Dynamical Behaviors of Stochastic Reaction-Diffusion Cohen-Grossberg Neural Networks with Delays

Abstract

This paper investigates dynamical behaviors of stochastic Cohen-Grossberg neural network with delays and reaction diffusion. By employing Lyapunov method, Poincaré inequality and matrix technique, some sufficient criteria on ultimate boundedness, weak attractor, and asymptotic stability are obtained. Finally, a numerical example is given to illustrate the correctness and effectiveness of our theoretical results.

1. Introduction

Cohen and Grossberg proposed and investigated Cohen-Grossberg neural networks in 1983 [1]. Hopfield neural networks, recurrent neural networks, cellular neural networks, and bidirectional associative memory neural networks are special cases of this model. Since then, the Cohen-Grossberg neural networks have been widely studied in the literature, see for example, [2–12] and references therein.

Strictly speaking, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields. Therefore, we must consider that the activations vary in space as well as in time. In [13–19], the authors gave some stability conditions of reaction-diffusion neural networks, but these conditions were independent of diffusion effects.

On the other hand, it has been well recognized that stochastic disturbances are ubiquitous and inevitable in various systems, ranging from electronic implementations to biochemical systems, which are mainly caused by thermal noise, environmental fluctuations, as well as different orders of ongoing events in the overall systems [20, 21]. Therefore, considerable attention has been paid to investigate the dynamics of stochastic neural networks, and many results on stability of stochastic neural networks have been reported in the literature, see for example, [22–38] and references therein.

The above references mainly considered the stability of equilibrium point of neural networks. What do we study when the equilibrium point does not exist? Except for stability property, boundedness and attractor are also foundational concepts of dynamical systems, which play an important role in investigating the uniqueness of equilibrium, global asymptotic stability, global exponential stability, the existence of periodic solution, and so on [39, 40]. Recently, ultimate boundedness and attractor of several classes of neural networks with time delays have been reported. In [41], the globally robust ultimate boundedness of integrodifferential neural networks with uncertainties and varying delays was studied. Some sufficient criteria on the ultimate boundedness of deterministic neural networks with both varying and unbounded delays were derived in [42]. In [43, 44], a series of criteria on the boundedness, global exponential stability, and the existence of periodic solution for nonautonomous recurrent neural networks were established. In [45, 46], some criteria on ultimate boundedness and attractor of stochastic neural networks were derived. To the best of our knowledge, there are few results on the ultimate boundedness and attractor of stochastic reaction-diffusion neural networks.

Therefore, the arising questions about the ultimate boundedness, attractor and stability for the stochastic reaction-diffusion Cohen-Grossberg neural networks with time-varying delays are important yet meaningful.

The rest of the paper is organized as follows: some preliminaries are in Section 2, main results are presented in Section 3, a numerical example and conclusions will be drawn in Sections 4 and 5, respectively.

2. Model Description and Assumptions

Consider the following stochastic Cohen-Grossberg neural networks with delays and diffusion terms: 𝑑𝑦_𝑖(𝑡,𝑥)=^𝑙_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑖𝑘𝜕𝑦_𝑖(𝑡,𝑥)𝜕𝑥_𝑘𝑑𝑡−𝑑_𝑖𝑦_𝑖()×𝑐𝑡,𝑥_𝑖𝑦_𝑖(𝑡,𝑥)−^𝑛_𝑗=1𝑎_𝑖𝑗𝑓_𝑗𝑦_𝑗(𝑡,𝑥)−^𝑛_𝑗=1𝑏_𝑖𝑗𝑔_𝑗𝑦_𝑗𝑡−𝜏_𝑗(𝑡),𝑥−𝐽_𝑖+𝑑𝑡_𝑚𝑗=1𝜎_𝑖𝑗𝑦_𝑗(𝑡,𝑥),𝑦_𝑗𝑡−𝜏_𝑗(𝑡),𝑥𝑑𝑤_𝑗(𝑡),𝑥∈𝑋,𝜕𝑦_𝑖𝜕𝑣∶=𝜕𝑦_𝑖𝜕𝑥₁,…,𝜕𝑦_𝑖𝜕𝑥_𝑙^𝑇𝑦=0,𝑥∈𝜕𝑋,_𝑖(𝑠,𝑥)=𝜉_𝑖(𝑠,𝑥),−𝜏≤𝑠≤0,𝑥∈𝑋, (2.1) for 1≤𝑖≤𝑛 and 𝑡≥0. In the above model, 𝑛≥2 is the number of neurons in the network; 𝑥_𝑖 is space variable; 𝑦_𝑖(𝑡,𝑥) is the state variable of the 𝑖th neuron at time 𝑡 and in space 𝑥; 𝑓_𝑗(𝑦_𝑗(𝑡,𝑥)) and 𝑔_𝑗(𝑦_𝑗(𝑡,𝑥)) denote the activation functions of the 𝑗th unit at time 𝑡 and in space 𝑥; constant 𝐷_𝑖𝑘≥0; 𝑑_𝑖(𝑦_𝑖(𝑡,𝑥)) presents an amplification function; 𝑐_𝑖(𝑦_𝑖(𝑡,𝑥)) is an appropriately behavior function; 𝑎_𝑖𝑗 and 𝑏_𝑖𝑗 denote the connection strengths of the 𝑗th unit on the 𝑖th unit, respectively; 𝜏_𝑗(𝑡) corresponds to the transmission delay and satisfies 0≤𝜏_𝑗(𝑡)≤𝜏;𝐽_𝑖 denotes the external bias on the 𝑖th unit; 𝜎_𝑖𝑗(⋅,⋅,𝑥) is the diffusion function; 𝑋 is a compact set with smooth boundary 𝜕𝑋 and measure mes𝑋>0 in 𝑅^𝑙;𝜉_𝑖(𝑠,𝑥) is the initial boundary value; 𝑤(𝑡)=(𝑤₁(𝑡),…,𝑤_𝑚(𝑡))^𝑇 is 𝑚-dimensional Brownian motion defined on a complete probability space (Ω,ℱ,ℙ) with a natural filtration {ℱ_𝑡}_𝑡≥0 generated by {𝑤(𝑠)∶0≤𝑠≤𝑡}, where we associate Ω with the canonical space generated by all {𝑤_𝑖(𝑡)} and denote by ℱ the associated 𝜎-algebra generated by {𝑤(𝑡)} with the probability measure ℙ.

