∑i=1nvixit2-eλt-τ0t1-τ˙0t∑i=1nvixit-τ0t2≤eλt∑i=1nvixit2-eλt-τ¯1-ρ∑i=1nvixit-τ0t2LV4t,xt=eλt∑i=1n∑j=1nwjixit2-eλt-τit1-τ˙it∑i=1n∑j=1nwjixit-τit2≤eλt∑i=1n∑j=1nwjixit2-eλt-τ¯1-ρ∑i=1n∑j=1nwjixit-τit2LV5t,xt=eλt+τ¯∑i=1n∑j=1nqiα¯iτ¯Qjdij1+∑k=1npikxjt2-∑i=1n∑j=1nqiα¯iQjdij1+∑k=1npik∫t-τ¯teλs+τ¯xjs2ds≤eλt+τ¯∑i=1n∑j=1nqiα¯iτ¯Qjdij1+∑k=1npikxjt2-eλt∑i=1n∑j=1nqiα¯iQjdij1+∑k=1npik∫t-τjttxjs2dsBy condition (6), there exists a sufficiently small constant λ>0 such that (14)LVt,xt≤eλt∑i=1n2λmax1≤i≤nqi-2qiα_iηi+qiα¯i+ri+vi+∑j=1nwji+∑j=1nqiα¯iBipij+Ljaij+Mjbij+Njcij+τ¯Qjdij+qjα¯jLiaji+μji+eλτ¯τ¯α¯jQidji+∑j=1n∑k=1nqjα¯jpjkLiaji+eλτ¯τ¯Qidjixit2+eλt∑i=1n2λmax1≤i≤nqiρ02-1-ρe-λτ¯vi+∑j=1nqjα¯jBjpji+α¯jNicji+α¯jpji+δji+∑j=1n∑k=1nqjα¯jpjiLkajk+Mkbjk+Nkcjk+τ¯Qkdjk+Nipjkcjixit-τ0t2+eλt∑i=1n-e-λτ¯ri+∑j=1nqjα¯jMibji+νji+∑j=1n∑k=1nqjα¯jMipjkbjixit-τ2+eλt∑i=1n∑k=1n∑j=1nqjξjik-1-ρe-λτ¯wkixit-τkt2+eλt∑i=1nqiα¯i1+∑j=1npijuit2≤eλt∑i=1nqiα¯i1+∑j=1npijuit2. Now, we need to define a stopping time as follows: ϱk:=inf{s≥0:xs≥k}. By means of the Dynkin formula, we can obtain that (15)EVt∧ϱk,xt∧ϱk-EV0,x0=E∫0t∧ϱkLVs,xsds,
that is(16)EVt∧ϱk,xt∧ϱk=EV0,x0+E∫0t∧ϱkLVs,xsds.Letting k→∞, then t∧ϱk→t. According to the monotone convergence theorem, we have (17)EVt,xt≤EV0,x0+eλt-1∑i=1nqiα¯1+∑j=1npiju∞2.
Noting that(18)EV0,x0≤2max1≤i≤nqi1+ρ02+max1≤i≤nriτ+max1≤i≤nviτ0+max1≤i≤n∑j=1nwjiτ¯+max1≤i≤n∑j=1nqjα¯iQjdij1+∑k=1npikτ¯eλτ¯τ¯Eϕ2=S0Eϕ2,then we have(19)EVx,xt≤S0Eϕ2+S1eλtu∞2,where S1=∑i=1nqiα¯i(1+∑j=1n|pij|). We can use the inequalities (m+n)2≤(1+ϵ)m2+(1+1/ϵ)n2, ϵ>0, and it follows that (20)min1≤i≤nqiEeλtxt2=min1≤i≤nqiEeλtxt-Pxt-τ0t+Pxt-τ0t2≤min1≤i≤nqieλtE1+ϵxt-Pxt-τ0t2+1+1ϵPxt-τ0t2≤1+ϵEVt,xt+min1≤i≤nqiρ021+1ϵEeλtxt-τ0t2≤1+ϵS0Eϕ2+S1eλtu∞2+min1≤i≤nqiρ021+1ϵEeλtxt-τ0t2. We can always find a large enough constant ϵ and a sufficiently small constant λ such that ρ02eλτ0(1+1/ϵ)<1. Therefore (21)sup-τ¯≤s≤tmin1≤i≤nqiEeλsxs2≤min1≤i≤nqiEϕ2+sup0≤s≤tmin1≤i≤nqiEeλsxs2≤min1≤i≤nqi+1+ϵS0Eϕ2+1+ϵS1eλtu∞2+sup-τ¯≤s≤tmin1≤i≤nqiρ02eλτ01+1ϵEeλsxs2,and we can easily derive (22)sup-τ¯≤s≤tEeλsxs2≤α0Eϕ2+γ0eλtu∞2,where α0=(min1≤i≤n{qi}+(1+ϵ)S0)/min1≤i≤nqi-min1≤i≤nqiρ021+1/ϵeλτ0 and γ0=(1+ϵ)S1/min1≤i≤nqi-min1≤i≤nqiρ021+1/ϵeλτ0. Furthermore, we get (23)Ext2≤α0e-λtEϕ2+γ0u∞2.According to Definition 8, the trivial solution of system (1) is mean-square exponentially input-to-state stable. Thus, we complete the proof.Remark 10.
