Stability Analysis of Stochastic Reaction-Diffusion Cohen-Grossberg Neural Networks with Time-Varying Delays

Abstract

This paper is concerned with 𝑝th moment exponential stability of stochastic reaction-diffusion Cohen-Grossberg neural networks with time-varying delays. With the help of Lyapunov method, stochastic analysis, and inequality techniques, a set of new suffcient conditions on 𝑝th moment exponential stability for the considered system is presented. The proposed results generalized and improved some earlier publications.

1. Introduction

Since the seminal work for Cohen-Grossberg neural networks by Cohen and Grossberg [1], theoretical understanding of neural network dynamics has advanced greatly. The model can be described by a system of ordinary differential equations

𝑥̇_𝑖(𝑡)=−𝛼_𝑖(𝑥_𝑖(𝑡))[𝛽_𝑖(𝑥_𝑖(𝑡))−^𝑛_𝑗=1𝑎_𝑖𝑗𝑔_𝑗(𝑥_𝑗(𝑡))], (1.1) where 𝑡≥0,𝑛≥2;𝑛 corresponds to the number of units in a neural network; 𝑥_𝑖(𝑡) denotes the potential (or voltage) of cell 𝑖 at time 𝑡; 𝑓_𝑗(⋅) denotes a nonlinear output function between cell 𝑖 and 𝑗; 𝛼_𝑖(⋅)>0 represents an amplification function; 𝛽_𝑖(⋅) represents an appropriately behaved function; the 𝑛×𝑛 connection matrix 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛 denotes the strengths of connectivity between cells, and if the output from neuron 𝑗 excites (resp., inhibits) neuron 𝑖, then 𝑎_𝑖𝑗≥0 (resp., 𝑎_𝑖𝑗≤0).

During hardware implementation, time delays do exist due to finite switching speed of the amplifiers and communication time and, thus, delays should be incorporated into the model equations of the network. For model (1.1), Ye et al. [2] introduced delays by considering the following delay differential equations:

𝑥̇_𝑖(𝑡)=−𝛼_𝑖(𝑥_𝑖(𝑡))[𝛽_𝑖(𝑥_𝑖(𝑡))−^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗(𝑥_𝑗(𝑡−𝜏_𝑗))]. (1.2)

Some other more detailed justifications for introducing delays into model equations of neural networks can be found in [3–13] and references therein. It is seen that (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 (see, e.g., [14–18]).

In addition to the delay effects, stochastic effects constitute another source of disturbances or uncertainties in real systems [19]. A lot of dynamical systems have variable structures subject to stochastic abrupt changes, which may result from abrupt phenomena such as stochastic failures and repairs of the components, changes in the interconnections of subsystems, and sudden environment changes. In the recent years, the stability investigation of stochastic Neural Networks is interesting to many investigators, and a large number of stability criteria of these systems have been reported [20–30]. The stochastic model can be described by a system of stochastic differential equations

d𝑥_𝑖(𝑡)=−𝛼_𝑖(𝑥_𝑖(𝑡))[𝛽_𝑖(𝑥_𝑖(𝑡))−^𝑛_𝑗=1𝑎_𝑖𝑗𝑔_𝑗(𝑥_𝑗(𝑡))−^𝑛_𝑗=1𝑏_𝑖𝑗𝑔_𝑗(𝑥_𝑗(𝑡−𝜏_𝑗+(𝑡)))]d𝑡_𝑛𝑗=1𝜎_𝑖𝑗(𝑡,𝑥_𝑖(𝑡),𝑥_𝑗(𝑡−𝜏_𝑗(𝑡)))d𝜔_𝑗(𝑡). (1.3)