System (2.1) has the following matrix form: 𝑑𝑦(𝑡,𝑥)=col^𝑙_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑖𝑘𝜕𝑦_𝑖(𝑡,𝑥)𝜕𝑥_𝑘𝑦()×𝑐𝑦()𝑦()𝑦((𝑡))𝑦()((𝑡))(𝑡)𝑑𝑡−𝑑𝑡,𝑥𝑡,𝑥−𝐴𝑓𝑡,𝑥−𝐵𝑔𝑡−𝜏,𝑥−𝐽𝑑𝑡+𝜎𝑡,𝑥,𝑦𝑡−𝜏,𝑥𝑑𝑤,𝑥∈𝑋, (2.2) where col^𝑙_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑖𝑘𝜕𝑦_𝑖(𝑡,𝑥)𝜕𝑥_𝑘=^𝑙_𝑘=1𝜕𝜕𝑥_𝑘𝐷_1𝑘𝜕𝑦₁(𝑡,𝑥)𝜕𝑥_𝑘,…,^𝑙_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑛𝑘𝜕𝑦_𝑛(𝑡,𝑥)𝜕𝑥_𝑘^𝑇,𝐴=𝑎_{𝑖𝑗𝑛×𝑛},𝐵=𝑏_{𝑖𝑗𝑛×𝑛},𝑓𝑦(𝑡,𝑥)=𝑓₁𝑦₁(𝑡,𝑥),…,𝑓_𝑛𝑦_𝑛(𝑡,𝑥)^𝑇,𝐽=𝐽₁,…,𝐽_𝑛^𝑇,𝑔𝑦(𝑡−𝜏(𝑡),𝑥)=diag𝑔₁𝑦₁𝑡−𝜏₁(𝑡),𝑥,…,𝑔_𝑛𝑦_𝑛𝑡−𝜏_𝑛(𝑡),𝑑𝑦()𝑑,𝑥𝑡,𝑥=diag₁𝑦₁(𝑡,𝑥),…,𝑑_𝑛𝑦_𝑛(),𝑐𝑦()𝑐𝑡,𝑥𝑡,𝑥=diag₁𝑦₁(𝑡,𝑥),…,𝑐_𝑛𝑦_𝑛(),𝜎𝑦()((𝑡))=𝜎𝑡,𝑥𝑡,𝑥,𝑦𝑡−𝜏,𝑥,𝑥_𝑖𝑗𝑦_𝑗(𝑡,𝑥),𝑦_𝑗𝑡−𝜏_𝑗(𝑡),𝑥,𝑥_𝑛×𝑚. (2.3)

Let 𝐿²(𝑋) be the space of real Lebesgue measurable functions on 𝑋 and a Banach space for the 𝐿₂-norm‖𝑢(𝑡)‖²₂=_𝑋𝑢²(𝑡,𝑥)𝑑𝑥. (2.4) Note that 𝜉={(𝜉₁(𝑠,𝑥),…,𝜉_𝑛(𝑠,𝑥))^𝑇:−𝜏≤𝑠≤0} is 𝐶([−𝜏,0]×𝑅^𝑙;𝑅^𝑛)-valued function and ℱ₀-measurable 𝑅^𝑛-valued random variable, where ℱ₀=ℱ_𝑠 on [−𝜏,0], 𝐶([−𝜏,0]×𝑅^𝑙;𝑅^𝑛) is the space of all continuous 𝑅^𝑛-valued functions defined on [−𝜏,0]×𝑅^𝑙 with a norm ‖𝜉_𝑖(𝑡)‖²₂=∫_𝑋𝜉²_𝑖(𝑡,𝑥)𝑑𝑥.

The following assumptions and lemmas will be used in establishing our main results.(A1) There exist constants 𝑙⁻_𝑖, 𝑙⁺_𝑖, 𝑚⁻_𝑖 and 𝑚⁺_𝑖 such that 𝑙⁻_𝑖≤𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)𝑢−𝑣≤𝑙⁺_𝑖,𝑚⁻_𝑖≤𝑔_𝑖(𝑢)−𝑔_𝑖(𝑣)𝑢−𝑣≤𝑚⁺_𝑖,∀𝑢,𝑣∈𝑅,𝑢≠𝑣,𝑖=1,…,𝑛. (2.5) (A2) There exist constants 𝜇 and 𝛾_𝑖>0 such that 𝜏̇_𝑖(𝑡)≤𝜇,𝑦_𝑖(𝑡,𝑥)𝑐_𝑖𝑦_𝑖(𝑡,𝑥)≥𝛾_𝑖𝑦²_𝑖(𝑡,𝑥),𝑥∈𝑋,𝑖=1,…,𝑛. (2.6) (A3)𝑑_𝑖 is bounded, positive, and continuous, that is, there exist constants 𝑑_𝑖, 𝑑_𝑖 such that 0<𝑑_𝑖≤𝑑_𝑖(𝑢)≤𝑑_𝑖, for 𝑢∈𝑅, 𝑖=1,2,…,𝑛.

