Exponential Stability for a Class of Stochastic Reaction-Diffusion Hopfield Neural Networks with Delays

Abstract

This paper studies the asymptotic behavior for a class of delayed reaction-diffusion Hopfield neural networks driven by finite-dimensional Wiener processes. Some new sufficient conditions are established to guarantee the mean square exponential stability of this system by using Poincaré’s inequality and stochastic analysis technique. The proof of the almost surely exponential stability for this system is carried out by using the Burkholder-Davis-Gundy inequality, the Chebyshev inequality and the Borel-Cantelli lemma. Finally, an example is given to illustrate the effectiveness of the proposed approach, and the simulation is also given by using the Matlab.

1. Introduction

Recently, the dynamics of Hopfield neural networks with reaction-diffusion terms have been deeply investigated because their various generations have been widely used in some practical engineering problems such as pattern recognition, associate memory, and combinatorial optimization (see [1–3]). However, under closer scrutiny, that a more realistic model would include some of the past states of the system, and theory of functional differential equations systems has been extensively developed [4, 5], meanwhile many authors have considered the asymptotic behavior of the neural networks with delays [6–9]. In fact random perturbation is unavoidable in any situation [3, 10]; if we include some environment noise in these systems, we can obtain a more perfect model of this situation [3, 11–16]. So, this paper is devoted to the exponential stability of the following delayed reaction-diffusion Hopfield neural networks driven by finite-dimensional Wiener processes:𝑑𝑢_𝑖(𝑡,𝑥)=^𝑙_𝑗=1𝜕𝜕𝑥_𝑗𝐷^𝑖𝑗(𝑥)𝜕𝑢^𝑖𝜕𝑥_𝑗−𝑎^𝑖𝑢_𝑖+^𝑛_𝑗=1+𝑐_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑡−𝑟,𝑥)𝑑𝑡_𝑚_𝑗=1𝜕𝑢_𝑖𝑔_𝑖𝑗𝑢_𝑖(𝑡−𝑟,𝑥)𝑑𝑊_𝑗,𝜕𝜈||||_𝜕𝒪𝑢=0,𝑡≥0,_𝑖(𝜃,𝑥)=𝜙_𝑖(𝜃,𝑥),𝑥∈𝒪∈ℝ^𝑙,𝜃∈[−𝑟,0],𝑖=1,2,…,𝑛.(1.1)

There are 𝑛 neural network units in this system and 𝑢_𝑖(𝑡,𝑥) denote the potential of the cell 𝑖 at 𝑡 and 𝑥. 𝑎_𝑖 are positive constants and denote the rate with which the 𝑖th unit will reset its potential to the resting state in isolation when it is disconnected from the network and external inputs at 𝑡, and 𝑐_𝑖𝑗 are the output connection weights from the 𝑗th neuron to the 𝑖th neuron. 𝑓_𝑗 are the active functions of the neural network. 𝑟 is the time delay of a neuron. 𝒪 denotes an open bounded and connected subset of ℝ^𝑙 with a sufficient regular boundary 𝜕𝒪, 𝜈 is the unit outward normal on 𝜕𝒪, 𝜕𝑢_𝑖/𝜕𝜈=(∇𝑢_𝑖,𝜈)_ℝ^𝑙, and 𝑔_𝑖𝑗 are noise intensities. Initial data 𝜙_𝑖 are ℱ₀-measurable and bounded functions, almost surely.

We denote (Ω,ℱ,ℙ) a complete probability space with filtration {ℱ_𝑡}_𝑡≥0 satisfying the usual conditions (see [10]). 𝑊_𝑖(𝑡),𝑖=1,2,…,𝑚, are scale standard Brownian motions defined on (Ω,ℱ,ℙ).

