Stochastic Dynamics of Nonautonomous Cohen-Grossberg Neural Networks

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 on 𝑝th 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.

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 of 𝑝th 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:d𝑥_𝑖(𝑡)=−ℎ_𝑖𝑥_𝑖(𝑡)𝑐_𝑖𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)−^𝑛_𝑗=1+𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)d𝑡_𝑛_𝑗=1𝜎_𝑖𝑗𝑥_𝑗(𝑡)d𝜔_𝑗(𝑡),𝑖=1,…,𝑛,(1.1) ord𝑥(𝑡)=−𝐻(𝑥(𝑡))𝐶(𝑥(𝑡))−𝐴(𝑡)𝐹(𝑥(𝑡))−𝐵(𝑡)𝐺𝑥_𝜏(𝑡)𝑑𝑡+𝜎(𝑥(𝑡))d𝜔(𝑡),(1.2) where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇, 𝐻(𝑥(𝑡))=diag(ℎ₁(𝑥₁(𝑡)),ℎ₂(𝑥₂(𝑡)),…,ℎ_𝑛(𝑥_𝑛(𝑡))), 𝐴(𝑡)=(𝑎_𝑖𝑗(𝑡))_𝑛×𝑛, 𝐵(𝑡)=(𝑏_𝑖𝑗(𝑡))_𝑛×𝑛, 𝐺(𝑥_𝜏(𝑡))=(𝑔₁(𝑥₁(𝑡−𝜏₁(𝑡))),…,𝑔_𝑛(𝑥_𝑛(𝑡−𝜏_𝑛(𝑡))))^𝑇, 𝐹(𝑥(𝑡))=(𝑓₁(𝑥₁(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡)))^𝑇, and 𝜎(𝑥(𝑡))=(𝜎_𝑖𝑗(𝑥_𝑗(𝑡)))_𝑛×𝑛. Here 𝑥_𝑖(𝑡) denotes the state variable associated with the 𝑖th neuron at time 𝑡; ℎ_𝑖(⋅) represent an amplification function; 𝑐_𝑖(⋅) is an appropriately behaved function; 𝑓_𝑗(⋅) and 𝑔_𝑗(⋅) are activation functions; 𝐴(𝑡)=(𝑎_𝑖𝑗(𝑡))_𝑛×𝑛 and 𝐵(𝑡)=(𝑏_𝑖𝑗(𝑡))_𝑛×𝑛 represents the strength of the neuron interconnection within the network; 𝜏_𝑗(𝑡) corresponds to the time delay required in processing, 0≤𝜏_𝑗(𝑡)≤𝜏; 𝜎(⋅)=(𝜎_𝑖𝑗(⋅))_𝑛×𝑛 is the diffusion coefficient matrix and 𝜔(𝑡)=(𝜔₁(𝑡),…,𝜔_𝑛(𝑡))^𝑇 is an n-dimensional Brownian motion defined on a complete probability space (Ω,ℱ,𝑃) with a natural filtration {ℱ_𝑡}_𝑡≥0 (i.e., ℱ _𝑡=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 Section 2, 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,{ℱ_𝑡}_𝑡≥0,𝑃) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ₀ contains all 𝑃-null sets). Let 𝒞=𝐶((−∞,0],𝑅^𝑛) be the Banach space of continuous functions which map into 𝑅^𝑛 with the topology of uniform convergence. For any 𝑥(𝑡)=(𝑥₁(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈𝑅^𝑛, we define ‖𝑥(𝑡)‖=‖𝑥(𝑡)‖_𝑝=(∑^𝑛𝑖=1|𝑥_𝑖(𝑡)|^𝑝)^1/𝑝, (1≤𝑝<∞).

The initial conditions for system (1.1) are 𝑥(𝑠)=𝜑(𝑠),−𝜏≤𝑠≤0,𝜑∈𝐿^𝑝ℱ₀((−𝜏,0],𝑅^𝑛); here 𝐿^𝑝ℱ₀((−𝜏,0],𝑅^𝑛) is 𝑅^𝑛-valued stochastic process 𝜑(𝑠),−𝜏≤𝑠≤0, 𝜑(𝑠) is ℱ₀-measurable, ∫^0−𝜏𝐸|𝜑(𝑠)|^𝑝d𝑠<∞. For the sake of convenience, throughout this paper, we assure 𝑓_𝑗(0)=𝑔_𝑗(0)=𝜎_𝑖𝑗(0)=0, which implies that system (1.1) admits an equilibrium solution 𝑥(𝑡)≡0.