The mean-square exponential input-to-state stability problem of stochastic Cohen-Grossberg neural networks was investigated in [1]. However, the delay is limited to a time-varying delay τ(t). In this paper, stochastic neutral-type Cohen-Grossberg neural networks discussed are generalized to the systems with mixed delays. They are more general than time-varying delay in [1]. Also, the mean-square exponential input-to-state stability problem for a class of stochastic recurrent neural networks with mixed delays was discussed in [11, 16]. But the neutral terms and Cohen-Grossberg models were not taken into account. Meanwhile, the Cohen-Grossberg neural networks include recurrent neural networks, Hopfield neural networks, and cellular neural networks. In other words, the systems considered in [11, 16] are special cases of system (1) in this paper.Remark 11.
Obviously, whenu(t)≡0, Theorem 9 reduces to the mean-square exponential stability of the neutral stochastic Cohen-Grossberg neural network. We can obtain the following corollary.Corollary 12.
Under Assumptions2–7, if the all conditions of Theorem 9 hold, the trivial solution of system (1) with u(t)≡0 is mean-square exponentially stable.Moreover, when we replaceτj(t),j=1,2,…,n, with τ1(t) in system (1), system (1) reduces to the following stochastic delayed Cohen-Grossberg neural network:(24)dxit-∑j=1npijxjt-τ0t=-αixitβixit-∑j=1naijfjxjt-∑j=1nbijgjxjt-τ-∑j=1ncijhjxjt-τ0t-∑j=1ndij∫t-τ1ttljxjsds+uitdt+∑j=1nσijxjt,xjt-τ,xjt-τ0t,xjt-τ1tdωjt.Moreover, we use the following assumption instead of Assumption 3.Assumption 13.
There exist nonnegative constantsμij, νij, δij, and ξij such that (25)σijx,y,z,k-σijx′,y′,z′,k′≤μijx-x′2+νijy-y′2+δijz-z′2+ξijk-k′2,for all x,x′,y,y′,z,z′,k,k′,i,j=1,2,…,n.Then we can use the similar way to the proof of Theorem9 to prove the next theorem. Thus, we get the following.Theorem 14.
Based on Assumptions2, 3, 4, 5, 6, and 13, the trivial solution of system (24) is mean-square exponentially input-to-state stable, if there exist positive constants qi, ri, vi, wi(i=1,2,…,n) such that the following conditions hold:(26)R1i=-2qiα_iηi+qiα¯i+ri+vi+wi+∑j=1nqiα¯iBipij+Ljaij+Mjbij+Njcij+τ¯Qjdij+qjα¯jLiaji+μji+eλτ¯τ¯α¯jQidji+∑j=1n∑k=1nqjα¯jpjkLiaji+eλτ¯τ¯Qidji<0R2i=-1-ρe-λτ¯vi+∑j=1nqjα¯jBjpji+α¯jNicji+α¯jpji+δji+∑j=1n∑k=1nqjα¯jpjiLkajk+Mkbjk+Nkcjk+τ¯Qkdjk+Nipjkcji<0R3i=-e-λτ¯ri+∑j=1nqjα¯jMibji+νji+∑j=1n∑k=1nqjα¯jMipjkbji<0R4i=∑j=1nqjξji-1-ρe-λτ¯wi<0.Corollary 15.