For convenience, we rewrite system (1.1) in the vector form: 𝑑𝑢=(∇⋅(𝐷(𝑥)∘∇𝑢)𝜕−𝑢(𝐴𝑡,𝑢𝑥+)𝐶𝑓(𝑢(𝑡−𝑟)))𝑑𝑡+𝐺(𝑢(𝑡−𝑟))𝑑𝑊,𝜕𝜈|||_𝜕𝒪𝑢(0,𝑥)==𝜙0,(𝑥)𝑡,≥0,(1.2) where 𝐶=(𝑐_𝑖𝑗)_𝑛×𝑛, 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)^𝑇, ∇𝑢=(∇𝑢₁,…,∇𝑢_𝑛)^𝑇, 𝑊=(𝑊₁,𝑊₂,…,𝑊_𝑚)^𝑇, 𝑓(𝑢)=(𝑓₁(𝑢₁),𝑓₂(𝑢₂),…,𝑓_𝑛(𝑢_𝑛))^𝑇, 𝐴=Diag(𝑎₁,𝑎₂,…,𝑎_𝑛), 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)^𝑇, 𝐺(𝑢)=(𝑔_𝑖𝑗(𝑢_𝑖))_𝑛×𝑚, 𝐷=(𝐷_𝑖𝑗)_𝑛×𝑙, and 𝐷∘∇𝑢=(𝐷_𝑖𝑗𝜕𝑢_𝑖/𝜕𝑥_𝑗)_𝑛×𝑙 is the Hadamard product of matrix 𝐷 and ∇𝑢; for the definition of divergence operator ∇⋅𝑢, we refer to [2, 3].

2. Preliminaries and Notations

In this paper, we introduce the following Hilbert spaces 𝐻≜𝐿²(𝒪), 𝑉≜𝐻¹(𝒪), according to [17–19], 𝑉⊂𝐻=𝐻′⊂𝑉′, where 𝐻′,𝑉′ denote the dual of the space 𝐻,𝑉, respectively, the injection is continuous, and the embedding is compact. ‖⋅‖,|‖⋅‖| represent the norm in 𝐻,𝑉, respectively.

𝑈≜(𝐿²(𝒪))^𝑛 is the space of vector-valued Lebesgue measurable functions on 𝒪, which is a Banach space under the norm ‖𝑢‖_𝑈=(∑^𝑛𝑖=1‖𝑢_𝑖(𝑥)‖²)^1/2.

𝐶≜𝐶([−𝑟,0],𝑈) is the Banach space of all continuous functions from [−𝑟,0] to 𝑈, when equipped with the sup-norm ‖𝜙‖_𝐶=sup_{−𝑟≤𝑠≤0}‖𝜙‖_𝑈.

With any continuous ℱ_𝑡-adapted 𝑈-valued stochastic process 𝑢(𝑡)∶Ω→𝑈, 𝑡≥−𝑟, we associate a continuous ℱ_𝑡-adapted 𝐶-valued stochastic process 𝑢_𝑡∶Ω→𝐶,𝑡>0, by setting 𝑢_𝑡(𝑠,𝑥)(𝜔)=𝑢(𝑡+𝑠,𝑥)(𝜔), 𝑠∈[−𝑟,0],𝑥∈𝒪.

𝐶^𝑏ℱ₀ denote the space of all bounded continuous processes 𝜙∶[−𝑟,0]×Ω→𝑈 such that 𝜙(𝜃,⋅) is ℱ₀-measurable for each 𝜃∈[−𝑟,0] and 𝐸‖𝜙‖_𝐶<∞.

ℒ(𝐾) is the set of all linear bounded operators from 𝐾 into 𝐾; when equipped with the operator norm, it becomes a Banach space.

In this paper, we assume the following.H1𝑓_𝑖 and 𝐺_𝑖𝑗 are Lipschitz continuous with positive Lipschitz constants 𝑘₁,𝑘₂ such that |𝑓_𝑖(𝑢)−𝑓_𝑖(𝑣)|≤𝑘₁|𝑢−𝑣| and |𝐺_𝑖𝑗(𝑢)−𝐺_𝑖𝑗(𝑣)|≤𝑘₂|𝑢−𝑣|,∀𝑢,𝑣∈ℝ, and 𝑓_𝑖(0)=0, 𝑔_𝑖𝑗(0)=0.H2 There exists 𝛼>0 such that 𝐷_𝑖𝑗(𝑥)≥𝛼/𝑙.H3 Let 𝜂=2𝛼𝛽²+2𝑘₃−𝑛𝑘²¹𝜎²𝑒^𝑟−𝑚𝑘²²𝑒^𝑟−2>0,𝑘₃=min{𝑎_𝑖}, 𝜎=max{|𝑐_𝑖𝑗|}.

Remark 2.1. We can infer from H1 that system (1.1) has an equilibrium 𝑢(𝑡,𝑥,𝜔)=0.
Let us define the linear operator as follows: 𝔄𝔄𝑢∶=Π∇(𝔄⋅)(𝐷∈(𝑈𝑥)⟶∘∇𝑢𝑈),,(2.1) and Π(𝔄)={𝑢∈𝐻²(𝒪)^𝑛,𝜕𝑢/𝜕𝜈|_𝜕𝒪=0}.