If 𝑉∈𝐶^2,1([−𝜏,∞)×𝑅^𝑛;𝑅₊), according to the Itô formula, define an operator ℒ𝑉 associated with (1.2) asℒ𝑉(𝑡,𝑥)=+𝑉_𝑡1+𝑉_𝑥−𝐻(𝑥(𝑡))𝐶(𝑥(𝑡))−𝐴(𝑡)𝐹(𝑥(𝑡))−𝐵(𝑡)𝐺𝑥_𝜏(𝑡)2trace𝜎^𝑇𝑉_𝑥𝑥𝜎,(2.1) where 𝑉_𝑡=𝜕𝑉(𝑡,𝑥)/𝜕𝑡, 𝑉_𝑥=(𝜕𝑉(𝑡,𝑥)/𝜕𝑥₁,…,𝜕𝑉(𝑡,𝑥)/𝜕𝑥_𝑛), and 𝑉_𝑥𝑥=(𝜕²𝑉(𝑡,𝑥)/𝜕𝑥_𝑖𝜕𝑥_𝑗)_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 constants ℎ_𝑖, ℎ_𝑖, such that 0<ℎ_𝑖≤ℎ_𝑖(𝑥)≤ℎ_𝑖<+∞,∀𝑥∈𝑅,𝑖=1,2,…,𝑛;(2.2)(𝐻₂)for each 𝑖=1,2,…,𝑛, there exist positive functions 𝛼_𝑖(𝑡)>0, such that 𝑥_𝑖(𝑡)𝑐_𝑖𝑥_𝑖(𝑡)≥𝛼_𝑖(𝑡)𝑥^2𝑖(𝑡);(2.3)(𝐻₃)there exist positive constants 𝛽_𝑗,𝛾_𝑗,𝑖,𝑗=1,2,…,𝑛, such that ||𝑓_𝑗(𝑢)−𝑓_𝑗(𝑣)||≤𝛽_𝑗|𝑢−𝑣|,||𝑔_𝑗(𝑢)−𝑔_𝑗(𝑣)||≤𝛾_𝑗|𝑢−𝑣|;(2.4)(𝐻₄)each 𝜎_𝑖𝑗(𝑥) satisfies the Lipschitz condition, and there exist positive constants 𝜇_𝑖,𝑖=1,2,…,𝑛, such that trace𝜎^𝑇(𝑥)𝜎(𝑥)≤^𝑛_𝑖=1𝜇_𝑖𝑥^2𝑖.(2.5)

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 functions 𝜏_𝑗(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,𝜏_𝑗(𝑡) is a differentiable function, namely, there exists 𝜉 such that 𝜏̇_𝑗(𝑡)≤𝜉<1.(2.6)

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 𝐸‖𝑥𝑡,𝑡₀,x₀‖^𝑝≤𝐶‖𝑥₀‖^𝑝𝑒^{−𝜆(𝑡−𝑡0)},on𝑡≥𝑡₀,∀𝑥₀∈𝑅^𝑛,(2.7) 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 limsup_𝑡→∞1𝑡log‖𝑥(𝑡)‖<0.(2.8)

Lemma 2.6 ([35] Burkholder-Davids-Gundy inequality). There exists a universal constant 𝐾_𝑝 for any 0<𝑝<∞ such that for every continuous local martingale 𝑀 vanishing at zero and any stopping time 𝜂, 𝐸sup_{0≤𝑠≤𝜂}||𝑀_𝑠||^𝑝≤𝐾^𝑝𝐸⟨𝑀,𝑀⟩_𝜂^𝑝/2,(2.9) where ⟨𝑀,𝑀⟩_𝜂 is the cross-variation of 𝑀. In particular, one may have 𝐾_𝑝=(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=4√2.

Lemma 2.7 ([35] Chebyshev's inequality). 𝑃𝜔∶||𝑋(𝜔)||≥𝑐≤𝑐^−𝑝𝐸||𝑋||^𝑝(2.10) if 𝑐>0, 𝑝>0, 𝑋∈𝐿^𝑝.

