Dynamical Behaviors of Stochastic Hopfield Neural Networks with Both Time-Varying and Continuously Distributed Delays

Abstract

This paper investigates dynamical behaviors of stochastic Hopfield neural networks with both time-varying and continuously distributed delays. By employing the Lyapunov functional theory and linear matrix inequality, some novel criteria on asymptotic stability, ultimate boundedness, and weak attractor are derived. Finally, an example is given to illustrate the correctness and effectiveness of our theoretical results.

1. Introduction

Hopfield neural networks [1] have been extensively studied in the past years and found many applications in different areas such as pattern recognition, associative memory, and combinatorial optimization. Such applications heavily depend on the dynamical behaviors such as stability, uniform boundedness, ultimate boundedness, attractor, bifurcation, and chaos. As it is well known, time delays are unavoidably encountered in the implementation of neural networks. Since time delays as a source of instability and bad performance always appear in many neural networks owing to the finite speed of information processing, the stability analysis for the delayed neural networks has received considerable attention. However, in these recent publications, most research on delayed neural networks has been restricted to simple cases of discrete delays. Since a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths, it is desired to model them by introducing distributed delays. Therefore, both discrete and distributed delays should be taken into account when modeling realistic neural networks [2, 3].

On the other hand, it has now been well recognized that stochastic disturbances are also ubiquitous owing to thermal noise in electronic implementations. Therefore, it is important to understand how these disturbances affect the networks. Many results on stochastic neural networks have been reported in [4–24]. Some sufficient criteria on the stability of uncertain stochastic neural networks were derived in [4–7]. Almost sure exponential stability of stochastic neural networks was studied in [8–10]. In [11–16], mean square exponential stability and 𝑝th moment exponential stability of stochastic neural networks were discussed. The stability of stochastic impulsive neural networks was discussed in [17–19]. The stability of stochastic neural networks with the Markovian jumping parameters was investigated in [20–22]. The passivity analysis for stochastic neural networks was discussed in [23, 24]. These references mainly considered the stability of equilibrium point of stochastic delayed neural networks. What do we study to understand the asymptotic behaviors when the equilibrium point does not exist?

Except for the stability property, boundedness and attractor are also foundational concepts of dynamical systems. They play an important role in investigating the uniqueness of equilibrium, global asymptotic stability, global exponential stability, and the existence of periodic solution, its control, and synchronization [25]. Recently, ultimate boundedness and attractor of several classes of neural networks with time delays have been reported in [26–33]. Some sufficient criteria were derived in [26, 27], but these results hold only under constant delays. Following, in [28], the globally robust ultimate boundedness of integrodifferential neural networks with uncertainties and varying delays were studied. After that, some sufficient criteria on the ultimate boundedness of neural networks with both varying and unbounded delays were derived in [29], but the concerned systems are deterministic ones. In [30, 31], a series of criteria on the boundedness, global exponential stability, and the existence of periodic solution for nonautonomous recurrent neural networks were established. In [32, 33], the ultimate boundedness and weak attractor of stochastic neural networks with time-varying delays were discussed. To the best of our knowledge, for stochastic neural networks with mixed time delays, there are few published results on the ultimate boundedness and weak attractor. Therefore, the arising questions about the ultimate boundedness, weak attractor, and asymptotic stability of the stochastic Hopfield neural networks with mixed time delays are important and meaningful.

The left paper is organized as follows. Some preliminaries are in Section 2, main results are presented in Section 3, a numerical example is given in Section 4, and conclusions are drawn in Section 5.

2. Preliminaries

Consider the following stochastic Hopfield neural networks with both time-varying and continuously distributed delays: 𝑑𝑥(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐷^𝑡_−∞𝐾()𝑔(𝑥(𝑠+𝜎𝑡−𝑠))𝑑𝑠+𝐽𝑑𝑡₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))𝑑𝑤(𝑡),(1) in which 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇 is a state vector associated with the neurons; 𝐶=diag(𝑐₁,…,𝑐_𝑛), 𝑐_𝑖>0 represents the rate with which the 𝑖th unit will reset its potential to the resting state in isolation when being disconnected from the network and the external stochastic perturbation; 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, and 𝐷=(𝑑_𝑖𝑗)_𝑛×𝑛 represent the connection weight matrix, the delayed connection weight matrix, and the distributively delayed connection weight matrix, respectively; 𝐽=(𝐽₁,…,𝐽_𝑛)^𝑇, 𝐽_𝑖 denotes the external bias on the ith unit; 𝑓(𝑥(𝑡))=(𝑓₁(𝑥₁(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡)))^𝑇, 𝑔(𝑥(𝑡))=(𝑔₁(𝑥₁(𝑡)),…,𝑔_𝑛(𝑥_𝑛(𝑡)))^𝑇, 𝑓_𝑗 and 𝑔_𝑗 denote activation functions, 𝐾(𝑡)=diag(𝑘₁(𝑡),…,𝑘_𝑛(𝑡)), and the delay kernel 𝑘_𝑗(𝑡) is a real-valued nonnegative continuous function defined on [0,∞); 𝜎₁ and 𝜎₂ are diffusion coefficient matrices; 𝑤(𝑡) is a one-dimensional Brownian motion or Winner process, which is defined on a complete probability space (Ω,ℱ,𝑃) with a natural filtration {ℱ_𝑡}_𝑡≥0 generated by {𝑤(𝑠)∶0≤𝑠≤𝑡}; 𝜏(𝑡) is the transmission delay, and the initial conditions associated with system (1) are of the following forms: 𝑥(𝑡)=𝜉(𝑡), −∞<𝑡≤0, where 𝜉(⋅) is a ℱ₀-measurable, bounded, and continuous 𝑅^𝑛-valued random variable defined on (−∞,0].