Lemma 2.2 (Poincaré’s inequality). Let 𝒪 be a bounded domain in 𝑅^𝑙 and 𝜙 belong to a collection of twice differentiable functions defined on 𝒪 into 𝑅; then ‖𝜙‖≤𝛽⁻¹||‖𝜙‖||,(2.2) where the constant 𝛽 depends on the size of 𝒪.

Lemma 2.3. Let us consider the equation 𝑑𝑢𝑑𝑡=𝑢(𝔄0)𝑢,=𝜙𝑡.≥0,(2.3) For every 𝜙∈𝑈, let 𝑢(𝑡)=𝑆(𝑡)𝜙 denote the solution of (2.3); then 𝑆(𝑡) is a contraction map in 𝑈.

Proof. Now we take the inner product of (2.3) with 𝑢(𝑡) in 𝑈; by employing the Gaussian theorem and condition H2, we get that (𝔄𝑢,𝑢)≤−𝛼‖|𝑢|‖^2𝐻_1(𝒪)𝑛, (⋅,⋅) is the inner product in 𝑈, ‖|𝑢|‖^2𝐻_1(𝒪)𝑛 denote the norm of 𝐻¹(𝒪)^𝑛 (see [3]), which means 12𝑑𝑑𝑡‖𝑢(𝑡)‖^2𝑈+𝛼‖‖||𝑢(𝑡)||‖‖^2𝐻_1(𝒪)𝑛≤0.(2.4) Thanks to the Poincaré inequality, one obtains 𝑑𝑑𝑡‖𝑢(𝑡)‖^2𝑈+2𝛼𝛽²‖𝑢(𝑡)‖^2𝑈≤0.(2.5) Multiplying 𝑒^{2𝛼𝛽2𝑡} in both sides of the inequality, we have 𝑑𝑑𝑡𝑒^{2𝛼𝛽2𝑡}‖𝑢(𝑡)‖^2𝑈≤0.(2.6) Integrating the above inequality from 0 to 𝑡, we obtain ‖𝑢(𝑡)‖^2𝑈≤𝑒^{−2𝛼𝛽2𝑡}‖𝜙‖^2𝑈.(2.7) By the definition of ‖𝑇(𝑡)‖_ℒ(𝑈), we have ‖𝑇(𝑡)‖_ℒ(𝑈)≤1.

Definition 2.4 (see [20–22]). A stochastic process 𝑢(𝑡)∶[−𝑟,+∞)×Ω→𝑈 is called a global mild solution of (1.1) if(i)𝑢(𝑡) is adapted to ℱ_𝑡(ii)𝑢(𝑡) is measurable with ∫^∞0‖𝑢(𝑡)‖^2𝑈𝑑𝑡<∞ almost surely and𝑢𝑢((𝑡𝑡))==𝑆𝜙(∈𝑡)𝐶𝜙−_𝑏ℱ0^𝑡0𝑆(𝑡−𝑠)Ads+^𝑡0𝑆(𝑡−𝑠)𝑓(𝑢(𝑠−𝑟))𝑑𝑠+^𝑡0𝑆(𝑡−𝑠)𝐺(𝑢(𝑠−𝑟))𝑑𝑊,,𝑡∈[−𝑟,0],(2.8) for all 𝑡∈[−𝑟,+∞) with probability one.

Definition 2.5. Equation (1.1) is said to be almost surely exponentially stable if, for any solution 𝑢(𝑡,𝑥,𝜔) with initial data 𝜙∈𝐶^𝑏ℱ₀, there exists a positive constant 𝜆 such that limsup_𝑡⟶∞ln‖‖𝑢_𝑡‖‖_𝐶≤−𝜆,𝑢_𝑡∈𝐶,almostsurely.(2.9)

Definition 2.6. System (1.1) is said to be exponentially stable in the mean square sense if there exist positive constants 𝜅 and 𝛼 such that, for any solution 𝑢(𝑡,𝑥,𝜔) with the initial condition 𝜙∈𝐶^𝑏ℱ₀, one has 𝐸‖𝑢(𝑡)‖^2𝐶≤𝜅𝑒^{−𝛼(𝑡−𝑡0)},𝑡≥𝑡₀,𝑢_𝑡∈𝐶.(2.10)

3. Main Result

Theorem 3.1. Suppose conditions H1–H3 hold; then (1.1) is exponentially stable in the mean square sense.