Lemma 2.8 ([36] Borel-Cantell's lemma). Let {𝐴_𝑛,𝑛≥1} be a sequence of events in some probability space, then (i)if ∑^∞𝑛=1𝑃(𝐴_𝑛)<∞, then 𝑃(𝐴_𝑛,𝑖.𝑜.)=0;(ii)moreover, if {𝐴_𝑛,𝑛≥1} are independent of each other, then ∑^∞𝑛=1𝑃(𝐴_𝑛)=∞ implies 𝑃𝐴_𝑛,𝑖.𝑜.=1,(2.11) where {𝐴_𝑛,𝑖.𝑜.} denotes occurring infinitely often within {𝐴_𝑛,𝑛≥1}, that is, {𝐴_𝑛,𝑖.𝑜.}=∩^∞𝑘=1∪^∞𝑛=𝑘𝐴_𝑛. “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,…,𝑚_𝑛) and two constants 0<𝑁₂, 0≤𝜇<1, such that 0<𝑁₂≤𝑁₂(𝑡)≤𝜇𝑁₁(𝑡),for𝑡≥𝑡₀,(3.1) where 𝑁₁(𝑡)=min_{1≤𝑖≤𝑛}𝑝ℎ^𝑖𝛼_𝑖(𝑡)−^𝑛_𝑗=1ℎ_𝑖(𝑝−1)||𝑎_𝑖𝑗(𝑡)||𝛽_𝑗−^𝑛_𝑗=1𝑚_𝑗𝑚_𝑖ℎ_𝑗||𝑎_𝑗𝑖(𝑡)||𝛽_𝑖−^𝑛_𝑗=1ℎ_𝑖(𝑝−1)||𝑏_𝑖𝑗(𝑡)||𝛾_𝑗−^𝑛_𝑗=1(𝑝−1)(𝑝−2)2𝜇^𝑗−^𝑛_𝑗=1𝑚_𝑗𝑚_𝑖(𝑝−1)𝜇_𝑖𝑁₂,(𝑡)=max_{𝑛1≤𝑖≤𝑛}_𝑗=1𝑚_𝑗𝑚_𝑖ℎ_𝑗||𝑏_𝑗𝑖(𝑡)||𝛾_𝑖,(3.2) 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: 𝑉(𝑥,𝑡)=^𝑛_𝑖=1𝑚_𝑖||𝑥^𝑝𝑖(𝑡)||.(3.3) As 𝑝≥2 denotes a positive constant, we can get the following inequality: if 𝑎 and 𝑏 denote nonnegative real numbers, then 𝑝𝑎^𝑝−1𝑏≤(𝑝−1)𝑎^𝑝+𝑏^𝑝,𝑎^𝑝−2𝑏²≤(𝑝−2)𝑎^𝑝/𝑝+2𝑏^𝑝/𝑝. Using this inequality, then the operator associated with system (1.1) has the form as follows: ℒ𝑉(𝑥,𝑡)=−𝑝^𝑛_𝑖=1+𝑝_𝑛𝑚_𝑖𝑥^𝑖𝑃−2(𝑡)𝑥_𝑖(𝑡)ℎ_𝑖𝑥_𝑖(𝑡)𝑐_𝑖𝑥_𝑖(𝑡)sgn𝑥_𝑖(𝑡)_𝑖=1𝑚_𝑖𝑥^𝑖𝑃−1(𝑡)ℎ_𝑖𝑥_𝑖(𝑡)^𝑛_𝑗=1+𝑝_𝑛𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)sgn𝑥_𝑖(𝑡)_𝑖=1𝑚_𝑖𝑥^𝑖𝑃−1(𝑡)ℎ_𝑖𝑥_𝑖(𝑡)^𝑛_𝑗=1+𝑝(𝑝−1)𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗sgn𝑥_𝑖(𝑡)2^𝑛_𝑖=1𝑚_𝑖𝑥^𝑖𝑃−2(𝑡)^𝑛_𝑗=1≤−𝑝_𝑛𝜎^2𝑖𝑗sgn𝑥_𝑖(𝑡)_𝑖=1+𝑝_𝑛𝑚_𝑖||𝑥^𝑖𝑃−2(𝑡)||ℎ_𝑖𝛼_𝑖(𝑡)𝑥^2𝑖(𝑡)_𝑖=1𝑚_𝑖||𝑥^𝑖𝑃−1(𝑡)||ℎ_𝑖𝑛_𝑗=1+𝑝_𝑛||𝑎_𝑖𝑗(𝑡)||𝛽_𝑗||𝑥_𝑗(𝑡)||_𝑖=1𝑚_𝑖||𝑥^𝑖𝑃−1(𝑡)||ℎ_𝑖𝑛_𝑗=1+𝑝(𝑝−1)||𝑏_𝑖𝑗(𝑡)||𝛾_𝑗||𝑥_𝑗𝑡−𝜏_𝑗(𝑡)||2^𝑛_𝑖=1𝑚_𝑖||𝑥^𝑖𝑃−2(𝑡)||^𝑛_𝑗=1≤−𝑁₁𝜇_𝑗𝑥^2𝑗(𝑡)(𝑡)𝑉(𝑥(𝑡),𝑡)+𝑁₂(𝑡)sup_{𝑡−𝜏≤𝑠≤𝑡}{𝑉(𝑥(𝑠),𝑠)},(3.4) where 𝑁₁(𝑡)=min_{1≤𝑖≤𝑛}𝑝ℎ^𝑖𝛼_𝑖(𝑡)−^𝑛_𝑗=1ℎ_𝑖(𝑝−1)||𝑎_𝑖𝑗(𝑡)||𝛽_𝑗−^𝑛_𝑗=1𝑚_𝑗𝑚_𝑖ℎ_𝑗||𝑎_𝑗𝑖(𝑡)||𝛽_𝑖−^𝑛_𝑗=1ℎ_𝑖(𝑝−1)||𝑏_𝑖𝑗(𝑡)||𝛾_𝑗−^𝑛_𝑗=1(𝑝−1)(𝑝−2)2𝜇^𝑗−^𝑛_𝑗=1𝑚_𝑗𝑚_𝑖(𝑝−1)𝜇_𝑖𝑁₂,(𝑡)=max_{𝑛1≤𝑖≤𝑛}_𝑗=1𝑚_𝑗𝑚_𝑖ℎ_𝑗||𝑏_𝑗𝑖(𝑡)||𝛾_𝑖.(3.5) The remaining part of the proof is similar to that of Theorem 3.3 in [33]; we omit it.