Assume that all the conditions of Theorem14 hold. Then, the trivial solution of system (24) with u(t)≡0 is mean-square exponentially stable.In particular, when the distributed delays are removed, system (1) reduces to the following neutral stochastic Cohen-Grossberg system with time-varying delay:(27)dxit-∑j=1npijxjt-τ0t=-αixitβixit-∑j=1naijfjxjt-∑j=1nbijgjxjt-τ-∑j=1ncijhjxjt-τ0t+uitdt+∑j=1nσijxjt,xjt-τ,xjt-τ0tdωjt.Therefore, we will replace Assumption 13 with the next assumption.Assumption 16.
There exist nonnegative constantsμij, νij, and δij such that (28)σijx,y,z-σijx′,y′,z′≤μijx-x′2+νijy-y′2+δijz-z′2, for all x,x′,y,y′,z,z′,i,j=1,2,…,n.Corollary 17.
Under Assumptions2, 3, 4, 5, 6, and 16, the trivial solution of system (24) is mean-square exponentially input-to-state stable, if there exist positive constants qi, ri, vi(i=1,2,…,n) such that the following conditions hold:(29)R1i=-2qiα_iηi+qiα¯i+ri+vi+∑j=1nqiα¯iBipij+Ljaij+Mjbij+Njcij+qjα¯jLiaji+μji+∑j=1n∑k=1nqjα¯jpjkLiaji<0R2i=-1-ρe-λτ¯vi+∑j=1nqjα¯jBjpji+α¯jNicji+α¯jpji+δji+∑j=1n∑k=1nqjα¯jpjiLkajk+Mkbjk+Nkcjk+Nipjkcji<0R3i=-e-λτ¯ri+∑j=1nqjα¯jMibji+νji+∑j=1n∑k=1nqjα¯jMipjkbji<0.Corollary 18.
Assume that all the conditions of Corollary17 hold. Then, the trivial solution of system (27) with u(t)≡0 is mean-square exponentially stable.Remark 19.
The systems are no longer stochastic neutral-type neural networks whenP=0; of course, we can also obtain the mean-square exponential input-to-state stability results from Theorems 9 and 14 when P=0.


## 4. Examples and Simulation

In this section, two numerical examples are given to illustrate the effectiveness of our theoretical conclusions.Example 20.
Consider the two-dimensional system (1) with f(x)=0.1cos⁡x, g(x)=0.1sin⁡x, h(x)=0.1sin⁡x, l(x)=0.2x, αi(xi(t))=0.4+0.3sin⁡(xi(t)), βi(xi(t))=6xi(t), τ¯=1, τ0(t)=0.5+0.3sin⁡t, τ1(t)=0.7+0.1cos⁡t, τ2(t)=0.6+0.2sin⁡t, u(t)=0.5cos⁡t; ω(t) is a two-dimensional Brownian motion.(30)σijxjt,xjt-τ,xjt-τ0t,xjt-τ1t,xjt-τ2t2×2=0.1x1t+x1t-τ+x1t-τ1t0.3x2t+x2t-τ0t0.2x1t+x1t-τ1t+x1t-τ2t0.3x2t-τ+x2t-τ0t+x2t-τ2taij2×2=-0.10.30.3-0.2,bij2×2=-0.10.5-0.3-0.4,cij2×2=0.30.20.40.4dij2×2=0.10.50.10.5,pij2×2=0.1-0.20.20.1,wij2×2=1.21.00.81.5μij2×2=0.100.150.020.00,νij2×2=0.120.000.000.12,δij2×2=0.000.150.000.12ξij12×2=0.120.100.050.00,ξij22×2=0.000.000.010.15