Proof. Let 𝑢 be the mild solution of (1.1); thanks to the Itô formula, we observe that 𝑑𝑒^𝜆𝑡𝑢^2𝑖=𝜆𝑒^𝜆𝑡𝑢^2𝑖𝑑𝑡+𝑒^𝜆𝑡2𝑢^𝑖^𝑙_𝑗=1𝜕𝜕𝑥_𝑖𝐷^𝑖𝑗𝜕𝑢_𝑖𝜕𝑥_𝑗−𝑎^𝑖𝑢_𝑖+^𝑛_𝑗=1+𝑒_𝜆𝑡𝑐_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑡−𝑟)𝑑𝑡𝐺_𝑖𝐺^𝑇𝑖𝑑𝑡+2𝑒^𝜆𝑡𝑢_𝑖𝐺_𝑖𝑑𝑊,𝐺_𝑖=𝐺_𝑖1,𝐺_𝑖2,…,𝐺_𝑖𝑚,(3.1) where 𝜆 is a positive constant that will be defined below. Then, by integration between 0 and 𝑡, we find that 𝑒^𝜆𝑡𝑢^2𝑖(𝑡)=𝜙_𝑖(0)²+^𝑡0𝜆𝑒^𝜆𝑠𝑢^2𝑖𝑑𝑠+2^𝑡0𝑒^𝜆𝑠𝑢^𝑖𝑙_𝑗=1𝜕𝜕𝑥_𝑗𝐷^𝑖𝑗𝜕𝑢_𝑖𝜕𝑥_𝑗𝑑𝑠−2^𝑡0+2_𝑡0𝑒^𝜆𝑠𝑎_𝑖𝑢^2𝑖𝑑𝑠𝑒^𝜆𝑠𝑢_𝑖𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑠−𝑟)𝑑𝑠+^𝑡0𝑒^𝜆𝑠𝐺_i𝐺^𝑇𝑖𝑑𝑠+2^𝑡0𝑒^𝜆𝑠𝑢_𝑖𝐺_𝑖𝑑𝑊.(3.2) Integrating the above equation over 𝒪, by virtue of Fubini’s theorem, we prove that 𝑒^𝜆𝑡‖‖𝑢^2𝑖‖‖²=‖‖𝜙_𝑖(0)‖‖²+𝜆^𝑡0𝑒^𝜆𝑠_𝒪𝑢^2𝑖𝑑𝑥𝑑𝑠+^𝑡0𝑒^𝜆𝑠_𝒪2𝑢_𝑖𝑙_𝑗=1𝜕𝜕𝑥_𝑗𝐷^𝑖𝑗𝜕𝑢_𝑖−2_𝑡0𝜕𝑥_𝑗𝑑𝑥𝑑𝑠𝑒^𝜆𝑠_𝒪𝑎_𝑖𝑢^2𝑖𝑑𝑥𝑑𝑠+2^𝑡0𝑒^𝜆𝑠_𝒪𝑢_𝑖𝑛_𝑗=1+𝑐_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑠−𝑟)𝑑𝑥𝑑𝑠_𝑡0𝑒^𝜆𝑠_𝒪𝐺_𝑖𝐺^𝑇𝑖𝑑𝑥𝑑𝑠+2^𝑡0𝑒^𝜆𝑠_𝒪𝑢_𝑖G_𝑖𝑑𝑥𝑑𝑊.(3.3) Taking the expectation on both sides of the last equation, by means of [3, 10, 16] 2𝐸^𝑡0_𝒪𝑒^𝜆𝑠𝑢_𝑖𝐺_𝑖𝑑𝑥𝑑𝑊=0.