However, besides delay and stochastic effects, diffusion effect cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields [31], so we must consider the activations vary in space as well as in time. In [32–36], authors have considered the stability of reaction-diffusion neural networks with constant or time-varying delays, which are expressed by partial differential equations. To the best of our knowledge, few authors have considered the problem of 𝑝th moment stability for stochastic Cohen-Grossberg neural networks with both time-varying delays and reaction-diffusion terms. Motivated by the above discussions, in this paper, we consider the stochastic reaction-diffusion Cohen-Grossberg neural networks with time-varying delays described by the following stochastic partial differential equations:

d𝑦_𝑖(𝑥,𝑡)=^𝑚_𝑘=1𝜕𝜕𝑥_𝑘(𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)d𝑡−𝛼_𝑖(𝑦_𝑖(𝑥,𝑡))×[𝛽_𝑖(𝑦_𝑖(𝑥,𝑡))−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑡)𝑔_𝑗(𝑦_𝑗(𝑥,𝑡))−^𝑛_𝑗=1𝑏_𝑖𝑗(𝑡)𝑔_𝑗(𝑦_𝑗(𝑥,𝑡−𝜏_𝑗+(𝑡)))]d𝑡_𝑛𝑗=1𝜎_𝑖𝑗(𝑡,𝑦_𝑖(𝑥,𝑡),𝑦_𝑗(𝑥,𝑡−𝜏_𝑗(𝑡)))d𝜔_𝑗(𝑡), (1.4) where

𝜕𝑦_𝑖𝜕𝑛∶=(𝜕𝑦_𝑖𝜕𝑥₁,…,𝜕𝑦_𝑖𝜕𝑥_𝑚)^𝑇𝑦=0,𝑡≥0,𝑖=1,…,𝑛,𝑥∈Ω,_𝑖=𝑦_𝑖(𝑥,𝑠)=𝜙_𝑖(𝑥,𝑠),−∞<𝑠≤0,𝑥∈Ω. (1.5) In the above model, 𝑛≥2 corresponds to the number of units in the neural network, 𝑥=(𝑥₁,…,𝑥_𝑚)^𝑇 is space variable, and 𝑦_𝑖(𝑥,𝑡) denotes the state variable of cell 𝑖 at time 𝑡 in space variable 𝑥; smooth function 𝐷_𝑖𝑘=𝐷_𝑖𝑘(𝑥,𝑦,𝑡)≥0 is a diffusion operator; Ω is a compact set with smooth boundary 𝜕Ω and the measure mesΩ>0 in 𝑅^𝑚, 𝜕𝑦_𝑖|𝜕𝑛_𝜕Ω=0, (1.6) and 𝜙_𝑖(𝑥,𝑠) are the boundary value and initial value, respectively; 𝑎_𝑖𝑗(𝑡) and 𝑏_𝑖𝑗(𝑡) denote the strengths of connectivity between cell 𝑖 and 𝑗 at time 𝑡, respectively; 𝜏_𝑗(𝑡) is time delay and satisfies 0≤𝜏_𝑗(𝑡)≤𝜏; 𝜎=(𝜎_𝑖𝑗(𝑡,𝑦_𝑖(𝑥,𝑡),𝑦_𝑗(𝑥,𝑡−𝜏_𝑗(𝑡))))_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 by standard Brownian motion {𝑤(𝑠)∶0≤𝑠≤𝑡}. As a standing hypothesis, we assume that 𝑔_𝑗(⋅) and 𝜎(𝑡,⋅,⋅) satisfy the Lipschitz condition and the linear growth condition and that (1.4) has a solution on 𝑡≥0 for the initial conditions.

The remainder of this paper is organized as follows. In Section 2, the basic notations and assumptions are introduced. In Section 3, criteria are proposed to determine 𝑝th moment exponential stability for the stochastic Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion term. An illustrative example is given to illustrate the effectiveness of the obtained results in Section 4. We also conclude this paper in Section 5.

2. Preliminaries

For any 𝑦(𝑥,𝑡)=(𝑦₁(𝑥,𝑡),…,𝑦_𝑛(𝑥,𝑡))^𝑇∈𝑅^𝑛, we define

‖𝑦_𝑖(𝑥,𝑡)‖_𝑝=[_Ω|𝑦_𝑖(𝑥,𝑡)|^𝑝d𝑥]^1/𝑝,‖𝑦(𝑥,𝑡)‖=[^𝑛_𝑗=1‖𝑦_𝑖(𝑥,𝑡)‖^𝑝_𝑝]^1/𝑝. (2.1) As usual, we will also assume that the following conditions are satisfied.