Takeλ=0.01, ρ=0.1, q1=1.2, q2=1.3, α_=0.2, α¯=0.5, η1=η2=10, r1=0.3, r2=0.4, v1=0.6, v2=1, B=L=M=N=Q=0.1, which satisfy Assumptions 2–7. Then it is uncomplicated to check the following conditions: (31)R11=-2q1α_1η1+q1α¯1+r1+v1+∑j=12wj1+∑j=12q1α¯1B1p1j+Lja1j+Mjb1j+Njc1j+τ¯Qjd1j+qjα¯jL1aj1+μj1+eλτ¯τ¯α¯jQ1dj1+∑j=12∑k=12qjα¯jpjkL1aj1+eλτ¯τ¯Q1dj1=-0.9604<0R12=-2q2α_2η2+q2α¯2+r2+v2+∑j=12wj2+∑j=12q2α¯2B2p2j+Lja2j+Mjb2j+Njc2j+τ¯Qjd2j+qjα¯jL2aj2+μj2+eλτ¯τ¯α¯jQ2dj2+∑j=12∑k=12qjα¯jpjkL2aj2+eλτ¯τ¯Q2dj2=-0.1591<0R21=-1-ρe-λτ¯v1+∑j=12qjα¯jBjpj1+α¯jN1cj1+α¯jpj1+δj1+∑j=12∑k=12qjα¯jpj1Lkajk+Mkbjk+Nkcjk+τ¯Qkdjk+N1pjkcj1=-0.2220<0R22=-1-ρe-λτ¯v2+∑j=12qjα¯jBjpj2+α¯jN2cj2+α¯jpj2+δj2+∑j=12∑k=12qjα¯jpj2Lkajk+Mkbjk+Nkcjk+τ¯Qkdjk+N2pjkcj2=-0.2600<0R31=-e-λτ¯r1+∑j=12qjα¯jM1bj1+νj1+∑j=12∑k=12qjα¯jM1pjkbj1=-0.1215<0R32=-e-λτ¯r2+∑j=12qjα¯jM2bj2+νj2+∑j=12∑k=12qjα¯jM2pjkbj2=-0.1705<0A11=∑j=11qjξj11-1-ρe-λτ¯w11=-0.8603<0A12=∑j=11qjξj21-1-ρe-λτ¯w12=-0.7710<0A21=∑j=11qjξj12-1-ρe-λτ¯w21=-0.6998<0A22=∑j=11qjξj22-1-ρe-λτ¯w22=-1.1416<0.
Therefore, all the conditions of Theorem9 are satisfied. We can obtain that the neutral stochastic Cohen-Grossberg system (1) is mean-square exponentially input-to-state stable. Figure 1(a) further shows that network (1) is mean-square exponentially input-to-state stable. When u1(t)=u2(t)=0, network (1) is mean-square exponentially stable; see Figure 1(b).Figure 1
Example20.
(a)
State response of network (1) (b)
State response of network (1) with u(t)=0Example 21.
Consider the three-dimensional system (24) with f(x)=0.1cos⁡x, g(x)=0.1sin⁡x, h(x)=0.1sin⁡x, l(x)=0.2x, αi(xi(t))=0.9+0.1cos⁡(xi(t)), βi(xi(t))=5xi(t), τ¯=1, τ0(t)=0.3+0.2sin⁡t, τ1(t)=0.8+0.1cos⁡t, u(t)=0.3sin⁡t; ω(t) is a three-dimensional Brownian motion.(32)σijxjt,xjt-τ,xjt-τ0t,xjt-τ1t3×3=0.1x1t+x1t-τ0.3x2t0.15x3t+x3t-τ0t0.2x1t0.2x2t+x2t-τ+x2t-τ0t0.1x3t-τ0.1x1t+x1t-τ1t0.1x2t+x2t-τ1t0.3x3t-τ+x3t-τ0taij3×3=0.20.2-0.20.10.20.00.0-0.1-0.2,bij3×3=0.3-0.20.20.30.20.0-0.10.50.1,cij3×3=0.2-0.10.1-0.20.00.10.10.10.0dij3×3=0.10.2-0.20.20.00.10.10.20.0,pij3×3=-0.40.00.00.00.20.00.00.0-0.2,μij3×3=0.120.150.000.100.000.200.120.000.00δij3×3=0.180.000.000.000.120.000.000.000.02,νij3×3=0.10.00.00.00.10.00.00.00.1,ξij3×3=0.120.000.000.000.030.000.000.000.12
Letλ=0.01, ρ=0.1, q1=1.0, q2=0.8, q2=1.2, α_=0.2, α¯=0.4, η1=η2=η3=10, r1=0.3, r2=0.4, r3=0.5, v1=0.6, v2=0.4, v3=0.6, w1=0.8, w2=0.6, w3=0.8, B=L=M=N=Q=0.1, which satisfy Assumptions 2, 3, 4, 5, 6, and 13. Then, a simple computation yields (33)R11=-2q1α_1η1+q1α¯1+r1+v1+w1+∑j=13q1α¯1B1p1j+Lja1j+Mjb1j+Njc1j+τ¯Qjd1j+qjα¯jL1aj1+μj1+eλτ¯τ¯α¯jQ1dj1+∑j=13∑k=13qjα¯jpjkL1aj1+eλτ¯τ¯Q1dj1=-1.4187<0R12=-2q2α_2η2+q2α¯2+r2+v2+w2+∑j=13q2α¯2B2p2j+Lja2j+Mjb2j+Njc2j+τ¯Qjd2j+qjα¯jL2aj2+μj2+eλτ¯τ¯α¯jQ2dj2+∑j=13∑k=13qjα¯jpjkL2aj2+eλτ¯τ¯Q2dj2=-1.2341<0R13=-2q3α_3η3+q3α¯3+r3+v3+w3+∑j=13q3α¯3B3p3j+Lja3j+Mjb3j+Njc3j+τ¯Qjd3j+qjα¯jL3aj3+μj3+eλτ¯τ¯α¯jQ3dj3+∑j=13∑k=13qjα¯jpjkL3aj3+eλτ¯τ¯Q3dj3=-2.1424<0R21=-1-ρe-λτ¯v1+∑j=13qjα¯jBjpj1+α¯jN1cj1+α¯jpj1+δj1+∑j=13∑k=13qjα¯jpj1Lkajk+Mkbjk+Nkcjk+τ¯Qkdjk+N1pjkcj1=-0.1188<0R22=-1-ρe-λτ¯v2+∑j=13qjα¯jBjpj2+α¯jN2cj2+α¯jpj2+δj2+∑j=13∑k=13qjα¯jpj2Lkajk+Mkbjk+Nkcjk+τ¯Qkdjk+N2pjkcj2=-0.1697<0R23=-1-ρe-λτ¯v3+∑j=13qjα¯jBjpj3+α¯jN3cj3+α¯jpj3+δj3+∑j=13∑k=13qjα¯jpj3Lkajk+Mkbjk+Nkcjk+τ¯Qkdjk+N3pjkcj3=-0.3812<0R31=-e-λτ¯r1+∑j=13qjα¯jM1bj1+νj1+∑j=13∑k=13qjα¯jM1pjkbj1=-0.1634<0R32=-e-λτ¯r2+∑j=13qjα¯jM2bj2+νj2+∑j=13∑k=13qjα¯jM2pjkbj2=-0.2728<0R33=-e-λτ¯r3+∑j=13qjα¯jM3bj3+νj3+∑j=13∑k=13qjα¯jM3pjkbj3=-0.3590<0R41=∑j=13qjξj1-1-ρe-λτ¯w1=-0.5928<0R42=∑j=13qjξj2-1-ρe-λτ¯w2=-0.5106<0R43=∑j=13qjξj3-1-ρe-λτ¯w3=-0.5688<0.
Therefore, all the conditions of Theorem14 are satisfied. Then, by Theorem 14, the neutral stochastic Cohen-Grossberg system (24) is mean-square exponentially input-to-state stable. Figure 2(a) further shows that network (24) is mean-square exponentially input-to-state stable. It is evident that network (24) is mean-square exponentially stable when u1(t)=u2(t)=u3(t)=0, which can be also seen from Figure 2(b).Figure 2
Example21.
(a)
State response of network (24) (b)
State response of network (24) with u(t)=0Remark 22.
According to the Example20, we can see that the state response of the neural network (1) only remains mean-square bounded but does not need to converge to the equilibrium point. Particularly, when u(t)=0, the state response of the neural network (1) can converge to the equilibrium point. And a similar conclusion can be drawn from the figures in Example 21.


## 5. Conclusions

In this paper, a class of stochastic Cohen-Grossberg neural networks with neutral terms and a constant delay, a time-varying delay, and multiple distributed time-varying delays has been researched. By constructing the Lyapunov-Krasovskii functional and applying stochastic analysis theory, Itô formula, Dynkin formula, and some sufficient conditions, we have got some new sufficient conditions to undertake the mean-square exponential input-to-state stability for the given models. The effectiveness of our conclusions has been illustrated through two numerical examples. The input-to-state stability is a hot topic. We will treat weakening conditions of system (1) and get stable results as one of our future prospects. Furthermore, we might continue to study the input-to-state stability of the neutral stochastic Cohen-Grossberg neural networks with Markovian switching and lévy noise.



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