(3.4) Then, by Fubini’s theorem, we have 𝑒^𝜆𝑡𝐸‖‖𝑢^2𝑖‖‖²=𝐸‖‖𝜙_𝑖(0)‖‖²+𝜆^𝑡0𝑒^𝜆𝑠_𝒪𝐸𝑢^2𝑖𝑑𝑥𝑑𝑠+2𝐸^𝑡0𝑒^𝜆𝑠_𝒪𝑢_𝑖𝑙_𝑗=1𝜕𝜕𝑥_𝑗𝐷^𝑖𝑗𝜕𝑢_𝑖−2_𝑡0𝜕𝑥_𝑗𝑑𝑥𝑑𝑠𝑒^𝜆𝑠_𝒪𝑎_𝑖𝐸𝑢^2𝑖𝑑𝑥𝑑𝑠+2𝐸^𝑡0𝑒^𝜆𝑠_𝒪𝑢_𝑖𝑛_𝑗=1+𝐸_𝑡0𝑐_𝑖𝑗𝑓_𝑗𝑢_𝑗(𝑠−𝑟)𝑑𝑥𝑑𝑠𝑒^𝜆𝑠_𝒪≜𝐼₁𝐺_𝑖𝐺^𝑇𝑖𝑑𝑥𝑑𝑠+𝐼₂+𝐼₃+𝐼₄+𝐼₅+𝐼₆.(3.5) We observe that 𝐼𝐼₁≜𝐸‖‖𝜙_𝑖(0)‖‖²≤sup_{[]𝜃∈−𝑟,0}𝐸‖‖𝜙_𝑖(𝜃)‖‖²,(3.6)₂≜𝜆^𝑡0_𝒪𝑒^𝜆𝑠𝐸𝑢^2𝑖𝑑𝑥𝑑𝑠=𝜆^𝑡0𝑒^𝜆𝑠𝐸‖‖𝑢_𝑖‖‖²𝑑𝑠.(3.7) From the Neumann boundary condition, by means of Green’s formula and H2 (see [3, 6, 7]), we know 𝐼₃≜2𝐸^𝑡0_𝒪𝑒^𝜆𝑠𝑢^𝑖𝑙_𝑗=1𝜕𝜕𝑥_𝑗𝐷^𝑖𝑗𝜕𝑢_𝑖=−2𝐸_𝑡0𝜕𝑥_𝑗𝑑𝑥𝑑𝑠_𝒪𝑒^𝑙𝜆𝑠_𝑗=1𝐷_𝑖𝑗𝜕𝜕𝑥𝑢^𝑖_𝑗²≤−2𝛼_𝑡0𝑑𝑥𝑑𝑠𝑒^𝜆𝑠𝐸||‖‖𝑢_𝑖‖‖||²𝑑𝑠≤−2𝛼𝛽²^𝑡0𝑒^𝜆𝑠𝐸‖‖𝑢_𝑖‖‖²𝑑𝑠.(3.8) Then, by using the positiveness of 𝑎_𝑖, one gets the relation 𝐼₄≜−2^𝑡0_𝒪𝑒^𝜆𝑠𝑎_𝑖𝐸𝑢^2𝑖𝑑𝑥𝑑𝑠≤−2𝑘₃^𝑡0𝑒^𝜆𝑠𝐸‖‖𝑢_𝑖‖‖²𝑑𝑠,(3.9) where 𝑘₃=min{𝑎₁,𝑎₂,…,𝑎_𝑛}>0. By using the Young inequality as well as condition H1, we have that 𝐼₅≜2𝐸^𝑡0_𝒪𝑒^𝜆𝑠𝑢_𝑖𝑛_𝑗=1≤𝑐_𝑖𝑗𝑓_𝑗𝑑𝑥𝑑𝑠_𝑡0_𝒪𝑒^𝜆𝑠⎛⎜⎜⎝𝐸||𝑢_𝑖||²+𝐸|||||^𝑛_𝑗=1𝑐_𝑖𝑗𝑓_𝑗≤|||||²⎞⎟⎟⎠𝑑𝑥𝑑𝑠_𝑡0_𝒪𝑒^𝜆𝑠𝐸||𝑢^𝑖||²+𝜎^2𝑛_𝑗=1𝐸||𝑓_𝑗𝑢_𝑗(𝑠−𝑟)||²≤𝑑𝑥𝑑𝑠_𝑡0_𝒪𝑒^𝜆𝑠𝐸||𝑢^𝑖||²+𝜎²𝑘^21𝑛_𝑗=1𝐸||𝑢_𝑗(𝑠−𝑟)||²≤𝑑𝑥𝑑𝑠_𝑡0𝑒^𝜆𝑠𝐸‖‖𝑢^𝑖‖‖²+𝜎²𝑘²¹𝐸‖𝑢(𝑠−𝑟)‖^2𝑈𝑑𝑠,(3.10) where 𝜎=max|𝑐_𝑖𝑗|, and 𝐼₆≜^𝑡0_𝒪𝑒^𝜆𝑠𝐸𝐺_𝑖𝐺^𝑇𝑖𝑑𝑥𝑑𝑠≤𝑚𝑘²²^𝑡0𝑒^𝜆𝑠𝐸‖‖𝑢_𝑖(𝑠−𝑟)‖‖²𝑑𝑠.