Throughout this paper, one always supposes that the following condition holds. (A1) For 𝑓(𝑥(𝑡)) and 𝑔(𝑥(𝑡)) in (1), there are always constants 𝑙⁻_𝑖, 𝑙⁺_𝑖, 𝑚⁻_𝑖, and 𝑚⁺_𝑖 such that 𝑙⁻_𝑖≤𝑓_𝑖(𝑥)−𝑓_𝑖𝑦𝑥−𝑦≤𝑙⁺_𝑖,𝑚⁻_𝑖≤𝑔_𝑖(𝑥)−𝑔_𝑖𝑦𝑥−𝑦≤𝑚⁺_𝑖,∀𝑥,𝑦∈𝑅.(2) Moreover, there exist constant 𝜈>0 and matrix 𝐾(𝜈)=diag(𝑘₁(𝜈),…,𝑘_𝑛(𝜈))>0 such that ^∞₀𝑘_𝑗(𝜃)𝑑𝜃=1,^∞₀𝑘_𝑗(𝜃)𝑒^𝜈𝜃𝑑𝜃=𝑘_𝑗(𝜈)<∞.(3)

Following, 𝐴>0 (resp., 𝐴≥0) means that matrix 𝐴 is a symmetric positive definite (resp., positive semi-definite). 𝐴^𝑇 and 𝐴⁻¹ denote the transpose and inverse of the matrix 𝐴. 𝜆_max(𝐴) and 𝜆_min(𝐴) represent the maximum and minimum eigenvalues of matrix 𝐴, respectively.

Definition 1. System (1) is said to be stochastically ultimately bounded if, for any 𝜀∈(0,1), there is a positive constant 𝐶=𝐶(𝜀) such that the solution 𝑥(𝑡) of system (1) satisfies limsup_𝑡→∞𝑃{‖𝑥(𝑡)‖≤𝐶}≥1−𝜀.(4)

Lemma 2 (see [34]). Let 𝑄(𝑥)=𝑄^𝑇(𝑥),𝑅(𝑥)=𝑅^𝑇(𝑥) and 𝑆(𝑥) depends affinely on 𝑥. Then, linear matrix inequality 𝑄(𝑥)𝑆(𝑥)𝑆^𝑇(𝑥)𝑅(𝑥)>0(5) is equivalent to (1)𝑅(𝑥)>0,𝑄(𝑥)−𝑆(𝑥)𝑅⁻¹(𝑥)𝑆^𝑇(𝑥)>0, (2)𝑄(𝑥)>0,𝑅(𝑥)−𝑆^𝑇(𝑥)𝑄⁻¹(𝑥)𝑆(𝑥)>0.

3. Main Results

Theorem 3. System (1) is stochastically ultimately bounded provided that 𝜏(𝑡) satisfies 0≤𝜏(𝑡)≤𝜏, 𝜏̇(𝑡)≤𝜇≤1, and there exist some matrices 𝑃>0, 𝑄_𝑖≥0, 𝑈_𝑖=diag(𝑢_𝑖1,…,𝑢_𝑖𝑛)≥0 (𝑖=1,2,3,4) such that the following linear matrix inequality holds: ΣΣ=¹¹0𝑃𝐴+𝐿₂𝑈₁𝑃𝐵𝑀₂𝑈₄𝑃𝐷𝜎^𝑇₁𝑃∗Σ₂₂0𝐿₂𝑈₂00𝜎^𝑇₂𝑃∗∗Σ₃₃0000∗∗∗Σ₄₄000∗∗∗∗Σ₅₅00∗∗∗∗∗−𝑈₃0∗∗∗∗∗−𝑃<0,(6) where ∗ denotes the corresponding symmetric terms, Σ₁₁=𝑄₁+𝜏𝑄₂−𝑃𝐶−𝐶𝑃−2𝐿₁𝑈₁−2𝑀₁𝑈₄,Σ₂₂=−1−𝜇𝑄₁−2𝐿₁𝑈₂,Σ₃₃=𝑄₃+𝜏𝑄₄−2𝑈₁,Σ₄₄=−1−𝜇𝑄₃−2𝑈₂,Σ₅₅=𝑈₃𝐾(𝜈)−2𝑈₄,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.(7)