In Theorem 3.1, 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 0<𝑁₂≤𝑁₂(𝑡)≤𝜇𝑁₁(𝑡),∀𝑡≥𝑡₀,(3.6) where 𝑁₁(𝑡)=min_{1≤𝑖≤𝑛}𝑝ℎ^𝑖𝛼_𝑖(𝑡)−ⁿ_𝑗=1ℎ_𝑖(𝑝−1)||𝑎_𝑖𝑗(𝑡)||𝛽_𝑗−^𝑛_𝑗=1ℎ_𝑗||𝑎_𝑗𝑖(𝑡)||𝛽_𝑖−^𝑛_𝑗=1ℎ_𝑖(𝑝−1)||𝑏_𝑖𝑗(𝑡)||𝛾_𝑗−^𝑛_𝑗=1(𝑝−1)(𝑝−2)2𝜇^𝑗−^𝑛_𝑗=1(𝑝−1)𝜇_𝑖𝑁₂,(𝑡)=max_{𝑛1≤𝑖≤𝑛}_𝑗=1ℎ_𝑗||𝑏_𝑗𝑖(𝑡)||𝛾_𝑖,(3.7) 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 𝑎_𝑖𝑗(𝑡), 𝑏_𝑖𝑗(𝑡), and 𝛼_𝑖(𝑡) are bounded functions for all 𝑖, 𝑗, then the trivial solution of (1.1) is almost sure exponential stability.

