# Stochastic Dynamics of Nonautonomous Cohen-Grossberg Neural

Authors: Chuangxia Huang; Jinde Cao
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This paper is devoted to the study of the stochastic stability of a class of
Cohen-Grossberg neural networks, in which the interconnections and delays are time-varying.
With the help of Lyapunov function, Burkholder-Davids-Gundy inequality,
and Borel-Cantell's theory, a set of novel sufficient conditions onpth moment exponential stability and almost sure exponential stability for the trivial solution
of the system is derived. Compared with the previous published results, our method
does not resort to the Razumikhin-type theorem and the semimartingale convergence
theorem. Results of the development as presented in this paper are more general than
those reported in some previously published papers. An illustrative example is also
given to show the effectiveness of the obtained results.


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## Body

## 1. Introduction

For decades, the studies of neural networks have attracted considerable multidisciplinary research interest. Ranging from signal processing, pattern recognition, programming problems, and static image processing, neural networks have witnessed a large amount of successful applications in many fields [1–7]. These applications rely crucially on the analysis of the dynamical behavior of the models [8–16]. Most existing literature on theoretical studies of neural networks is predominantly concerned with deterministic differential equations.Recently, studies have been intensively focused on stochastic models [17–24]; it has been realized that the synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes, and it is of great significance to consider stochastic effects on the stability of neural networks described by stochastic functional differential equations, see [25–34].In [17], Liao and Mao studied mean square exponential stability and instability of cellular neural networks (CNNs). In [18, 26], the authors continued their research to discuss almost sure exponential stability for a class of stochastic neural networks with discrete delays by using the nonnegative semimartingale convergence theorem. In [25], exponential stability of stochastic Cohen-Grossberg neural networks (CGNNs) with time-varying delays via Razumikihin-type technique were investigated. In [19], Wan and Sun investigated mean square exponential stability of stochastic delayed Hopfield neural networks (HNNs) by using the method of variation of constants. Also with the help of the method of variation of constants, Sun and Cao in [29] investigated pth moment exponential stability of stochastic recurrent neural networks with time-varying delays.However, to the best of our knowledge, few authors have considered the problem ofpth moment exponential stability and almost sure exponential stability of stochastic nonautonomous Cohen-Grossberg neural networks. In fact, in the process of the electronic circuits’ applications, assuring constant connection matrix and delays are unrealistic. Therefore, in this sense, time-varying connection matrix and delays will be better candidates for modeling neural information processing.Motivated by the above discussions, in this paper, we consider the stochastic Cohen-Grossberg Neural Networks (SCGNN) with time-varying connection matrix and delays described by the following non-autonomous stochastic functional differential equations:(1.1)dxi(t)=-hi(xi(t))[ci(xi(t))-∑j=1naij(t)fj(xj(t))-∑j=1nbij(t)gj(xj(t-τj(t)))]dt+∑j=1nσij(xj(t))dωj(t),i=1,…,n,
or(1.2)dx(t)=-H(x(t))[C(x(t))-A(t)F(x(t))-B(t)G(xτ(t))]dt+σ(x(t))dω(t),