Lemma 2.1 (Poincaré inequality, [47]). Assume that a real-valued function 𝑤(𝑥)∶𝑋→𝑅 satisfies 𝑤(𝑥)∈𝐷={𝑤(𝑥)∈𝐿²(𝑋),(𝜕𝑤/𝜕𝑥_𝑖)∈𝐿²(𝑋)(1≤𝑖≤𝑙),(𝜕𝑤(𝑥)/𝜕𝑣)|_𝜕𝑋=0}, where 𝑋 is a bounded domain of 𝑅^𝑙 with a smooth boundary 𝜕𝑋. Then, 𝜆_1𝑋|𝑤(𝑥)|²𝑑𝑥≤_𝑋|∇𝑤(𝑥)|²𝑑𝑥, (2.7) which 𝜆₁ is the lowest positive eigenvalue of the Neumann boundary problem: (𝑥)(𝑥),(𝑥)−Δ𝑢=𝜆𝑢𝜕𝑢𝜕𝑣_𝜕𝑋=0,𝑥∈𝑋, (2.8)∇=(𝜕/𝜕𝑥₁,…,𝜕/𝜕𝑥_𝑚) is the gradient operator, Δ=∑^𝑚_𝑘=1(𝜕²/𝜕𝑥²_𝑘) is the Laplace operator.

Remark 2.2. Assumption (A1) is less conservative than that in [26, 28], since the constants 𝑙⁻_𝑖, 𝑙⁺_𝑖, 𝑚⁻_𝑖, and 𝑚⁺_𝑖 are allowed to be positive, negative, or zero, that is to say, the activation function in (A1) is assumed to be neither monotonic, differentiable, nor bounded. Assumption (A2) is weaker than those given in [23, 27, 30] since 𝜇 is not required to be zero or smaller than 1 and is allowed to take any value.

Remark 2.3. According to the eigenvalue theory of elliptic operators, the lowest eigenvalue 𝜆₁ is only determined by 𝑋 [47]. For example, if 𝑋=[0,𝐿], then 𝜆₁=(𝜋/𝐿)²; if 𝑋=(0,𝑎)×(0,𝑏), then 𝜆₁=min{(𝜋/𝑎)²,(𝜋/𝑏)²}.

The notation 𝐴>0 (resp., 𝐴≥0) means that matrix 𝐴 is symmetric-positive definite (resp., positive semidefinite). 𝐴^𝑇 denotes the transpose of the matrix 𝐴. 𝜆_min(𝐴) represents the minimum eigenvalue of matrix 𝐴. ‖𝑦(𝑡)‖²=∫_𝑋𝑦^𝑇(𝑡,𝑥)𝑦(𝑡,𝑥)𝑑𝑥=∑^𝑛_𝑖=1‖𝑦_𝑖(𝑡)‖²₂.

3. Main Results

Theorem 3.1. Suppose that assumptions (A1)–(A3) hold and there exist some matrices 𝑃=diag(𝑝₁,…,𝑝_𝑛)>0, 𝑄_𝑖≥0, 𝜎_𝑖>0, 𝑉_𝑖=diag(𝑣_𝑖1,…,𝑣_𝑖𝑛)≥0(𝑖=1,2), 𝑈_𝑗=diag(𝑢_𝑗1,…,𝑢_𝑗𝑛)≥0(𝑗=1,2,3), and 𝜎₃ such that the following linear matrix inequality hold:(A4)ΣΣ=¹𝜎₃𝐿₂𝑈₁𝑀₂𝑈₃0∗Σ₂00𝑀₂𝑈₂∗∗Σ₃00∗∗∗Σ₄0∗∗∗Σ₅𝜎<0,trace^𝑇𝑦(𝑡,𝑥),𝑦(𝑡−𝜏(𝑡),𝑥)𝑃𝜎𝑦(𝑡,𝑥),𝑦(𝑡−𝜏(𝑡),𝑥)≤𝑦^𝑇(𝑡,𝑥)𝜎₁𝑦(𝑡,𝑥)+𝑦^𝑇(𝑡−𝜏(𝑡),𝑥)𝜎₂𝑦(𝑡−𝜏(𝑡),𝑥)+2𝑦^𝑇(𝑡,𝑥)𝜎₃𝑦(𝑡−𝜏(𝑡),𝑥), (3.1)where 𝑥∈𝑋,∗ means the symmetric term, Σ₁=−2𝜆₁𝑃𝐷−2𝛾𝑑𝑃+3𝑑²𝑃+𝑀₃𝑉₁𝑀₃+𝜎₁+𝑄₁−2𝐿₁𝑈₁−2𝑀₁𝑈₃,Σ₂=𝑀₃𝑉₂𝑀₃+𝜎₂−1−𝜇𝑄₁−2𝑀₁𝑈₂,Σ₃=𝐴^𝑇𝑃𝐴−2𝑈₁,Σ₄=𝑄₂−𝑉₁−2𝑈₃,Σ₅=𝐵^𝑇𝑃𝐵−1−𝜇𝑄₂−𝑉₂−2𝑈₂,𝐷=diag𝐷₁,…,𝐷_𝑛,𝐷_𝑖=min_{1≤𝑘≤𝑙}𝐷_𝑖𝑘,𝛾=diag𝛾₁,…,𝛾_𝑛,𝑑=diag𝑑₁,…,𝑑_𝑛,𝑑=diag𝑑₁,…,𝑑_𝑛,𝐿₁=diag𝑙⁻₁𝑙⁺₁,…,𝑙⁻_𝑛𝑙⁺_𝑛,𝐿₂=diag𝑙⁻₁+𝑙⁺₁,…,𝑙⁻_𝑛+𝑙⁺_𝑛,𝑀₁=diag𝑚⁻₁𝑚⁺₁,…,𝑚⁻_𝑛𝑚⁺_𝑛,𝑀₂=diag𝑚⁻₁+𝑚⁺₁,…,𝑚⁻_𝑛+𝑚⁺_𝑛,𝑀₃=diagmax𝑚⁻₁,𝑚⁺₁,…,max𝑚⁻_𝑛,𝑚⁺_𝑛. (3.2)
Then system (2.1) is stochastically ultimately bounded, that is, if for any 𝜀∈(0,1), there is a positive constant 𝐶=𝐶(𝜀) such that the solution 𝑦(𝑡,𝑥) of system (2.1) satisfies limsup_𝑡→∞𝑃𝑦(𝑡)≤𝐶≥1−𝜀. (3.3)