Proof. Let 𝑁 be an integer such that 𝑁−𝜏≥1 and 𝐼_𝑁=[𝑁,𝑁+1]; consider the following Lyapunov function: 𝑉(𝑥(𝑡))=‖𝑥(𝑡)‖^𝑝.(3.8) Using the Itô formula, we have d‖𝑥(𝑡)‖^𝑝=^𝑛_𝑖=1−𝑝||𝑥ⁱ×𝑐(𝑡)||^𝑝−2𝑥_𝑖(𝑡)ℎ_𝑖𝑥_𝑖(𝑡)_𝑖+𝑝(𝑝−1)𝑥_𝑖(𝑡)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑡)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)d𝑡2^𝑛_𝑖=1+||𝑥_𝑖(𝑡)||^𝑝−2trace𝜎^𝑇(𝑥(𝑡))𝜎(𝑥(𝑡))d𝑡_𝑛_𝑛𝑖=1_𝑗=1𝑝||𝑥_𝑖(𝑡)||^𝑝−2𝑥_𝑖(𝑡)𝜎_𝑖𝑗𝑥_𝑗(𝑡)d𝜔_𝑗(𝑡).(3.9) Calculating the integral of (3.9) from 𝑁 to 𝑡, we have ‖𝑥(𝑡)‖^𝑝−‖𝑥(𝑁)‖^𝑝=^𝑡𝑁^𝑛_𝑖=1×𝑐−𝑝||𝑥_𝑖(𝑠)||^𝑝−2𝑥_𝑖(𝑠)ℎ_𝑖𝑥_𝑖(𝑠)_𝑖+𝑝(𝑝𝑥_𝑖−(𝑠1))−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑓_𝑗𝑥_𝑗(𝑠)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑗(𝑡)2^𝑛_𝑖=1+||𝑥_𝑖(𝑠)||^𝑝−2trace𝜎^𝑇(𝑥(𝑠))𝜎(𝑥(𝑠))d𝑠_𝑡𝑁𝑛_𝑛𝑖=1_𝑗=1𝐸sup_{𝑡∈𝐼𝑁}𝑝||𝑥_𝑖(𝑠)||^𝑝−2𝑥_𝑖(𝑠)𝜎_𝑖𝑗𝑥_𝑗(𝑠)d𝜔_𝑗(𝑠),(3.10)‖𝑥(𝑡)‖^𝑝=𝐸sup_{𝑡∈𝐼𝑁}‖𝑥(𝑁)‖^𝑝+^𝑡𝑁^𝑛_𝑖=1×𝑐−𝑝||𝑥_𝑖(𝑠)||^𝑝−2𝑥_𝑖(𝑠)𝑥_𝑖(𝑠)ℎ_𝑖𝑥_𝑖(𝑠)_𝑖−𝑥_𝑖(𝑠)−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗(𝑠)_𝑛_𝑗=1+𝑝(𝑝−1)𝑏_𝑖𝑗(𝑠)𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑗(𝑠)2^𝑛_𝑖=1+||𝑥_𝑖(𝑠)||^𝑝−2trace𝜎^𝑇(𝑥(𝑠))𝜎(𝑥(𝑠))d𝑠_𝑡𝑁𝑛_𝑛𝑖=1_𝑗=1≤𝐸‖𝑥(𝑁)‖_𝑝𝑝||𝑥_𝑖(𝑠)||^𝑝−2𝑥_𝑖(𝑠)𝜎_𝑖𝑗𝑥_𝑗(𝑠)d𝜔_𝑗(𝑠)+𝑀₁+𝑀₂,(3.11) where 𝑀₁=𝐸sup_{𝑡∈𝐼𝑁}^𝑡𝑁^𝑛_𝑖=1×𝑐−𝑝||𝑥_𝑖(𝑠)||^𝑝−2𝑥_𝑖(𝑠)ℎ_𝑖(𝑥(𝑠))_𝑖+𝑝(𝑝𝑥−_𝑖(𝑠1))−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑓_𝑗𝑥_𝑗(𝑠)−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑠)𝑔_𝑗𝑥_𝑗𝑠−𝜏_𝑗(𝑠)2^𝑛_𝑖=1𝑀||𝑥_𝑖(𝑠)||^𝑝−2trace𝜎^𝑇(𝑥(𝑠))𝜎(𝑥(𝑠))d𝑠,₂=𝐸sup_{𝑡∈𝐼𝑁}^𝑡𝑁𝑛_𝑛𝑖=1_𝑗=1𝑝||𝑥_𝑖(s)||^𝑝−2𝑥_𝑖(𝑠)𝜎_𝑖𝑗𝑥_𝑗(𝑠)d𝜔_𝑗(𝑠).(3.12) From Theorem 3.1, there exists a pair of positive constants 𝜆 and 𝛿, such that 𝐸‖𝑥(𝑡)‖^𝑝≤𝛿‖‖𝑥₀‖‖^𝑝𝑒^{−𝜆(𝑡−𝑡0)},on𝑡≥𝑡₀.