(3.11) We infer from (3.6)–(3.11) that 𝑒^𝜆𝑡𝐸‖‖𝑢_𝑖(𝑡)‖‖²≤+s𝜎u₂p_{𝜃∈[−𝑟,0]}𝐸‖‖𝜙_𝑖(𝜃)‖‖²−2𝛼𝛽²+2𝑘₃−1−𝜆^𝑡0𝑒^𝜆𝑠𝐸‖‖𝑢_𝑖‖‖²𝑑𝑠𝑘²¹^𝑡0𝑒^𝜆𝑠‖𝑢(𝑡−𝑟)‖^2𝑈𝑑𝑠+𝑚𝑘²²^𝑡0𝑒^𝜆𝑠𝐸‖‖𝑢_𝑖(𝑠−𝑟)‖‖²𝑑𝑠.(3.12) Adding (3.12) from 𝑖=1 to 𝑖=𝑛, we obtain 𝑒^𝜆𝑡𝐸‖𝑢‖^2𝑈≤+𝐸‖𝑛𝜙𝑘‖^2𝐶₂₁−2𝛼𝛽²+2𝑘₃−1−𝜆^𝑡0𝑒^𝜆𝑠𝐸‖𝑢‖^2𝑈𝑑𝑠𝜎²+𝑚𝑘²²^𝑡0𝑒^𝜆𝑠𝐸‖𝑢(𝑠−𝑟)‖^2𝑈𝑑𝑠,(3.13) due to ^𝑡0𝑒^𝜆𝑠𝐸‖𝑢(𝑠−𝑟)‖^2𝑈𝑑𝑠≤𝑒^𝜆𝑟^𝑡−𝑟≤𝑒_2𝜆𝑟𝑒^𝜆𝑠𝐸‖𝑢(𝑠)‖^2𝑈𝑑𝑠^0−𝑟𝐸‖𝜙(𝑠)‖^2𝑈𝑑𝑠+𝑒^𝜆𝑟^𝑡0≤𝑟𝑒_2𝜆𝑟𝑒^𝜆𝑠𝐸‖𝑢(𝑠)‖^2𝑈𝑑𝑠𝐸‖𝜙‖^2𝐶+𝑒^𝜆𝑟^𝑡0𝑒^𝜆𝑠𝐸‖𝑢(𝑠)‖^2𝑈𝑑𝑠;(3.14) we induce from the previous equations that 𝑒^𝜆𝑡𝐸‖𝑢‖^2𝑈≤−𝑐₁^𝑡0𝑒^𝜆𝑠𝐸‖𝑢‖^2𝑈𝑑𝑠+𝑐₂,(3.15) where 𝑐₁=2𝛼𝛽²+2𝑘₃−1−𝑛𝑘²¹𝜎²𝑒^𝜆𝑟−𝑚𝑘²²𝑒^𝜆𝑟−𝜆 and 𝑐₂=(1+𝑚𝑘²²𝑟𝑒^2𝜆𝑟+𝑛𝑘²¹𝜎²𝑟𝑒^2𝜆𝑟)𝐸‖𝜙‖^2𝐶; so we choose 𝜆=1 such that 𝑐₁=𝜂>0. By using the classical Gronwall inequality we see that 𝑒^𝜆𝑡𝐸‖𝑢‖^2𝑈≤𝑐₂𝑒^−𝜂𝑡;(3.16) in other words, we get 𝐸‖𝑢‖^2𝑈≤𝑐₂𝑒^{−(𝜂+1)𝑡}.(3.17) So, for 𝑡+𝜃≥𝑡/2≥0, we also have 𝐸‖𝑢(𝑡+𝜃)‖^2𝑈≤≤𝑐𝑐₂₂𝑒^{−(𝜂+1)(𝑡+𝜃)},𝑒^−𝜅𝑡,𝜃∈[−𝑟,0],𝜅=(𝜂+1)2,(3.18) and we can conclude that 𝐸‖‖𝑢_𝑡‖‖^2𝐶≤𝑐₂𝑒^−𝜅𝑡.(3.19)