Proof. If 𝜇≤1, then it follows from (A4) that there exists a sufficiently small 𝜆>0 such that ΔΔ=¹𝜎₃𝐿₂𝑈₁𝑀₂𝑈₃0∗Δ₂00𝑀₂𝑈₂∗∗Δ₃00∗∗∗Δ₄0∗∗∗Δ₅<0, (3.4) where Δ₁=−2𝜆₁𝑃𝐷−2𝛾𝑑𝑃+𝜆𝑃+3𝑑²𝑃+2𝜆𝐼+𝑀₃𝑉₁𝑀₃+𝜎₁+𝑄₁−2𝐿₁𝑈₁−2𝑀₁𝑈₃,Δ₂=𝜆𝐼+𝑀₃𝑉₂𝑀₃+𝜎₂−1−𝜇𝑒^−𝜆𝜏𝑄₁−2𝑀₁𝑈₂,Δ₃=𝜆𝐼+𝐴^𝑇𝑃𝐴−2𝑈₁,Δ₄=𝜆𝐼+𝑄₂−𝑉₁,Δ₅=𝜆𝐼+𝐵^𝑇𝑃𝐵−1−𝜇𝑒^−𝜆𝜏𝑄₂−𝑉₂−2𝑈₂. (3.5)
If 𝜇>1, then it follows from (A4) that there exists a sufficiently small 𝜆>0 such that ΔΔ=¹𝜎₃𝐿₂𝑈₁𝑀₂𝑈₃0∗Δ₂00𝑀₂𝑈₂∗∗Δ₃00∗∗∗Δ₄0∗∗∗Δ₅<0, (3.6) where Δ₁, Δ₃, and Δ₄ are the same as in (3.4), Δ₂=𝜆𝐼+𝑀₃𝑉₂𝑀₃+𝜎₂−1−𝜇𝑄₁−2𝑀₁𝑈₂,Δ₅=𝜆𝐼+𝐵^𝑇𝑃𝐵−1−𝜇𝑄₂−𝑉₂−2𝑈₂. (3.7)
Consider the following Lyapunov functional: 𝑉𝑦(𝑡)=_𝑋𝑒^𝜆𝑡𝑦^𝑇()()+𝑡,𝑥𝑃𝑦𝑡,𝑥𝑑𝑥_{𝑋𝑡(𝑡)𝑡−𝜏}𝑒^𝜆𝑠𝑦^𝑇(𝑠,𝑥)𝑄₁𝑦(𝑠,𝑥)+𝑔^𝑇𝑦(𝑠,𝑥)𝑄₂𝑔𝑦(𝑠,𝑥)𝑑𝑠𝑑𝑥. (3.8)
Applying Itô formula in [48] to 𝑉(𝑦(𝑡)) along (2.2), one obtains 𝑑𝑉𝑦(𝑡)=_𝑋𝜆𝑒^𝜆𝑡𝑦^𝑇(𝑡,𝑥)𝑃𝑦(𝑡,𝑥)𝑑𝑥𝑑𝑡+2^𝑛_𝑖=1𝑝_𝑖𝑒^𝜆𝑡_𝑋𝑦_𝑖(𝑡,𝑥)^𝑙_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘𝑑𝑥𝑑𝑡−2𝑒^𝜆𝑡_𝑋𝑦^𝑇(𝑡,𝑥)𝑃𝑑𝑦(𝑡,𝑥)𝑐𝑦(𝑡,𝑥)−𝐴𝑓𝑦(𝑡,𝑥)−𝐵𝑔𝑦(𝑡−𝜏(𝑡),𝑥)−𝐽𝑑𝑥𝑑𝑡+𝑒^𝜆𝑡_𝑋trace𝜎^𝑇𝑦(𝑡,𝑥),𝑦(𝑡−𝜏(𝑡),𝑥),𝑥𝑃𝜎𝑦(𝑡,𝑥),𝑦(𝑡−𝜏(𝑡),𝑥),𝑥𝑑𝑥𝑑𝑡+2𝑒^𝜆𝑡_𝑋𝑦^𝑇()𝑦()((𝑡))(𝑡)+𝑡,𝑥𝑃𝜎𝑡,𝑥,𝑦𝑡−𝜏,𝑥,𝑥𝑑𝑥𝑑𝑤_𝑋𝑒^𝜆𝑡𝑦^𝑇(𝑡,𝑥)𝑄₁𝑦(𝑡,𝑥)+𝑔^𝑇𝑦(𝑡,𝑥)𝑄₂𝑔𝑦()−(𝜏̇(𝑡𝑒𝑡,𝑥𝑑𝑥𝑑𝑡1−))^{𝜆(𝑡−𝜏(𝑡))}𝑦^𝑇(𝑡−𝜏(𝑡),𝑥)𝑄₁𝑦(𝑡−𝜏(𝑡),𝑥)+𝑔^𝑇𝑦(𝑡−𝜏(𝑡),𝑥)𝑄₂𝑔𝑦(𝑡−𝜏(𝑡),𝑥)𝑑𝑥𝑑𝑡. (3.9)