(3.13) Furthermore, from (𝐻₁)–(𝐻₄) and inequality (3.13), we have 𝑀₁≤𝐸^𝑁𝑁+1^𝑛_𝑖=1−𝑝||𝑥^𝑖(𝑠)||^𝑝ℎ_𝑖𝛼_𝑖(𝑡)+^𝑛_𝑗=1+𝑝||𝑥_𝑖(𝑠)||^𝑝−2||𝑎_𝑖𝑗(𝑠)||ℎ_𝑖𝛽_𝑗||𝑥_𝑖(𝑠)𝑥_𝑗(𝑠)_𝑛||_𝑗=1+𝑝(𝑝−1)𝑝||𝑥_𝑖(𝑠)||^𝑝−2||𝑏_𝑖𝑗(𝑠)||ℎ_𝑖𝛾_𝑗||𝑥_𝑖(𝑠)𝑥_𝑗𝑠−𝜏_𝑗(𝑠)||2^𝑛_𝑖=1≤𝐸_𝑁𝑁+1𝜇_i||𝑥^𝑝𝑖(𝑠)||d𝑠^𝑛_𝑖=1−𝑝ℎ^𝑖𝛼_𝑖(𝑡)||𝑥_𝑖(𝑠)||^𝑝+^𝑛_𝑗=1+_𝑛||𝑎_𝑖𝑗(𝑠)||ℎ_𝑖𝛽_𝑗(𝑝−1)||𝑥_𝑖(𝑠)||^𝑝+||𝑥_𝑗(𝑠)||^𝑝_𝑗=1+𝑝(𝑝−1)||𝑏_𝑖𝑗(𝑠)||ℎ_𝑖𝛾_𝑗(𝑝−1)||𝑥_𝑖(𝑠)||^𝑝+||𝑥_𝑗𝑠−𝜏_𝑗(𝑠)||^𝑝2^𝑛_𝑖=1𝜇_𝑖||𝑥_𝑖(𝑠)||^𝑝≤d𝑠_𝑁𝑁+1^𝑛_𝑖=1𝑝ℎ^𝑖𝛼_𝑖(𝑡)+^𝑛_𝑗=1𝑝||𝑎_𝑖𝑗(𝑠)||ℎ_𝑖𝛽_𝑗+^𝑛_𝑗=1(𝑝−1)||𝑏_𝑖𝑗(𝑠)||ℎ_𝑖𝛾_𝑗+𝑝(𝑝−1)2^𝑛_𝑖=1𝜇_𝑖+𝐸‖𝑥(𝑠)‖^𝑝d𝑠_𝑁𝑁+1^𝑛_𝑛𝑖=1_𝑗=1||𝑏_𝑖𝑗(𝑠)||ℎ_𝑖𝛾_𝑗𝐸‖‖𝑥𝑠−𝜏_𝑗(𝑠)‖‖^𝑝≤d𝑠_𝑁𝑁+1^𝑛_𝑖=1𝑝ℎ^𝑖𝛼_𝑖(𝑡)+𝑝^𝑛_𝑗=1||𝑎_𝑖𝑗(𝑠)||ℎ_𝑖𝛽_𝑗+^𝑛_𝑗=1||𝑏_𝑖𝑗(𝑠)||ℎ_𝑖𝛾_𝑗𝑝−1+𝑒^𝜆𝜏+𝑝(𝑝−1)2𝜇^𝑖𝑒^𝜆𝑡0𝛿‖‖𝑥⁰≤𝐼₁‖‖^𝑝𝑒^−𝜆𝑠d𝑠𝛿‖𝑥₀‖^𝑝𝑒^−𝜆𝑁,(3.14) where 𝐼₁=sup_𝑡∈𝑅^𝑛_𝑖=1𝑝ℎ^𝑖𝛼_𝑖(𝑡)+𝑝^𝑛_𝑗=1||𝑎_𝑖𝑗(𝑡)||ℎ_𝑖𝛽_𝑗+^𝑛_𝑗=1||𝑏_𝑖𝑗(𝑡)||ℎ_𝑖𝛾_𝑗𝑝−1+𝑒^𝜆𝜏+𝑝(𝑝−1)2𝜇^𝑖𝑒^𝜆𝑡0.(3.15) For any two different norms ‖⋅‖₂, ‖⋅‖_𝑝−2, (1<𝑝<∞),forall𝑥∈𝑅^𝑛, as the space 𝑅^𝑛 is a finite dimensional space, there exist two positive constants 𝜁₁, 𝜁₂, such that ‖𝑥‖‖𝑥‖²²_{p−2𝑝−2}≤𝜁₁‖𝑥‖^2𝑝,≤𝜁₂‖𝑥‖^p𝑝−2.(3.16) As 𝑀=∫^𝑡𝑁∑^𝑛𝑖=1∑^𝑛𝑗=1𝑝|𝑥_𝑖(𝑠)|^𝑝−2𝑥_𝑖(𝑠)𝜎_𝑖𝑗(𝑥_𝑗(𝑠))d𝜔_𝑗(𝑠) is continuous local martingale, then from Lemma 2.6, (𝐻₄), and (3.16), it follows that 𝑀₂≤𝐾₁𝐸⎧⎪⎨⎪⎩^{𝑁𝑛𝑁+1}_𝑗=1^𝑛_𝑖=1𝑝||𝑥_𝑖(𝑠)||^𝑝−1||𝜎_𝑖𝑗𝑥_𝑗(𝑠)||²d𝑠⎫⎪⎬⎪⎭^1/2≤4√2𝐸^{𝑁𝑛𝑁+1}_𝑗=1^𝑛_𝑖=1||𝑥_𝑖(𝑠)||^2𝑝−2^𝑛_𝑖=1𝑝²𝜎^2𝑖𝑗𝑥_𝑗(𝑠)d𝑠^1/2≤4√2𝐸^𝑁𝑁+1𝑝^2𝑛_𝑗=1𝜇_𝑗||𝑥_𝑗(𝑡)||²^𝑛_𝑖=1||𝑥_𝑖(𝑠)||^2𝑝−2d𝑠^1/2≤4√2𝐸^𝑁𝑁+1𝑝²max_{1≤𝑖≤𝑛}𝜇_𝑖𝜁₁‖𝑥‖^2𝑝𝜁₂‖𝑥‖^p𝑝−2d𝑠^1/2≤2𝐸12sup_{𝑠∈𝐼𝑁}‖𝑥(𝑠)‖^𝑝^𝑁𝑁+116𝑝²𝜁₁𝜁₂max_{1≤𝑖≤𝑛}𝜇_𝑖‖𝑥(𝑠)‖^𝑝d𝑠^1/2≤12𝐸sup_{𝑡∈𝐼𝑁}‖𝑥(𝑡)‖^𝑝+16𝑝²𝜁₁𝜁₂max_{1≤𝑖≤𝑛}𝜇_𝑖^𝑁𝑁+1𝐸‖𝑥(𝑠)‖^𝑝𝑑𝑠.