(𝐻₁) There exist positive constants 𝛼_𝑖,𝛼_𝑖, such that 𝛼_𝑖≤𝑎_𝑖(𝑦_𝑖(𝑥,𝑡))≤𝛼_𝑖. (2.2)(𝐻₂)For each 𝑖∈{1,…,𝑛}, there exists positive constant 𝐺_𝑖, such that |𝑔_𝑖(𝑢)−𝑔_𝑖(𝑣)|≤𝐺_𝑖|𝑢−𝑣|,∀𝑢,𝑣,∈𝑅. (2.3)(𝐻₃) There exist positive functions 𝛾_𝑗(𝑡), such that 𝑦_𝑗(𝑥,𝑡)𝛽_𝑗(𝑦_𝑗(𝑥,𝑡))≥𝛾_𝑗(𝑡)𝑦²_𝑗(𝑥,𝑡). (2.4)(𝐻^′₃)There exists positive constant 𝛾_𝑗, such that 𝑦_𝑗(𝑥,𝑡)𝛽_𝑗(𝑦_𝑗(𝑥,𝑡))≥𝛾_𝑗𝑦²_𝑗(𝑥,𝑡). (2.5)(𝐻₄) There are nonnegative functions 𝑐⁰_𝑖𝑗(𝑡),𝑐¹_𝑖𝑗(𝑡), for all 𝑡,𝑢,𝑣∈𝑅, such that 𝜎²_𝑖𝑗(𝑡,𝑢,𝑣)≤𝑐⁰_𝑖𝑗(𝑡)𝑢²+𝑐¹_𝑖𝑗(𝑡)𝑣². (2.6)(𝐻^′₄) There are nonnegative constants 𝑐⁰_𝑖𝑗,𝑐¹_𝑖𝑗, for all 𝑡,𝑢,𝑣∈𝑅, such that 𝜎²_𝑖𝑗(𝑡,𝑢,𝑣)≤𝑐⁰_𝑖𝑗𝑢²+𝑐¹_𝑖𝑗𝑣². (2.7)(𝐻₅) The following inequality holds: _Ω𝐷_𝑖𝑘(𝜕𝑦_𝑖𝜕𝑥_𝑘)^𝑚_𝑘=1⋅∇𝑦_𝑖^𝑝−1d𝑥>0. (2.8)

Definition 2.1. The trivial solution of (1.4) is said to be 𝑝th moment exponentially stable if there is a pair of positive constants 𝜆 and 𝐺 such that 𝐸‖𝑦(𝜙,𝑡)‖^𝑝<𝐺𝐸‖𝜙‖^𝑝𝑒^{−𝜆(𝑡−𝑡0)},on𝑡≥𝑡₀ (2.9) for any 𝜙, where 𝜆 also called as convergence rate. When 𝑝=2, it is usually said to be exponentially stable in mean square.

Definition 2.2. Let ℎ∶𝑅→𝑅 be a continuous function, 𝑑⁺ℎ; the upper right Dini-derivative of ℎ is defined as 𝑑⁺ℎ(𝑡)=limsup_𝛿→0+ℎ(𝑡+𝛿)−ℎ(𝑡)𝛿. (2.10)

The following lemmas are important in our approach.