From assumptions (A1)–(A4), one obtains 2_𝑋𝑦^𝑇(𝑡,𝑥)𝑃𝑑𝑦(𝑡,𝑥)𝑐𝑦(𝑡,𝑥)𝑑𝑥≥2_𝑋𝑦^𝑇(𝑡,𝑥)𝑃𝑑()2𝛾𝑦𝑡,𝑥𝑑𝑥,_𝑋𝑦^𝑇(𝑡,𝑥)𝑃𝑑𝑦(𝑡,𝑥)𝐴𝑓𝑦(𝑡,𝑥)𝑑𝑥=2_𝑋𝑦^𝑇()𝑑𝑦()𝑦()≤𝑡,𝑥𝑡,𝑥𝑃𝐴𝑓𝑡,𝑥𝑑𝑥_𝑋𝑦^𝑇(𝑡,𝑥)𝑑²𝑦(𝑡,𝑥)𝑃𝑦(𝑡,𝑥)+𝑓^𝑇𝑦(𝑡,𝑥)𝐴^𝑇𝑦()≤𝑃𝐴𝑓𝑡,𝑥𝑑𝑥_𝑋𝑦^𝑇(𝑡,𝑥)𝑑²𝑃𝑦(𝑡,𝑥)+𝑓^𝑇𝑦(𝑡,𝑥)𝐴^𝑇𝑃𝐴𝑓𝑦(𝑡,𝑥)𝑑𝑥,2_𝑋𝑦^𝑇()𝑦()𝑦((𝑡))≤𝑡,𝑥𝑃𝑑𝑡,𝑥𝐵𝑔𝑡−𝜏,𝑥𝑑𝑥_𝑋𝑦^𝑇(𝑡,𝑥)𝑑²𝑃𝑦(𝑡,𝑥)+𝑔^𝑇𝑦(𝑡−𝜏(𝑡),𝑥)𝐵^𝑇𝑦((𝑡))2𝑃𝐵𝑔𝑡−𝜏,𝑥𝑑𝑥,_𝑋𝑦^𝑇(𝑡,𝑥)𝑃𝑑𝑦(𝑡,𝑥)𝐽𝑑𝑥≤_𝑋𝑦^𝑇(𝑡,𝑥)𝑑²𝑃𝑦(𝑡,𝑥)+𝐽^𝑇𝑃𝐽𝑑𝑥. (3.10)
From the boundary condition and Lemma 2.1, one obtains ^𝑙_𝑘=1𝑋𝑦_𝑖𝜕𝜕𝑥_𝑘𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘=𝑑𝑥_𝑋𝑦_𝑖∇⋅𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘^𝑙_𝑘=1=𝑑𝑥_𝑋∇⋅𝑦_𝑖𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘^𝑙_𝑘=1𝑑𝑥−_𝑋𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘^𝑙_𝑘=1⋅∇𝑦_𝑖=𝑑𝑥_{𝑙𝑘=1𝜕𝑋}𝑦_𝑖𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘^𝑙_𝑘=1⋅𝑑𝑠−^𝑙_𝑘=1𝑋𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘²𝑑𝑥=−^𝑙_𝑘=1𝑋𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘²𝑑𝑥≤^𝑙_𝑘=1𝑋𝐷_𝑖𝜕𝑦_𝑖𝜕𝑥_𝑘²𝑑𝑥=−𝐷_𝑖𝑋∇𝑦_𝑖²𝑑𝑥≤−𝜆₁𝐷_𝑖𝑋𝑦_𝑖²𝑑𝑥=−𝜆₁𝐷_𝑖𝑦_𝑖²₂, (3.11) where “·” is inner product, 𝐷_𝑖=min_{1≤𝑘≤𝑙}{𝐷_𝑖𝑘}, 𝐷_𝑖𝑘𝜕𝑦_𝑖𝜕𝑥_𝑘^𝑙_𝑘=1=𝐷_𝑖1𝜕𝑦_𝑖𝜕𝑥₁,…,𝐷_𝑖𝑙𝜕𝑦_𝑖𝜕𝑥_𝑙^𝑇. (3.12)