(3.17) According to (3.11), (3.13), (3.14), and (3.17), we have the following inequality: 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≤+116𝑝²𝜁₁𝜁₂max_{1≤𝑖≤𝑛}𝜇_𝑖^𝑁𝑁+1𝛿‖‖𝑥₀‖‖^𝑝𝑒^{−𝜆(𝑠−𝑡0)}𝑑𝑠2𝐸sup_{𝑡∈𝐼𝑁}‖𝑥(𝑡)‖^𝑝+𝑒^𝜆𝑡0+𝐼₁+16𝑝²𝜁₁𝜁₂𝜆max^{1≤𝑖≤𝑛}𝜇_𝑖𝑒^𝜆𝑡0𝛿𝑒^−𝜆𝑁.(3.18) Therefore, 𝐸sup_{𝑡∈𝐼𝑁}‖𝑥(𝑡)‖^𝑝≤𝜃𝑒^−𝜆𝑁,(3.19) where 𝜃=2𝑒^𝜆𝑡0+𝐼₁+16𝑝²𝜁₁𝜁₂𝜆max^{1≤𝑖≤𝑛}𝜇_𝑖𝑒^𝜆𝑡0𝛿.(3.20) For each integer 𝑁, we set 𝜀_𝑁=𝑒^{−𝜆𝑁/2}. Then, from Lemma 2.7, we have 𝑃𝜔∶sup_{𝑡∈𝐼𝑁}‖𝑥(𝑡)‖^𝑝>𝜀_𝑁≤𝐸sup^{𝑡∈𝐼𝑁}‖𝑥(𝑡)‖^𝑝𝜀_𝑁≤𝜃𝑒^{−𝜆𝑁/2}.(3.21) Therefore, in view of Lemma 2.8, for almost all 𝜔∈Ω, we have that sup_{𝑡∈𝐼𝑁}‖𝑥(𝑡)‖^𝑝≤𝑒^{−𝜆𝑁/2},(3.22) 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, 1𝑡log‖𝑥(𝑡)‖=𝑝1𝑡log‖𝑥(𝑡)‖^𝑝≤−𝜆2𝑝,(3.23) if 𝑁≤𝑡≤𝑁+1, 𝑁≥𝑁₀. Hence limsup_𝑡→∞1𝑡log‖𝑥(𝑡)‖=𝑝1𝑡≤−𝜆𝑝a.s.(3.24) 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 Theorems 3.1 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 of 𝑎_𝑖𝑗(𝑡), 𝑏_𝑖𝑗(𝑡), and 𝛼_𝑖(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 𝜌_𝑖𝑗, 𝜌_𝑖𝑗, 𝜗_𝑖𝑗 such that for any 𝑡, we have ||𝑎_𝑖𝑗(𝑡)||||𝑏_𝑖𝑗(𝑡)||𝛼_𝑖(𝑡)≤𝜌_𝑖𝑗𝑡^𝜌𝑖𝑗+𝜗_𝑖𝑗,(3.25) 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 ℎ_𝑖(𝑥_𝑖(𝑡))=1, 𝑐_𝑖(𝑥_𝑖(𝑡))=𝑐_𝑖𝑥_𝑖(𝑡)(𝑐_𝑖>0), and 𝑎_𝑖𝑗(𝑡)=𝑎_𝑖𝑗,𝑏_𝑖𝑗(𝑡)=𝑏_𝑖𝑗, then it turns out to be following stochastic delayed recurrent neural networks with time-varying delays d𝑥_𝑖(𝑡)=−𝑐_𝑖𝑥_𝑖(𝑡)𝑑𝑡+^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑥_𝑗(𝑡)𝑑𝑡+^𝑛_𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑥_𝑗𝑡−𝜏_𝑗(𝑡)d𝑡+^𝑛_𝑗=1𝜎_𝑖𝑗𝑥_𝑗(𝑡)d𝜔_𝑗(𝑡).