Lemma 2.3 (Hardy inequality [4]). Assume there exist constants 𝑎_𝑘≥0,𝑝_𝑘>0(𝑘=1,…,𝑚+1), then the following inequality holds: (^𝑚+1_𝑘=1𝑎^𝑝𝑘_𝑘)^1/𝑆𝑚+1≤(^𝑚+1_𝑘=1𝑝_𝑘𝑎^𝑟_𝑘)^1/𝑟𝑆^−1/𝑟_𝑚+1, (2.11) where 𝑟>0 and 𝑆_𝑚+1=∑^𝑚+1_𝑘=1𝑝_𝑘. In (2.11), if one lets 𝑝_𝑚+1=1,𝑟=𝑆_𝑚+1=∑^𝑚_𝑘=1𝑝_𝑘+1, one will get (^𝑚_𝑘=1𝑎^𝑝𝑘_𝑘)𝑎_𝑚+1≤1𝑟(^𝑚_𝑘=1𝑝_𝑘𝑎^𝑟_𝑘1)+𝑟𝑎^𝑟_𝑚+1, (2.12) if one lets 𝑝_𝑚+1=2,𝑟=𝑆_𝑚+1=∑^𝑚_𝑘=1𝑝_𝑘+2, one will get (^𝑚_𝑘=1𝑎^𝑝𝑘_𝑘)𝑎_𝑚+1≤1𝑟(^𝑚_𝑘=1𝑝_𝑘𝑎^𝑟_𝑘2)+𝑟𝑎^𝑟_𝑚+1. (2.13)

Lemma 2.4 (generalized Halanay inequality [37]). For two positive-valued functions 𝑎(𝑡) and 𝑏(𝑡) defined on [𝑡₀∞), assume there exists a constant number 0≤𝜇<1 satisfying 0<𝑎₀≤𝑎(𝑡),0<𝑏(𝑡)≤𝜇𝑎(𝑡) hold for all 𝑡≥𝑡₀; 𝑦(𝑡) is nonnegative continuous function on [𝑡₀−𝜏,∞) and satisfies the following inequality: 𝑑⁺𝑦(𝑡)≤−𝑎(𝑡)𝑦(𝑡)+𝑏(𝑡)𝑦(𝑡)for𝑡≥𝑡₀, (2.14) where 𝑦(𝑡)=sup_{𝑡−𝜏≤𝑠≤𝑡}𝑦(𝑠); 𝜏≥0 is constant. Then one has 𝑦(𝑡)≤𝑦(𝑡₀)𝑒^{−𝜆∗(𝑡−𝑡0)}, (2.15) where 𝜆^∗>0 is defined as 𝜆^∗=inf_𝑡≥𝑡0{𝜆(𝑡)∶𝜆=𝑎(𝑡)−𝑏(𝑡)𝑒^𝜆𝜏}. (2.16)