where x(t)=(x1(t),x2(t),…,xn(t))T, H(x(t))=diag(h1(x1(t)),h2(x2(t)),…,hn(xn(t))), A(t)=(aij(t))n×n, B(t)=(bij(t))n×n, G(xτ(t))=(g1(x1(t-τ1(t))),…,gn(xn(t-τn(t))))T, F(x(t))=(f1(x1(t)),…,fn(xn(t)))T, and σ(x(t))=(σij(xj(t)))n×n. Here xi(t) denotes the state variable associated with the ith neuron at time t; hi(·) represent an amplification function; ci(·) is an appropriately behaved function; fj(·) and gj(·) are activation functions; A(t)=(aij(t))n×n and B(t)=(bij(t))n×n represents the strength of the neuron interconnection within the network; τj(t) corresponds to the time delay required in processing, 0≤τj(t)≤τ; σ(·)=(σij(·))n×n is the diffusion coefficient matrix and ω(t)=(ω1(t),…,ωn(t))T is an n-dimensional Brownian motion defined on a complete probability space (Ω,ℱ,P) with a natural filtration {ℱt}t≥0 (i.e., ℱt=σ{w(s):0≤s≤t}).Obviously, model (1.1) or (1.2) is quite general, and it includes several well-known neural networks models as its special cases such as Hopfield neural networks, cellular neural networks, and bidirectional association memory neural networks [10, 16, 27, 28]. There are at least three different types of stochastic stability to describe limiting behaviors of stochastic differential equations: stability in probability, moment stability and almost sure stability (see [35]). When designing an associative memory neural network, we should make convergence speed as high as possible to ensure the quick convergence of the network operation. Therefore, pth moment (p≥2) exponential stability and almost sure exponential stability are most useful concepts as they imply that the solutions will tend to the trivial solution exponentially fast. This motivates us to study pth moment exponential stability, and almost sure exponential stability for System (1.1) in this paper.The remainder of this paper is organized as follows. In Section2, the basic assumptions and preliminaries are introduced. After establishing the criteria for the pth moment (p≥2) exponential stability and almost sure exponential stability for System (1.1) by using the Lyapunov function method, Burkholder-Davids-Gundy inequality and Borel-Cantell’s theory in Section 3, an illustrative example and its simulations are given in Section 4.


## 2. Preliminaries

Throughout this article, we let(Ω,ℱ,{ℱt}t≥0,P) be a complete probability space with a filtration {ℱt}t≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ0 contains all P-null sets). Let 𝒞=C((-∞,0],Rn) be the Banach space of continuous functions which map into Rn with the topology of uniform convergence. For any x(t)=(x1(t),…,xn(t))T∈Rn, we define ∥x(t)∥=∥x(t)∥p=(∑i=1n|xi(t)|p)1/p, (1≤p<∞).The initial conditions for system (1.1) are x(s)=φ(s),-τ≤s≤0,φ∈Lℱ0p((-τ,0],Rn); here Lℱ0p((-τ,0],Rn) is Rn-valued stochastic process φ(s),-τ≤s≤0, φ(s) is ℱ0-measurable, ∫-τ0E|φ(s)|pds<∞. For the sake of convenience, throughout this paper, we assure fj(0)=gj(0)=σij(0)=0, which implies that system (1.1) admits an equilibrium solution x(t)≡0.IfV∈C2,1([-τ,∞)×Rn;R+), according to the Itô formula, define an operator ℒV associated with (1.2) as(2.1)LV(t,x)=Vt+Vx{-H(x(t))[C(x(t))-A(t)F(x(t))-B(t)G(xτ(t))]}+12trace[σTVxxσ],
where Vt=∂V(t,x)/∂t, Vx=(∂V(t,x)/∂x1,…,∂V(t,x)/∂xn), and Vxx=(∂2V(t,x)/∂xi∂xj)n×n.To establish the main results of the model given in (1.1), some of the standing assumptions are formulated as follows: (H1)
there exist positive constantsh̲i, h¯i, such that(2.2)0<h̲i≤hi(x)≤h¯i<+∞,∀x∈R,i=1,2,…,n;(H2)
for eachi=1,2,…,n, there exist positive functions αi(t)>0, such that(2.3)xi(t)ci(xi(t))≥αi(t)xi2(t);(H3)
there exist positive constantsβj,γj,i,j=1,2,…,n, such that(2.4)|fj(u)-fj(v)|≤βj|u-v|,|gj(u)-gj(v)|≤γj|u-v|;(H4)
eachσij(x) satisfies the Lipschitz condition, and there exist positive constants μi,i=1,2,…,n, such that(2.5)trace{σT(x)σ(x)}≤∑i=1nμixi2.Remark 2.1.