(3.26) 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: d𝑥(𝑡)=3−cos𝑥¹0(𝑡)3−sin0𝑥₂×−5+00.002𝑡5+00.003𝑡(𝑡)𝑥₁𝑥(𝑡)₂(𝑡)+0−.2601.40.2tanh𝑥¹0.2tanh𝑥₂(𝑡)+−01.8−220.2tanh𝑥₁(𝑡)0.2tanh𝑥₂𝑡−𝜏₁(𝑡)𝑡−𝜏₂(𝑡)d𝑡+𝜎(𝑥(𝑡))d𝑤(𝑡),𝑡≥0,(4.1) where 𝜏(𝑡)=(𝜏₁(𝑡),𝜏₂(𝑡))^𝑇and 𝜏_𝑖(𝑡) is any bounded positive function for 𝑖=1,2. Each 𝜎_𝑖𝑗(𝑥) satisfies the Lipschitz condition, and there exist positive constants 𝜇₁=𝜇₂=2, such that trace𝜎^𝑇(𝑥)𝜎(𝑥)≤2𝑥²¹+𝑥²².(4.2) In the example, let 𝑝=3; by simple computation, we obtain 𝑁₁(𝑡)=min_{1≤𝑖≤𝑛}𝑝ℎ^𝑖𝛼_𝑖(𝑡)−^𝑛_𝑗=1ℎ_𝑖(𝑝−1)||𝑎_𝑖𝑗(𝑡)||𝛽_𝑗−^𝑛_𝑗=1𝑚_𝑗𝑚_𝑖ℎ_𝑗||𝑎_𝑗𝑖(𝑡)||𝛽_𝑖−^𝑛_𝑗=1ℎ_𝑖(𝑝−1)||𝑏_𝑖𝑗(𝑡)||𝛾_𝑗−^𝑛_𝑗=1(𝑝−1)(𝑝−2)2𝜇^𝑗−^𝑛_𝑗=1𝑚_𝑗𝑚_𝑖(𝑝−1)𝜇_𝑖𝑁=11.6+0.008𝑡,₂(𝑡)=max_{𝑛1≤𝑖≤𝑛}_𝑗=1𝑚_𝑗𝑚_𝑖ℎ_𝑗||𝑏_𝑗𝑖(𝑡)||𝛾_𝑖=8.8+0.0008𝑡.(4.3) Choosing 𝜇=8/9, one can easily get that 0<𝑁₂≤𝑁₂(𝑡)≤𝜇𝑁₁(𝑡),for𝑡≥0.(4.4) 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 solution 𝐸(𝑥31(𝑡)) of system (4.1).

Figure 2 

Numerical solution 𝐸(𝑥32(𝑡)) of system (4.1).

Figure 3 

Numerical solution 𝑥1(𝑡) of system (4.1).

Figure 4 

Numerical solution 𝑥2(𝑡) of system (4.1).

Remark 4.2. Let 𝜏₁(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).

Acknowledgments

The authors are extremely grateful to Professor Yong Zhou and the anonymous reviewers for their constructive and valuable comments, which have contributed much to the improved presentation of this paper. This work was jointly supported by the Foundation of Chinese Society for Electrical Engineering (2008), the Excellent Youth Foundation of Educational Committee of Hunan Provincial (10B002), the Key Project of Chinese Ministry of Education (211118), and the National Natural Science Foundation of China (11072059).