Theorem 3.2. If the system (1.1) satisfies hypotheses H1–H3, then it is almost surely exponentially stable.

Proof. Let 𝑢(𝑡) be the mild solution of (1.1). By Definition 2.4 as well as the inequality (∑^𝑛𝑖=1𝑎_𝑖)²≤𝑛∑^𝑛𝑖=1𝑎^2𝑖,𝑎_𝑖∈ℝ, we have 𝐸sup_{𝑁≤𝑡≤𝑁+1}‖𝑢(𝑡)‖^2𝑈≤4sup_{𝑁≤𝑡≤𝑁+1}‖𝑆(𝑡−𝑁+1)𝑢(𝑁)‖^2𝑈+4sup_{𝑁≤𝑡≤𝑁+1}‖‖‖^𝑡𝑁−1−𝐴𝑆(𝑡−𝑠)𝑢𝑑𝑠‖‖‖^2𝑈+4sup_{𝑁≤𝑡≤𝑁+1}‖‖‖^𝑡𝑁−1𝑆(𝑡−𝑠)𝐶𝑓(𝑢(𝑠−𝑟))𝑑𝑠‖‖‖^2𝑈+4sup_{𝑁≤𝑡≤𝑁+1}‖‖‖^𝑡𝑁−1𝑆(𝑡−𝑠)𝐺(𝑢(𝑠−𝑟))𝑑𝑊‖‖‖^2𝑈≜𝐼₁+𝐼₂+𝐼₃+𝐼₄.(3.20) Using the contraction of the map 𝑆(𝑡) and the result of Theorem 3.1, we find 𝐼₁≜4sup_{𝑁≤𝑡≤𝑁+1}𝐸‖(𝑆(𝑡−𝑁+1)𝑢(𝑁−1))‖^2𝑈≤4sup_{𝑁≤𝑡≤𝑁+1}𝐸‖‖𝑢_𝑁−1‖‖^2𝐶≤4𝑐₂𝑒^{−𝜅(𝑁−1)}.(3.21) By the Hölder inequality, we obtain 𝐼₂≜4sup_{𝑁≤𝑡≤𝑁+1}𝐸‖‖‖^𝑡𝑁−1−𝐴𝑆(𝑡−𝑠)𝑢𝑑𝑠‖‖‖^2𝑈≤≤48ssuupp_{𝑁≤𝑡≤𝑁+1𝑁≤𝑡≤𝑁+1}(𝑡−𝑁+1)^𝑡𝑁−1𝐸‖−𝐴𝑆(𝑡−𝑠)𝑢‖^2𝑈𝑑𝑠^t_𝑁−1≤8𝑘₂₄𝐸‖𝐴𝑢‖^2𝑈𝑑𝑠^{𝑁+1𝑁−1}𝐸‖𝑢‖^2𝑈𝑑𝑠≤8𝑘²⁴^{𝑁+1𝑁−1}≤8𝑘₂₄𝐸‖‖𝑢_𝑠‖‖^2𝐶𝑑𝑠𝑐₂^{𝑁+1𝑁−1}𝑒^−𝜅𝑠𝑑𝑠≤8𝑘²⁴𝜌₁𝑒^{−𝜅(𝑁−1)},(3.22) where 𝜌₁=𝑐₂/𝜅,𝑘₄=max{𝑎₁,𝑎₂,…,𝑎_𝑛}.
By virtue of Theorem 3.1, Hölder inequality, and H1, we have 𝐼₃≜4sup_{𝑁≤𝑡≤𝑁+1}‖‖‖𝐸^𝑡𝑁−1𝑆(𝑡−𝑠)𝐶𝑓(𝑢(𝑠−𝑟))𝑑𝑠‖‖‖^2𝑈≤≤48𝜎₂sup_{𝑁≤𝑡≤𝑁+1}(𝑡−𝑁+1)𝐸^𝑡𝑁−1‖𝐶𝑓(𝑢(𝑠−𝑟))‖^2𝑈𝑑𝑠≤8𝑘₂₁sup_{𝑁≤𝑡≤𝑁+1}𝐸^𝑡𝑁−1‖𝑓(𝑢(𝑠−𝑟))‖^2𝑈𝑑𝑠𝜎²^{𝑁+1𝑁−1}𝐸‖𝑢(𝑠−𝑟)‖^2𝑈𝑑𝑠≤8𝑘²¹𝜎²^{𝑁+1𝑁−1}≤8𝑘₂₁𝐸‖‖𝑢_𝑠‖‖^2𝐶𝑑𝑠𝑐₂^{𝑁+1𝑁−1}𝑒^−𝜅𝑠𝑑𝑠≤8𝑘²¹𝜌₁𝑒^{−𝜅(𝑁−1)}.(3.23) Then, by the Burkholder-Davis-Gundy inequality (see [18, 22]), there exists 𝑐₃ such that 𝐼₄≜4sup_{𝑁≤𝑡≤𝑁+1}𝐸‖‖‖^𝑡𝑁−1𝑆(𝑡−𝑠)𝐺(𝑢(𝑠−𝑟))𝑑𝑊‖‖‖^2𝑈≤≤44𝑐𝑐₃₃sup_{𝑁≤𝑡≤𝑁+1}𝐸^𝑡𝑁−1‖𝑆(𝑡−𝑠)𝐺(𝑢(𝑠−𝑟))𝐼‖^2𝑈𝑑𝑠𝑘²²sup_{𝑁≤𝑡≤𝑁+1}^𝑡𝑁−1𝐸‖𝑢(𝑠−𝑟)‖^2𝑈𝑑𝑠≤4𝑐₃𝑘²²^{𝑁+1𝑁−1}≤4𝑐₃𝐸‖‖𝑢_𝑠‖‖^2𝐶𝑑𝑠𝑘²²𝑐₂^{𝑁+1𝑁−1}𝑒^−𝜅𝑠𝑑𝑠≤4𝑐₃𝑘²²𝜌₁𝑒^{−𝜅(𝑁−1)},(3.24) where 𝐼=(1,1,…,1)^𝑇 is an 𝑚-dimensional vector.
We can deduce from (3.21)–(3.24) that 𝐸sup_{𝑁≤𝑡≤𝑁+1}‖𝑢(𝑡)‖^2𝑈≤𝜌₂𝑒^{−𝜅(𝑁−1)},(3.25) where 𝜌₂=4𝑐₂+(8𝑘²⁴+8𝑘²¹+4𝑐₃𝑘²²)𝜌₁.
Thus, for any positive constants 𝜀_𝑁, thanks to the Chebyshev inequality we have that 𝑃sup_{𝑁≤𝑡≤𝑁+1}‖𝑢(𝑡)‖_𝑈>𝜀_𝑁≤𝜀1^2𝑁sup_{𝑁≤𝑡≤𝑁+1}𝐸‖𝑢(𝑡)‖^2𝑈≤1𝜀^2𝑁𝜌₂𝑒^{−𝜅(𝑁−1)}.(3.26) Due to the Borel-Cantelli lemma, we see that limsup_𝑡⟶∞ln‖𝑢(𝑡)‖^2𝑈𝑡≤−𝜅,almostsurely.(3.27) This completes the proof of the theorem.