3. Main Results

Theorem 3.1. Under assumptions (𝐻₁),(𝐻₂),(𝐻₃),(𝐻₄),(𝐻₅), if there exist a positive diagonal matrix 𝑄=diag(𝑞₁,…,𝑞_𝑛), constants 𝜈_𝑘>0(𝑘=1,…,𝑘₁), 𝜇_𝑘>0(𝑘=1,…,𝑘₂), 𝜌_𝑘>0(𝑘=1,…,𝑘₃), 𝜉_𝑖𝑗,𝜉^∗_𝑖𝑗,𝜂_𝑖𝑗,𝜂^∗_𝑖𝑗,𝑚_𝑖𝑗,𝑚^∗_𝑖𝑗,𝑝_𝑖𝑗,𝑝^∗_𝑖𝑗,𝑞_𝑖𝑗,𝑞^∗_𝑖𝑗∈ℝ, 0<𝜆₂, and 0≤𝜇<1, such that 0<𝜆₂≤𝜆₂(𝑡)≤𝜇𝜆₁(𝑡),holdsforall𝑡≥𝑡₀, (3.1) where 𝜆₁(𝑡)=min_{1≤𝑖≤𝑛}{𝑝𝛼_𝑖𝛾_𝑖(𝑡)−^𝑛_𝑗=1𝛼_𝑖^𝑘1_𝑘=1𝜈_𝑘|𝑎_𝑖𝑗(𝑡)|^{𝑝𝜉𝑖𝑗/𝜈𝑘}𝐺^{𝑝𝜂𝑖𝑗/𝜈𝑘}_𝑗−1𝑞_𝑖^𝑛_𝑗=1𝑞_𝑗𝛼_𝑗|𝑎_𝑗𝑖(𝑡)|^{𝑝𝜉∗𝑗𝑖}𝐺^{𝑝𝜂∗𝑗𝑖}_𝑖−^𝑛_𝑗=1𝛼_𝑖^𝑘2_𝑘=1𝜇_𝑘|𝑏_𝑖𝑗(𝑡)|^{𝑝𝑝𝑖𝑗/𝜇𝑘}𝐺^{𝑝𝑞𝑖𝑗/𝜇𝑘}_𝑗−𝑝(𝑝−1)2^𝑛_𝑗=1𝑐⁰_𝑖𝑗(𝑡)−𝑝−12^𝑛𝑘_𝑗=1³_𝑘=1𝜌_𝑘|𝑐¹_𝑖𝑗(𝑡)|^{𝑝𝑚𝑖𝑗/𝜌𝑘}𝜆},₂(𝑡)=max_{1≤𝑖≤𝑛}[^𝑛_𝑗=1𝑞_𝑗𝑞_𝑖𝛼_𝑗|𝑏_𝑗𝑖(𝑡)|^{𝑝𝑝∗𝑗𝑖}𝐺^{𝑝𝑞∗𝑗𝑖}_𝑖+(𝑝−1)^𝑛_𝑗=1𝑞_𝑗𝑞_𝑖(𝑐¹_𝑗𝑖(𝑡))^{𝑝𝑚∗𝑗𝑖/2}] (3.2)𝑝=∑^𝑘1_𝑗=1𝜈_𝑘+1=∑^𝑘2_𝑗=1𝜇_𝑘+1=∑^𝑘3_𝑗=1𝜌_𝑘+2, 𝑘₁𝜉_𝑖𝑗+𝜉^∗_𝑖𝑗=1,𝑘₁𝜂_𝑖𝑗+𝜂^∗_𝑖𝑗=1, 𝑘₂𝑝_𝑖𝑗+𝑝^∗_𝑖𝑗=1,𝑘₂𝑞_𝑖𝑗+𝑞^∗_𝑖𝑗=1, 𝑘₃𝑚_𝑖𝑗+𝑚^∗_𝑖𝑗=1,𝑘₃𝑚_𝑖𝑗+𝑚^∗_𝑖𝑗=1, then for all 𝜉∈𝐿^𝑝_ℱ0([−𝜏,0],𝑅^𝑛), the trivial solution of system (1.4) is 𝑝th moment exponentially stable, where 𝑝≥2 is a constant.