The activation functions are typically assumed to be continuous, differentiable, and monotonically increasing, such as the functions of sigmoid type. These restrictive conditions are no longer needed in this paper. Instead, only the Lipschitz condition is imposed in Assumption(H3). Note that the type of activation functions in (H3) have already been used in numerous papers, see [5, 10] and references therein.Remark 2.2.
We remark here that non-autonomous conditions (H2)–(H4) replace the usual autonomous conditions which is more useful for practical purpose; please refer to [4, 13] and references therein.Remark 2.3.
The delay functionsτj(t) considered in this paper only needed to be bounded; they can be time-varying, nondifferentiable functions. This generalized some recently published results in [4, 13, 26–29]. Different from the models considered in [4, 13, 29], in this paper, we have removed the following condition: (H0) For each j=1,2,…,n,τj(t) is a differentiable function, namely, there exists ξ such that
(2.6)τ̇j(t)≤ξ<1.Definition 2.4 (see [35]).
The trivial solution of (1.1) is said to be pth moment exponential stability if there is a pair of positive constants λ and C such that
(2.7)E∥x(t,t0,x0)∥p≤C∥x0∥pe-λ(t-t0),on⁡t≥t0,∀x0∈Rn,
where p≥2 is a constant; when p=2, it is usually said to be exponential stability in mean square.Definition 2.5 (see [35]).
The trivial solution of (1.1) is said to be almost sure exponential stability if for almost all sample paths of the solution x(t), we have
(2.8)limsup⁡t→∞1tlog⁡‖x(t)‖<0.Lemma 2.6 ([35] Burkholder-Davids-Gundy inequality).
There exists a universal constantKp for any 0<p<∞ such that for every continuous local martingale M vanishing at zero and any stopping time η,
(2.9)E(sup⁡0≤s≤η|Ms|p)≤KpE(〈M,M〉η)p/2,
where 〈M,M〉η is the cross-variation of M. In particular, one may have Kp=(32/p)p/2 if 0<P<2 and K2=4 if p=2; although they may not be optimal, for example, one could have K1=42.Lemma 2.7 ([35] Chebyshev's inequality).
(2.10)P{ω:|X(ω)|≥c}≤c-pE|X|p
if c>0, p>0, X∈Lp.Lemma 2.8 ([36] Borel-Cantell's lemma).
Let{An,n≥1} be a sequence of events in some probability space, then(i)
if∑n=1∞P(An)<∞, then P(An,i.o.)=0;(ii)
moreover, if{An,n≥1} are independent of each other, then ∑n=1∞P(An)=∞ implies
(2.11)P(An,i.o.)=1,
where {An,i.o.} denotes occurring infinitely often within {An,n≥1}, that is, {An,i.o.}=∩k=1∞∪n=k∞An. “i.o." is the abbreviation of “infinitely often".


## 3. Main Results

Theorem 3.1.
Under the assumptions (H1)–(H4), if there are a positive diagonal matrix M=diag(m1,…,mn) and two constants 0<N2, 0≤μ<1, such that
(3.1)0<N2≤N2(t)≤μN1(t),fort≥t0,
where
(3.2)N1(t)=min⁡1≤i≤n{ph̲iαi(t)-∑j=1nh¯i(p-1)|aij(t)|βj-∑j=1nmjmih¯j|aji(t)|βi-∑j=1nh¯i(p-1)|bij(t)|γj-∑j=1n(p-1)(p-2)2μj-∑j=1nmjmi(p-1)μi},N2(t)=max⁡1≤i≤n∑j=1nmjmih¯j|bji(t)|γi,
then the trivial solution of system (1.1) is pth moment exponential stability, where p≥2 denotes a positive constant. When p=2, the trivial solution of system (1.1) is exponential stability in mean square.Proof.
Consider the following Lyapunov function:(3.3)V(x,t)=∑i=1nmi|xip(t)|.