Combining (3.10) and (3.11) into (3.9), we have 𝑑𝑉𝑦(𝑡)≤_𝑋𝑒^𝜆𝑡𝑦^𝑇(𝑡,𝑥)𝜆𝑃−2𝜆₁𝑃𝐷−2𝑃𝑑𝛾+3𝑑²𝑃𝑦()+𝑡,𝑥𝑑𝑥𝑑𝑡_𝑋𝑒^𝜆𝑡𝑓^𝑇𝑦(𝑡,𝑥)𝐴^𝑇𝑃𝐴𝑓𝑦(𝑡,𝑥)+𝑔^𝑇𝑦(𝑡−𝜏(𝑡),𝑥)𝐵^𝑇𝑃𝐵𝑔𝑦(𝑡−𝜏(𝑡),𝑥)𝑑𝑥𝑑𝑡+_𝑋𝑒^𝜆𝑡𝐽^𝑇𝑃𝐽𝑑𝑥+_𝑋𝑒^𝜆𝑡𝑦^𝑇(𝑡,𝑥)𝜎₁𝑦(𝑡,𝑥)+𝑦^𝑇(𝑡−𝜏(𝑡),𝑥)𝜎₂𝑦(𝑡−𝜏(𝑡),𝑥)+2𝑦^𝑇(𝑡,𝑥)𝜎₃𝑦(𝑡−𝜏(𝑡),𝑥)𝑑𝑥𝑑𝑡+_𝑋2𝑒^𝜆𝑡𝑦^𝑇()𝑦()((𝑡))(𝑡)+𝑡,𝑥𝑃𝜎𝑡,𝑥,𝑦𝑡−𝜏,𝑥,𝑥𝑑𝑥𝑑𝑤_𝑋𝑒^𝜆𝑡𝑦^𝑇(𝑡,𝑥)𝑄₁𝑦(𝑡,𝑥)+𝑔^𝑇𝑦(𝑡,𝑥)𝑄₂𝑔𝑦(𝑡,𝑥)−1−𝜇ℎ𝜇𝑒^𝜆𝑡×𝑦^𝑇(𝑡−𝜏(𝑡),𝑥)𝑄₁𝑦(𝑡−𝜏(𝑡),𝑥)+𝑔^𝑇𝑦(𝑡−𝜏(𝑡),𝑥)𝑄₂𝑔𝑦(𝑡−𝜏(𝑡),𝑥)𝑑𝑥𝑑𝑡, (3.13) where ℎ(𝜇)=𝑒^−𝜆𝜏(𝜇≤1) or 1(𝜇>1).
In addition, it follows from (A1) that𝑦^𝑇(𝑡,𝑥)𝑀₃𝑉₁𝑀₃𝑦(𝑡,𝑥)−𝑔^𝑇𝑦(𝑡,𝑥)𝑉₁𝑔𝑦(𝑦𝑡,𝑥)≥0,_𝑇(𝑡−𝜏(𝑡),𝑥)𝑀₃𝑉₂𝑀₃𝑦(𝑡−𝜏(𝑡),𝑥)−𝑔^𝑇𝑦(𝑡−𝜏(𝑡),𝑥)𝑉₂𝑔𝑦(𝑡−𝜏(𝑡),𝑥)≥0,0≤−2^𝑛_𝑖=1𝑢_1𝑖𝑓_𝑖𝑦_𝑖(𝑡,𝑥)−𝑓_𝑖(0)−𝑙⁺_𝑖𝑦_𝑖(𝑡,𝑥)𝑓_𝑖𝑦_𝑖(𝑡,𝑥)−𝑓_𝑖(0)−𝑙⁻_𝑖𝑦_𝑖(𝑡,𝑥)=−2^𝑛_𝑖=1𝑢_1𝑖𝑓_𝑖𝑦_𝑖(𝑡,𝑥)−𝑙⁺_𝑖𝑦_𝑖(𝑡,𝑥)𝑓_𝑖𝑦_𝑖(𝑡,𝑥)−𝑙⁻_𝑖𝑦_𝑖(𝑡,𝑥)−2^𝑛_𝑖=1𝑢_1𝑖𝑓²_𝑖(0)+2^𝑛_𝑖=1𝑢_1𝑖𝑓_𝑖(0)2𝑓_𝑖𝑦_𝑖(𝑡,𝑥)−𝑙⁺_𝑖+𝑙⁻_𝑖𝑦_𝑖(𝑡,𝑥)≤−2^𝑛_𝑖=1𝑢_1𝑖𝑓_𝑖𝑦_𝑖(𝑡,𝑥)−𝑙⁺_𝑖𝑦_𝑖(𝑡,𝑥)𝑓_𝑖𝑦_𝑖(𝑡,𝑥)−𝑙⁻_𝑖𝑦_𝑖(+𝑡,𝑥)_𝑛𝑖=1𝜆𝑓²_𝑖𝑦_𝑖(𝑡,𝑥)+4𝜆⁻¹𝑓²_𝑖(0)𝑢²_1𝑖+𝜆𝑦²_𝑖(𝑡,𝑥)+𝜆⁻¹𝑓²_𝑖(0)𝑢²_1𝑖𝑙⁺_𝑖+𝑙⁻_𝑖². (3.14) Similarly, one obtains 