4. Simulation

Consider two-dimensional stochastic reaction-diffusion recurrent neural networks with delay as follows:𝑑𝑑𝑢𝑢₁₂(𝑡,𝑥)=10Δ𝑢₁−7𝑢₁+1.3tanh𝑢₁(𝑡−1,𝑥)𝑑𝑡+𝑢₁(𝑡−1,𝑥)𝑑𝑊,(𝑡,𝑥)=+1𝑢0₂Δ𝑢₂−7𝑢₂+tanh𝑢₁(𝑡−1,𝑥)−tanh𝑢₂(𝑡−1,𝑥)𝑑𝑡𝜕𝑢_𝑖(𝑡−1,𝑥)𝑑𝑊(𝑡,0)𝜕𝑥=𝜕𝑢^𝑖(𝑡,20)𝑢𝜕𝑥=0,𝑡≥0,₁(𝜃,𝑥)=cos(0.2𝜋𝑥),𝑢₂(𝜃,𝑥)=cos(0.1𝜋𝑥),𝑥∈[0,20],𝜃∈[−1,0].(4.1)

Δ is the Laplace operator. We have 𝛽≥1/20, 𝛼≥10, 𝑘₁=1, 𝑘₂=1, 𝑘₃=7, 𝜎=1.3, 𝑛=2, and 𝜂>0; by Theorems 3.1 and 3.2, this system is mean square exponentially stable as well as almost surely exponentially stable. The results can be shown in Figures 1, 2 and 3.

(a)

(b)

Figure 1 

Figure 2 

Figure 3 

We use the forward Euler method to simulate this example [23–25]. We choose the time step Δ𝑡=0.01 and space step Δ𝑥=1, and 𝛿=Δ𝑡/Δ𝑥²=0.01.

Acknowledgment

The authors wish to thank the referees for their suggestions and comments. We are also indebted to the editors for their help. This work was supported by the National Natural Science Foundation of China (no. 11171374), Natural Science Foundation of Shandong Province (no. ZR2011AZ001).