As p≥2 denotes a positive constant, we can get the following inequality: if a and b denote nonnegative real numbers, then pap-1b≤(p-1)ap+bp,ap-2b2≤(p-2)ap/p+2bp/p. Using this inequality, then the operator associated with system (1.1) has the form as follows:
(3.4)LV(x,t)=-p∑i=1nmixiP-2(t)xi(t)hi(xi(t))ci(xi(t))sgn⁡{xi(t)}+p∑i=1nmixiP-1(t)hi(xi(t))∑j=1naij(t)fj(xj(t))sgn⁡{xi(t)}+p∑i=1nmixiP-1(t)hi(xi(t))∑j=1nbij(t)gj(xj(t-τj))sgn⁡{xi(t)}+p(p-1)2∑i=1nmixiP-2(t)∑j=1nσij2sgn⁡{xi(t)}≤-p∑i=1nmi|xiP-2(t)|h̲iαi(t)xi2(t)+p∑i=1nmi|xiP-1(t)|h¯i∑j=1n|aij(t)|βj|xj(t)|+p∑i=1nmi|xiP-1(t)|h¯i∑j=1n|bij(t)|γj|xj(t-τj(t))|+p(p-1)2∑i=1nmi|xiP-2(t)|∑j=1nμjxj2(t)≤-N1(t)V(x(t),t)+N2(t)sup⁡t-τ≤s≤t{V(x(s),s)},
where
(3.5)N1(t)=min⁡1≤i≤n{ph̲iαi(t)-∑j=1nh¯i(p-1)|aij(t)|βj-∑j=1nmjmih¯j|aji(t)|βi-∑j=1nh¯i(p-1)|bij(t)|γj-∑j=1n(p-1)(p-2)2μj-∑j=1nmjmi(p-1)μi},N2(t)=max⁡1≤i≤n∑j=1nmjmih¯j|bji(t)|γi.
The remaining part of the proof is similar to that of Theorem 3.3 in [33]; we omit it.In Theorem3.1, if we let M be the identity matrix, we can easily obtain the following corollary.Corollary 3.2.
Under the assumptions (H1)–(H4), if there are two constants 0<N2,0≤μ<1, such that
(3.6)0<N2≤N2(t)≤μN1(t),∀t≥t0,
where
(3.7)N1(t)=min⁡1≤i≤n{ph̲iαi(t)-∑j=1nh¯i(p-1)|aij(t)|βj-∑j=1nh¯j|aji(t)|βi-∑j=1nh¯i(p-1)|bij(t)|γj-∑j=1n(p-1)(p-2)2μj-∑j=1n(p-1)μi},N2(t)=max⁡1≤i≤n∑j=1nh¯j|bji(t)|γi,
then the trivial solution of system (1.1) is pth moment exponentially stability.Remark 3.3.
Compared with [10, 12], our method does not resort to the Razumikhin-type theorem or Halanay inequality.Theorem 3.4.
Suppose system (1.1) satisfies assumptions (H1)–(H4) and the inequality (3.1) hold; if aij(t), bij(t), and αi(t) are bounded functions for all i, j, then the trivial solution of (1.1) is almost sure exponential stability.Proof.
LetN be an integer such that N-τ≥1 and IN=[N,N+1]; consider the following Lyapunov function:
(3.8)V(x(t))=‖x(t)‖p.
Using the Itô formula, we have
(3.9)d‖x(t)‖p=∑i=1n{∑j=1n-p|xi(t)|p-2xi(t)hi(xi(t))×[ci(xi(t))-∑j=1naij(t)fj(xj(t))-∑j=1nbij(t)gj(xj(t-τj(t)))]}dt+p(p-1)2∑i=1n|xi(t)|p-2trace(σT(x(t))σ(x(t)))dt+∑i=1n∑j=1np|xi(t)|p-2xi(t)σij(xj(t))dωj(t).