0≤−2^𝑛_𝑖=1𝑢_2𝑖𝑔_𝑖𝑦_𝑖𝑡−𝜏_𝑖(𝑡),𝑥−𝑔_𝑖(0)−𝑚⁺_𝑖𝑦_𝑖𝑡−𝜏_𝑖(𝑡)×𝑔,𝑥_𝑖𝑦_𝑖𝑡−𝜏_𝑖(𝑡),𝑥−𝑔_𝑖(0)−𝑚⁻_𝑖𝑦_𝑖𝑡−𝜏_𝑖(𝑡),𝑥≤−2^𝑛_𝑖=1𝑢_2𝑖𝑔_𝑖𝑦_𝑖𝑡−𝜏_𝑖(𝑡),𝑥−𝑚⁺_𝑖𝑦_𝑖𝑡−𝜏_𝑖(𝑡)×𝑔,𝑥_𝑖𝑦_𝑖𝑡−𝜏_𝑖(𝑡),𝑥−𝑚⁻_𝑖𝑦_𝑖𝑡−𝜏_𝑖(𝑡)+,𝑥_𝑛𝑖=1𝜆𝑔²_𝑖𝑦_𝑖𝑡−𝜏_𝑖(𝑡),𝑥+4𝜆⁻¹𝑔²_𝑖(0)𝑢²_2𝑖+𝜆𝑦²_𝑖𝑡−𝜏_𝑖(𝑡),𝑥+𝜆⁻¹𝑔²_𝑖(0)𝑢²_2𝑖𝑚⁺_𝑖+𝑚⁻_𝑖²,0≤−2^𝑛_𝑖=1𝑢_3𝑖𝑔_𝑖𝑦_𝑖(𝑡,𝑥)−𝑔_𝑖(0)−𝑚⁺_𝑖𝑦_𝑖(𝑡,𝑥)𝑔_𝑖𝑦_𝑖(𝑡,𝑥)−𝑔_𝑖(0)−𝑚⁻_𝑖𝑦_𝑖(𝑡,𝑥)≤−2^𝑛_𝑖=1𝑢_3𝑖𝑔_𝑖𝑦_𝑖(𝑡,𝑥)−𝑚⁺_𝑖𝑦_𝑖(𝑡,𝑥)𝑔_𝑖𝑦_𝑖(𝑡,𝑥)−𝑚⁻_𝑖𝑦_𝑖(+𝑡,𝑥)_𝑛𝑖=1𝜆𝑔²_𝑖𝑦_𝑖(𝑡,𝑥)+4𝜆⁻¹𝑔²_𝑖(0)𝑢²_3𝑖+𝜆𝑦²_𝑖(𝑡,𝑥)+𝜆⁻¹𝑔²_𝑖(0)𝑢²_3𝑖𝑚⁺_𝑖+𝑚⁻_𝑖². (3.15)
From (3.13)–(3.15), one derives 𝑑𝑉𝑦(𝑡)≤_𝑋2𝑒^𝜆𝑡𝑦^𝑇()𝑦()((𝑡))(𝑡)+𝑡,𝑥𝑃𝜎𝑡,𝑥,𝑦𝑡−𝜏,𝑥,𝑥𝑑𝑥𝑑𝑤_𝑋𝑒^𝜆𝑡𝜂^𝑇(𝑡,𝑥)Δ𝜂(𝑡,𝑥)𝑑𝑥+𝑒^𝜆𝑡𝐶₁, (3.16) or 𝑑𝑉𝑦(𝑡)≤_𝑋2𝑒^𝜆𝑡𝑦^𝑇()𝑦()((𝑡))(𝑡)+𝑡,𝑥𝑃𝜎𝑡,𝑥,𝑦𝑡−𝜏,𝑥,𝑥𝑑𝑥𝑑𝑤_𝑋𝑒^𝜆𝑡𝜂^𝑇(𝑡,𝑥)Δ𝜂(𝑡,𝑥)𝑑𝑥+𝑒^𝜆𝑡𝐶₁, (3.17) where 𝜂(𝑡,𝑥)=(𝑦^𝑇(𝑡,𝑥),𝑦^𝑇(𝑡−𝜏(𝑡),𝑥),𝑓^𝑇(𝑦(𝑡,𝑥)),𝑔^𝑇(𝑦(𝑡,𝑥)),𝑔^𝑇(𝑦(𝑡−𝜏(𝑡),𝑥)))^𝑇, 𝐶₁=_𝑋𝐽^𝑇𝑃𝐽+^𝑛_𝑖=14𝜆⁻¹𝑓²_𝑖(0)𝑢²_1𝑖+𝜆⁻¹𝑓²_𝑖(0)𝑢²_1𝑖𝑙⁺_𝑖+𝑙⁻_𝑖²+4𝜆⁻¹𝑔²_𝑖(0)𝑢²_2𝑖+𝑢²_3𝑖+𝜆⁻¹𝑔²_𝑖(0)𝑢²_2𝑖+𝑢²_3𝑖𝑚⁺_𝑖+𝑚⁻_𝑖²𝑑𝑥. (3.18) Thus, one obtains 𝜆_min(𝑃)𝑒^𝜆𝑡𝐸𝑦(𝑡)²≤𝐸𝑉𝑦(𝑡)≤𝐸𝑉𝑦(0)+𝜆⁻¹𝑒^𝜆𝑡𝐶₁,𝐸𝑦(𝑡)²≤𝑒^−𝜆𝑡𝐸𝑉𝑦(0)+𝜆⁻¹𝐶₁𝜆_min(𝑃). (3.19)(3.20)
For any 𝜀>0, set 𝐶=𝜆⁻¹𝐶₁/𝜆_min(𝑃)𝜀. By Chebyshev’s inequality and (3.20), we obtain limsup_𝑡→∞𝑃𝑦(𝑡)>𝐶≤limsup_𝑡→∞𝐸𝑦(𝑡)²𝐶²=𝜀, (3.21) which implies limsup_𝑡→∞𝑃𝑦(𝑡)≤𝐶≥1−𝜀. (3.22) The proof is completed.