Proof. Consider the following Lyapunov function: 𝑉(𝑡,𝑦(𝑡))=_Ω^𝑛_𝑖=1𝑞_𝑖|𝑦_𝑖(𝑥,𝑡)|^𝑝d𝑥. (3.3) Applying Itô formula to 𝑉(𝑡,𝑦(𝑡)), for 𝛿>0, we can get =𝑉(𝑡+𝛿,𝑦(𝑡+𝛿))−𝑉(𝑡,𝑦(𝑡))_{𝑡𝑡+𝛿}𝑉_𝑡+(𝑠,𝑦(𝑠))d𝑠_{𝑡𝑡+𝛿}𝑉_𝑦(𝑠,𝑦(𝑠)){^𝑚_𝑘=1𝜕𝜕𝑥_𝑘(𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)−𝛼_𝑖(𝑦_𝑖(𝑥,𝑠))[𝛽_𝑖(𝑦_𝑖(𝑥,𝑠))−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑔_𝑗(𝑦_𝑗−(𝑥,𝑠))_𝑛𝑗=1𝑏_𝑖𝑗(𝑠)𝑔_𝑗(𝑦_𝑗(𝑥,𝑠−𝜏_𝑗+(𝑠)))]}d𝑠_{𝑡𝑡+𝛿}𝑉_𝑦(𝑠,𝑦(𝑠))^𝑛_𝑗=1𝜎_𝑖𝑗(𝑠,𝑦_𝑖(𝑥,𝑠),𝑦_𝑗(𝑥,𝑠−𝜏_𝑗(𝑠)))d𝜔_𝑗(𝑠)+_𝑡^𝑡+𝛿12trace(𝜎^𝑇𝑉_𝑦𝑦≤𝜎)d𝑠_{𝑡𝑡+𝛿Ω}𝑝^𝑛_𝑖=1𝑞_𝑖|𝑦_𝑖^𝑝−2(𝑥,𝑠)|𝑦_𝑖(𝑥,𝑠)×{^𝑚_𝑘=1𝜕𝜕𝑥_𝑘(𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)−𝛼_𝑖(𝑦_𝑖(𝑥,𝑠))[𝛽_𝑖(𝑦_𝑖(𝑥,𝑠))−^𝑛_𝑗=1𝑎_𝑖𝑗(𝑠)𝑔_𝑗(𝑦_𝑗−(𝑥,𝑠))_𝑛𝑗=1𝑏_𝑖𝑗(𝑠)𝑔_𝑗(𝑦_𝑗(𝑥,𝑠−𝜏_𝑗+(𝑠)))]}d𝑥d𝑠_{𝑡𝑡+𝛿}𝑉_𝑦(𝑠,𝑦(𝑠))^𝑛_𝑗=1𝜎_𝑖𝑗(𝑠,𝑦_𝑖(𝑥,𝑠),𝑦_𝑗(𝑥,𝑠−𝜏_𝑗(𝑠)))d𝜔_𝑗(𝑠)+_𝑡^𝑡+𝛿12trace(𝜎^𝑇𝑉_𝑦𝑦𝜎)d𝑠≤_𝑡^𝑡+𝛿_Ω𝑝{^𝑛_𝑖=1𝑞_𝑖|𝑦_𝑖^𝑝−1(𝑥,𝑠)|^𝑚_𝑘=1𝜕𝜕𝑥_𝑘(𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)−^𝑛_𝑖=1𝑞_𝑖𝛼_𝑖𝛾_𝑖(𝑠)|𝑦^𝑝_𝑖+(𝑥,𝑠)|_{𝑛𝑛𝑖=1𝑗=1}𝑞_𝑖𝛼_𝑖𝐺_𝑗|𝑎_𝑖𝑗(𝑠)𝑦_𝑖^𝑝−1(𝑥,𝑠)𝑦_𝑗+(𝑥,𝑠)|_{𝑛𝑛𝑖=1𝑗=1}𝑞_𝑖𝛼_𝑖𝐺_𝑗|𝑏_𝑖𝑗(𝑠)𝑦_𝑖^𝑝−1(𝑥,𝑠)𝑦_𝑗(𝑥,𝑠−𝜏_𝑗(𝑠))|}d𝑥d𝑠+𝑝_𝑡^𝑡+𝛿_Ω^𝑛𝑛_𝑖=1𝑗=1𝑞_𝑖|𝑦_𝑖^𝑝−1(𝑥,𝑠)|𝜎_𝑖𝑗(𝑠,𝑦_𝑖(𝑥,𝑠),𝑦_𝑗(𝑥,𝑠−𝜏_𝑗(𝑠)))d𝑥d𝜔_𝑗+(𝑠)_{𝑡𝑡+𝛿Ω}𝑝(𝑝−1)2𝑞_𝑖^𝑛𝑛_𝑖=1𝑗=1[𝑐⁰_𝑖𝑗(𝑠)|𝑦^𝑝_𝑖(𝑥,𝑠)|+𝑐¹_𝑖𝑗(𝑠)|𝑦_𝑖^𝑝−2(𝑥,𝑠)𝑦²_𝑗(𝑥,𝑠−𝜏_𝑗(𝑠))|]d𝑥d𝑠. (3.4) From the boundary condition, we get ^𝑚_𝑘=1Ω𝑦_𝑖^𝑝−1𝜕𝜕𝑥_𝑘(𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)d𝑥=_Ω𝑦_𝑖^𝑝−1∇⋅(𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)^𝑚_𝑘=1=d𝑥_Ω∇⋅(𝑦_𝑖^𝑝−1𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)^𝑚_𝑘=1d𝑥−_Ω(𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)^𝑚_𝑘=1⋅∇𝑦_𝑖^𝑝−1=d𝑥_𝜕Ω((𝑦_𝑖^𝑝−1𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘))^𝑚_𝑘=1⋅d𝑆−_Ω(𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)^𝑚_𝑘=1⋅∇𝑦_𝑖^𝑝−1d𝑥=−_Ω(𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)^𝑚_𝑘=1⋅∇𝑦_𝑖^𝑝−1d𝑥, (3.5) in which, ∇=(𝜕/𝜕𝑥₁,…,𝜕/𝜕𝑥_𝑚)^𝑇 is the gradient operator, and (𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘)^𝑚_𝑘=1=(𝐷_𝑖1𝜕𝑦_𝑖𝜕𝑥₁,…,𝐷_𝑖𝑚𝜕𝑦_𝑖𝜕𝑥_𝑚)^𝑇. (3.6) On the other hand, from Lemma 2.3, we have 𝑝𝐺_𝑗|𝑎_𝑖𝑗(𝑠)𝑦_𝑖^𝑝−1(𝑥,𝑠)𝑦_𝑗(𝑥,𝑠)|=𝑝^𝑘1_𝑘=1(|𝑎_𝑖𝑗(𝑠)|^{𝜉𝑖𝑗/𝜈𝑘}𝐺^{𝜂𝑖𝑗/𝜈𝑘}_𝑗|𝑦_𝑖(𝑥,𝑠)|)^𝜈𝑘|𝑎_𝑖𝑗(𝑠)|^{𝜉∗𝑖𝑗}𝐺^{𝜂∗𝑖𝑗}_𝑗|𝑦_𝑗≤(𝑥,𝑠)|_𝑘1𝑘=1𝜈_𝑘|𝑎_𝑖𝑗(𝑠)|^{𝑝𝜉𝑖𝑗/𝜈𝑘}𝐺^{𝑝𝜂𝑖𝑗/𝜈𝑘}_𝑗|𝑦_𝑖(𝑥,𝑠)|^𝑝+|𝑎_𝑖𝑗(𝑠)|^{𝑝𝜉∗𝑖𝑗}𝐺^{𝑝𝜂∗𝑖𝑗}_𝑗|𝑦_𝑗(𝑥,𝑠)|^𝑝,𝑝𝐺_𝑗|𝑏_𝑖𝑗(𝑠)𝑦_𝑖^𝑝−1(𝑥,𝑠)𝑦_𝑗(𝑥,𝑠−𝜏_𝑗(𝑠))|=𝑝^𝑘2_𝑘=1(|𝑏_𝑖𝑗(𝑠)|^{𝑝𝑖𝑗/𝜇𝑘}𝐺^{𝑞𝑖𝑗/𝜇𝑘}_𝑗|𝑦_𝑖(𝑥,𝑠)|)^𝜇𝑘|𝑏_𝑖𝑗(𝑠)|^{𝑝∗𝑖𝑗}𝐺^{𝑞∗𝑖𝑗}_𝑗|𝑦_𝑗(𝑥,𝑠−𝜏_𝑗≤(𝑠))|_𝑘2𝑘=1𝜇_𝑘|𝑏_𝑖𝑗(𝑠)|^{𝑝𝑝𝑖𝑗/𝜇𝑘}𝐺^{𝑝𝑞𝑖𝑗/𝜇𝑘}_𝑗|𝑦_𝑖(𝑥,𝑠)|^𝑝+|𝑏_𝑖𝑗(𝑠)|^{𝑝𝑝∗𝑖𝑗}𝐺^{𝑝𝑞∗𝑖𝑗}_𝑗|𝑦_𝑗(𝑥,𝑠−𝜏_𝑗(𝑠))|^𝑝,𝑝(𝑝−1)2𝑐¹_𝑖𝑗(𝑠)|𝑦_𝑖^𝑝−2(𝑥,𝑠)𝑦²_𝑗(𝑥,𝑠−𝜏_𝑗=(𝑠))|𝑝(𝑝−1)2^𝑘3_𝑘=1(|𝑐¹_𝑖𝑗(𝑠)|^{𝑚𝑖𝑗/𝜌𝑘}|𝑦_𝑖(𝑥,𝑠)|)^𝜌𝑘(|𝑐¹_𝑖𝑗(𝑠)|^{𝑚∗𝑖𝑗/2}|𝑦_𝑗(𝑥,𝑠−𝜏_𝑗(𝑠))|)²≤𝑝−12^𝑘3_𝑘=1𝜌_𝑘|𝑐¹_𝑖𝑗(𝑠)|^{𝑝𝑚𝑖𝑗/𝜌𝑘}|𝑦_𝑖(𝑥,𝑠)|^𝑝+(𝑝−1)|𝑐¹_𝑖𝑗(𝑠)|^{𝑝𝑚∗𝑖𝑗/2}|𝑦_𝑗(𝑥,𝑠−𝜏_𝑗(𝑠))|^𝑝. (3.7)(3.8)(3.9) using Corollary 3.2, system (4.1) is 4th moment exponential stable.