Calculating the integral of (3.9) from N to t, we have
(3.10)‖x(t)‖p-‖x(N)‖p=∫Nt{∑i=1n-p|xi(s)|p-2xi(s)hi(xi(s))×[ci(xi(s))-∑j=1naij(t)fj(xj(s))-∑j=1nbij(t)gj(xj(s-τj(t)))]+p(p-1)2∑i=1n|xi(s)|p-2trace(σT(x(s))σ(x(s)))}ds+∫Nt∑i=1n∑j=1np|xi(s)|p-2xi(s)σij(xj(s))dωj(s),(3.11)Esup⁡t∈IN‖x(t)‖p=Esup⁡t∈IN{‖x(N)‖p+∫Nt{∑i=1n-p|xi(s)|p-2xi(s)xi(s)hi(xi(s))×[ci(xi(s))-∑j=1naij(s)fj(xj(s))-∑j=1nbij(s)gj(xj(s-τj(s)))]+p(p-1)2∑i=1n|xi(s)|p-2trace(σT(x(s))σ(x(s)))}ds+∫Nt∑i=1n∑j=1np|xi(s)|p-2xi(s)σij(xj(s))dωj(s)}≤E‖x(N)‖p+M1+M2,
where
(3.12)M1=Esup⁡t∈IN∫Nt{∑i=1n-p|xi(s)|p-2xi(s)hi(x(s))×[ci(xi(s))-∑j=1naij(s)fj(xj(s))-∑j=1nbij(s)gj(xj(s-τj(s)))]+p(p-1)2∑i=1n|xi(s)|p-2trace(σT(x(s))σ(x(s)))}ds,M2=Esup⁡t∈IN∫Nt∑i=1n∑j=1np|xi(s)|p-2xi(s)σij(xj(s))dωj(s).
From Theorem 3.1, there exists a pair of positive constants λ and δ, such that
(3.13)E‖x(t)‖p≤δ‖x0‖pe-λ(t-t0),on⁡t≥t0.
Furthermore, from (H1)–(H4) and inequality (3.13), we have
(3.14)M1≤E∫NN+1{∑i=1n[∑j=1n-p|xi(s)|ph̲iαi(t)+∑j=1np|xi(s)|p-2|aij(s)|h¯iβj|xi(s)xj(s)|+∑j=1np|xi(s)|p-2|bij(s)|h¯iγj|xi(s)xj(s-τj(s))|]+p(p-1)2∑i=1nμi|xip(s)|}ds≤E∫NN+1{∑i=1n[∑j=1n-ph̲iαi(t)|xi(s)|p+∑j=1n|aij(s)|h¯iβj[(p-1)|xi(s)|p+|xj(s)|p]+∑j=1n|bij(s)|h¯iγj[(p-1)|xi(s)|p+|xj(s-τj(s))|p]]+p(p-1)2∑i=1nμi|xi(s)|p}ds≤∫NN+1{∑i=1n[ph̲iαi(t)+∑j=1np|aij(s)|h¯iβj+∑j=1n(p-1)|bij(s)|h¯iγj+p(p-1)2∑i=1nμi]E‖x(s)‖p}ds+∫NN+1{∑i=1n∑j=1n|bij(s)|h¯iγjE‖x(s-τj(s))‖p}ds≤∫NN+1{∑i=1n[ph̲iαi(t)+p∑j=1n|aij(s)|h¯iβj+∑j=1n|bij(s)|h¯iγj(p-1+eλτ)+p(p-1)2μi]eλt0}δ‖x0‖pe-λsds≤I1δ∥x0∥pe-λN,
where
(3.15)I1=sup⁡t∈R{∑i=1n[ph̲iαi(t)+p∑j=1n|aij(t)|h¯iβj+∑j=1n|bij(t)|h¯iγj(p-1+eλτ)+p(p-1)2μi]eλt0}.
For any two different norms ∥·∥2, ∥·∥p-2, (1<p<∞),for allx∈Rn, as the space Rn is a finite dimensional space, there exist two positive constants ζ1, ζ2, such that
(3.16)‖x‖22≤ζ1‖x‖p2,‖x‖p-2p-2≤ζ2‖x‖pp-2.