Theorem 3.1 shows that there exists 𝑡₀>0 such that for any 𝑡≥𝑡₀, 𝑃{‖𝑦(𝑡)‖≤𝐶}≥1−𝜀. Let 𝐵_𝐶 be denoted by 𝐵_𝐶=𝑦∣𝑦(𝑡)≤𝐶,𝑡≥𝑡₀. (3.23) Clearly, 𝐵_𝐶 is closed, bounded, and invariant. Moreover, limsup_𝑡→∞inf_{𝑧∈𝐵𝐶}𝑦(𝑡)−𝑧=0 (3.24) with no less than probability 1−𝜀, which means that 𝐵_𝐶 attracts the solutions infinitely many times with no less than probability 1−𝜀, so we may say that 𝐵_𝐶 is a weak attractor for the solutions.

Theorem 3.2. Suppose that all conditions of Theorem 3.1 hold. Then there exists a weak attractor 𝐵_𝐶 for the solutions of system (2.1).

Theorem 3.3. Suppose that all conditions of Theorem 3.1 hold and 𝑐(0)=𝑓(0)=𝑔(0)=𝐽=0. Then zero solution of system (2.1) is mean square exponential stability.

Remark 3.4. Assumption (A4) depends on 𝜆₁ and 𝜇, so the criteria on the stability, ultimate boundedness, and weak attractor depend on diffusion effects and the derivative of the delays and are independent of the magnitude of the delays.

4. An Example

In this section, a numerical example is presented to demonstrate the validity and effectiveness of our theoretical results.

Example 4.1. Consider the following system 𝑑𝑦(𝑡,𝑥)=col^𝑙_𝑘=1𝜕𝜕𝑥_𝑘𝐷_𝑖𝑘𝜕𝑦_𝑖(𝑡,𝑥)𝜕𝑥_𝑘𝑦()×𝑐𝑦()𝑦()𝑦((𝑡))+()((𝑡)(𝑡)𝑑𝑡−𝑑𝑡,𝑥𝑡,𝑥−𝐴𝑓𝑡,𝑥−𝐵𝑔𝑡−𝜏,𝑥−𝐽𝑑𝑡𝐺𝑦𝑡,𝑥+𝐻𝑦𝑡−𝜏,𝑥)𝑑𝑤,𝑥∈𝑋, (4.1) where 𝑛=2, 𝑙=𝑚=1, 𝑋=[0,𝜋], 𝐷₁₁=𝐷₂₁=0.5, 𝑑₁(𝑦₁(𝑡))=0.3+0.1cos𝑦₁(𝑡), 𝑑₂(𝑦₂(𝑡))=0.3+0.1sin𝑦₂(𝑡), 𝑐(𝑦(𝑡))=𝛾𝑦(𝑡), 𝑓(𝑦)=𝑔(𝑦)=0.1tanh(𝑦), ,,𝐴=−0.50.40.2−0.5,𝐵=0.4−0.7−0.80.4,𝛾=1001𝐽=0.010.01,𝐺=𝐻=0.20.10.10.2 (4.2)𝑤(𝑡) is one-dimensional Brownian motion. Then we compute that 𝜆₁=1, 𝐷=diag(0.5,0.5), 𝐿₁=𝑀₁=0, 𝐿₂=𝑀₂=𝑀₃=diag(0.1,0.1), 𝑑=diag(0.2,0.2), 𝑑=diag(0.4,0.4), 𝜎₁=𝐺^𝑇𝑃𝐺, 𝜎₂=𝐻^𝑇𝑃𝐻, and 𝜎₃=𝐺^𝑇𝑃𝐻. By using the Matlab LMI Toolbox, for 𝜇=0.1, based on Theorem 3.1, such system is stochastically ultimately bounded when 𝑃=23.94090024.5531,𝑈₁=,𝑈13.87010015.0659₂=7.5901006.4378,𝑈₃=𝑄11.80080011.6500₁=13.7292−0.0345−0.034513.9274,𝑄₂=,𝑉16.9580−4.6635−4.663516.5060₁=15.18440015.1109,𝑉₂=.13.07770012.4917 (4.3)

5. Conclusion

In this paper, new results and sufficient criteria on the ultimate boundedness, weak attractor, and stability are established for stochastic reaction-diffusion Cohen-Grossberg neural networks with delays by using Lyapunov method, Poincaré inequality and matrix technique. The criteria depend on diffusion effect and derivative of the delays and are independent of the magnitude of the delays.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 11271295, 10926128, 11047114, and 71171152), Science and Technology Research Projects of Hubei Provincial Department of Education (nos. Q20111607 and Q20111611) and Young Talent Cultivation Projects of Guangdong (LYM09134).