Remark 4.2. One can find that models considered in [20, 22–29] are special cases of model (1.4). To the best of our knowledge, few authors have considered the 𝑝th moment exponential stability for Stochastic reaction-diffusion Neural Networks with time-varying connection matrix and delays. It is assumed in [22, 23, 25, 26] that delays are constants, the delay functions appear in [29] are differential and their derivatives are simultaneously required to be not greater than 1, the activation functions appear in [22, 26] are bounded. Obviously, we have dropped out these basic assumptions in this paper.

It is obvious that the results in [20–30] and the references therein cannot be applicable to system (4.1) even if we remove the reaction-diffusion terms from the system for the connection matrix and delays considered in this example are time-varying. This implies that the results of this paper are essentially new. Just choose 𝑥≡constant these conclusions can be verified by the numerical simulations shown in Figure 1.

Figure 1 

Numerical solution 𝐸(𝑦41(𝑡)),𝐸(𝑦42(𝑡)) of system (4.1).

5. Conclusions

In this paper, stochastic Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion have been investigated. All features of stochastic systems (especially the connection matrices and delays are time-varying) reaction-diffusion systems have been taken into account in the neural networks. Without requiring the differential and monotonicity of the activation functions and the symmetry of the connection matrices, a set of new sufficient conditions for checking 𝑝th moment exponential stability of the trivial solution of the considered system is presented by using of Lyapunov function, stochastic analysis technique, and the generalized Halanay inequality. The proposed results generalized and improved some of the earlier published results greatly. The results obtained in this paper are independent of the magnitude of delays and diffusion effect, which implies that strong self-regulation is dominant in the networks. In addition, the methods used in this paper are also applicable to other neural networks, such as stochastic Hopfield neural networks with time-varying delays and reaction-diffusion terms and stochastic bidirectional associative memory (BAM) neural networks with time-varying delays and reaction-diffusion terms. If we remove the noise and reaction-diffusion terms from the system, the derived conditions for stability of general deterministic neural networks can be viewed as byproducts of our results.

Acknowledgments

The authors are extremely grateful to two anonymous reviewers, and particularly to Professor Yong Zhou for their valuable comments and suggestions, which have contributed a lot to the improved presentation of this paper. This work was supported in part by the National Natural Science Foundation of China under Grants no. 50925727, no. 10971240, and no. 60876022, the Foundation of Chinese Society for Electrical Engineering, the Foundation of Yunnan Provincial Education Department under Grant no. 07Y10085, the Scientific Research Fund of Yunnan Province under Grant no. 2008CD186.