As M=∫Nt∑i=1n∑j=1np|xi(s)|p-2xi(s)σij(xj(s))dωj(s) is continuous local martingale, then from Lemma 2.6, (H4), and (3.16), it follows that
(3.17)M2≤K1E{∫NN+1∑j=1n[∑i=1np|xi(s)|p-1|σij(xj(s))|]2ds}1/2≤42E{∫NN+1∑j=1n{[∑i=1n|xi(s)|2p-2][∑i=1np2σij2(xj(s))]}ds}1/2≤42E{∫NN+1p2∑j=1nμj|xj(t)|2[∑i=1n|xi(s)|2p-2]ds}1/2≤42E{∫NN+1p2max⁡1≤i≤n{μi}ζ1‖x‖p2ζ2‖x‖pp-2ds}1/2≤2E{[12sup⁡s∈IN‖x(s)‖p]∫NN+116p2ζ1ζ2max⁡1≤i≤n{μi}‖x(s)‖pds}1/2≤12Esup⁡t∈IN‖x(t)‖p+16p2ζ1ζ2max⁡1≤i≤n{μi}∫NN+1E{‖x(s)‖p}ds.
According to (3.11), (3.13), (3.14), and (3.17), we have the following inequality:
(3.18)Esup⁡t∈IN‖x(t)‖p≤E‖x(N)‖p+M1+M2≤δ‖x0‖pe-λ(N-t0)+I1δ‖x0‖pe-λN+12Esup⁡t∈IN‖x(t)‖p+16p2ζ1ζ2max⁡1≤i≤n{μi}∫NN+1δ‖x0‖pe-λ(s-t0)ds≤12Esup⁡t∈IN‖x(t)‖p+{eλt0+I1+16p2ζ1ζ2λmax⁡1≤i≤n{μi}eλt0}δe-λN.
Therefore,
(3.19)Esup⁡t∈IN‖x(t)‖p≤θe-λN,
where
(3.20)θ=2{eλt0+I1+16p2ζ1ζ2λmax⁡1≤i≤n{μi}eλt0}δ.
For each integer N, we set ɛN=e-λN/2. Then, from Lemma 2.7, we have
(3.21)P{ω:sup⁡t∈IN‖x(t)‖p>ɛN}≤Esup⁡t∈IN‖x(t)‖pɛN≤θe-λN/2.
Therefore, in view of Lemma 2.8, for almost all ω∈Ω, we have that
(3.22)sup⁡t∈IN‖x(t)‖p≤e-λN/2,
holds for all but finitely many N. Hence, there exists an N0=N0(ω), for all ω∈Ω excluding a P-null set, for which (3.21) holds whenever N≥N0. Consequently, for almost all ω∈Ω,
(3.23)1tlog⁡‖x(t)‖=1ptlog⁡‖x(t)‖p≤-λ2p,
if N≤t≤N+1, N≥N0. Hence
(3.24)limsup⁡t→∞1tlog⁡‖x(t)‖=1pt≤-λpa.s.
Therefore, the trivial solution of (1.1) is almost sure exponential stability.Remark 3.5.
Compared with [26, 32], our method does not resort to the semimartingale convergence theorem. Since system (1.1) does not require the delays to be constants, furthermore, the model is non-autonomous, it is clear that the results obtained in [19, 25–32, 34] cannot be applicable to system (1.1). This implies that the results of this paper are essentially new and complement some corresponding ones already known.Remark 3.6.
By Theorems3.1 and 3.4, the stability of system (1.1) is dependent on the magnitude of noise, and therefore, stochastic noise fluctuation is one of the very important aspects in designing a stable network and should to be considered adequately.
It should be noted that the assumptions of the boundedness ofaij(t), bij(t), and αi(t) in Theorem 3.4 are not necessary; we use these assumptions just to simplify the process of the proof. In fact, in view of (3.15), (3.20), (3.21), and (3.22), similar to the proof of Theorem 3.4, we have the following theorem.Theorem 3.7.
Suppose system (1.1) satisfies assumptions (H1)–(H4) and the inequality (3.1) hold, if there exist positive constants ρij, ρij, ϑij such that for any t, we have
(3.25)|aij(t)|⋁|bij(t)|⋁αi(t)≤ρijtρij+ϑij,
then the trivial solution of (1.1) is almost sure exponential stability.Remark 3.8.
Furthermore, the derived conditions for stability of the following stochastic delayed recurrent neural networks can be viewed as byproducts of our results. The significant of this paper does offer a wider selection on the networks parameters in order to achieve some necessary convergence in practice.Remark 3.9.
For system (1.1), when hi(xi(t))=1, ci(xi(t))=cixi(t)(ci>0), and aij(t)=aij,bij(t)=bij, then it turns out to be following stochastic delayed recurrent neural networks with time-varying delays
(3.26)dxi(t)=-cixi(t)dt+∑j=1naijfj(xj(t))dt+∑j=1nbijgj(xj(t-τj(t)))dt+∑j=1nσij(xj(t))dωj(t).
Using Theorems 3.1 and 3.4, one can easily get a set of similar corollary for checking the pth moment exponential stability and almost sure exponential stability for the trivial solution of this system.


## 4. An Illustrative Example

In this section, an example is presented to demonstrate the correctness and effectiveness of the main obtained results.Example 4.1.
Consider the following stochastic Cohen-Grossberg neural networks with time-varying delays:(4.1)dx(t)=(3-cos⁡(x1(t))003-sin(x2(t)))×[-(5+0.002t005+0.003t)(x1(t)x2(t))+(-20.40.61)(0.2tanh(x1(t))0.2tanh(x2(t)))+(-0.821-2)(0.2tanh(x1(t-τ1(t)))0.2tanh(x2(t-τ2(t))))]dt+σ(x(t))dw(t),t≥0,
where τ(t)=(τ1(t),τ2(t))Tand τi(t) is any bounded positive function for i=1,2. Each σij(x) satisfies the Lipschitz condition, and there exist positive constants μ1=μ2=2, such that
(4.2)trace{σT(x)σ(x)}≤2(x12+x22).
In the example, let p=3; by simple computation, we obtain
(4.3)N1(t)=min⁡1≤i≤n{ph̲iαi(t)-∑j=1nh¯i(p-1)|aij(t)|βj-∑j=1nmjmih¯j|aji(t)|βi-∑j=1nh¯i(p-1)|bij(t)|γj-∑j=1n(p-1)(p-2)2μj-∑j=1nmjmi(p-1)μi}=11.6+0.008t,N2(t)=max⁡1≤i≤n∑j=1nmjmih¯j|bji(t)|γi=8.8+0.0008t.
Choosing μ=8/9, one can easily get that
(4.4)0<N2≤N2(t)≤μN1(t),fort≥0.
Thus, it follows Theorem 3.7 that system (4.1) is the third moment exponentially stable and also almost sure exponentially stable. These conclusions can be verified by the following numerical simulations (Figures 1, 2, 3, and 4).Figure 1
Numerical solutionE(x13(t)) of system (4.1).Figure 2
Numerical solutionE(x23(t)) of system (4.1).Figure 3
Numerical solutionx1(t) of system (4.1).Figure 4
Numerical solutionx2(t) of system (4.1).Remark 4.2.
Letτ1(t)=τ2(t)=0.5sint+1; we can find that [29, Theorem 1] is not satisfied; therefore, they fail to conclude whether system (4.1) is pth moment exponentially stable even when the delay functions are differential and their derivatives are simultaneously required to be not greater than 1. It is obvious that the results in [19, 25–32, 34] and the references therein cannot be applicable to system (4.1).



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*Source: 297147-2011-05-29.xml*