Proof. The key of the proof is to prove that there exists a positive constant 𝐶_∗, which is independent of the initial data, such that limsup_𝑡→∞𝐸‖𝑥(𝑡)‖²≤𝐶_∗.(8) If (8) holds, then it follows from Chebyshev's inequality that for any 𝜀>0 and 𝐶=𝐶_∗/𝜀, limsup_𝑡→∞𝑃{‖𝑥(𝑡)‖>𝐶}≤limsup_𝑡→∞𝐸‖𝑥(𝑡)‖²𝐶²=𝜀,(9) which implies that (4) holds. Now, we begin to prove that (8) holds.
From Σ<0 and Lemma 2, one may obtain Σ₁₁0𝑃𝐴+𝐿₂𝑈₁𝑃𝐵𝑀₂𝑈₄𝑃𝐷∗Σ₂₂0𝐿₂𝑈₂00∗∗Σ₃₃000∗∗∗Σ₄₄00∗∗∗∗Σ₅₅0∗∗∗∗∗−𝑈₃+𝜎^𝑇₁𝑃𝜎^𝑇₂𝑃0000𝑃⁻¹𝜎^𝑇₁𝑃𝜎^𝑇₂𝑃0000^𝑇<0.(10)
Hence, there exists a sufficiently small 𝜆∈(0,𝜈) such that ΓΓ=¹¹0𝑃𝐴+𝐿₂𝑈₁𝑃𝐵𝑀₂𝑈₄𝑃𝐷∗Γ₂₂0𝐿₂𝑈₂00∗∗Γ₃₃000∗∗∗Γ₄₄00∗∗∗∗Γ₅₅0∗∗∗∗∗−𝑈₃+𝜎^𝑇₁𝑃𝜎^𝑇₂𝑃0000𝑃⁻¹𝜎^𝑇₁𝑃𝜎^𝑇₂𝑃0000^𝑇<0,(11) where 𝐼 is identity matrix, Γ₁₁=𝑒^𝜆𝜏𝑄₁+𝜏𝑄₂−𝑃𝐶−𝐶𝑃−2𝐿₁𝑈₁−2𝑀₁𝑈₄Γ+2𝜆𝑃+2𝜆𝐼,₂₂=Σ₂₂+𝜆𝐼,Γ₃₃=𝜆𝐼+𝑒^𝜆𝜏𝑄₃+𝜏𝑄₄−2𝑈₁,Γ₄₄=Σ₄₄+𝜆𝐼,Γ₅₅=Σ₅₅+𝜆𝐼.(12)
Consider the Lyapunov-Krasovskii function as follows: 𝑉(𝑥(𝑡),𝑡)=𝑉₁(𝑥(𝑡),𝑡)+𝑉₂(𝑥(𝑡),𝑡)+𝑉₃(𝑥(𝑡),𝑡)+𝑉₄(𝑥(𝑡),𝑡),(13) where 𝑉₁(𝑥(𝑡),𝑡)=𝑒^𝜆𝑡𝑥^𝑇(𝑡)(𝑡),𝑉𝑃𝑥₂(𝑥(𝑡),𝑡)=^𝑛_𝑗=1𝑢_3𝑗^∞₀𝑘_𝑗(𝜃)^𝑡_𝑡−𝜃𝑒^{𝜆(𝑠+𝜃)}𝑔²_𝑗𝑥_𝑗(𝑠)𝑉𝑑𝑠𝑑𝜃,₃(𝑥(𝑡))=,𝑡_{𝑡(𝑡)𝑡−𝜏}𝑒^{𝜆(𝑠+𝜏)}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+𝑓^𝑇(𝑥(𝑠))𝑄₃𝑓(𝑥(𝑠𝑉))𝑑𝑠,₄(𝑥(𝑡))=,𝑡_{𝑡𝑡−𝜏(𝑡)𝑡𝑠}𝑒^𝜆𝜃𝑥^𝑇(𝜃)𝑄₂𝑥(𝜃)+𝑓^𝑇(𝑥(𝜃))𝑄₄𝑓(𝑥(𝜃))𝑑𝜃𝑑𝑠.(14)
Then, it can be obtained by Ito's formula in [35] that 𝑑𝑉(𝑥(𝑡),𝑡)=𝑒^𝜆𝑡2𝑥^𝑇(𝑡)𝑃𝜎₁𝑥(𝑡)+𝜎₂𝑥((𝑡(𝑡)+𝑡−𝜏))𝑑𝑤ℒ𝑉₁(𝑥(𝑡))+𝜕𝑉,𝑡₂(𝑥(𝑡),𝑡)+𝑉₃(𝑥(𝑡),𝑡)+𝑉₄(𝑥(𝑡),𝑡)𝜕𝑡𝑑𝑡,(15) where 𝜕𝑉₂(𝑥(𝑡),𝑡)=𝜕𝑡_𝑛𝑗=1𝑢_3𝑗^∞₀𝑘_𝑗(𝜃)𝑒^{𝜆(𝑡+𝜃)}𝑔²_𝑗𝑥_𝑗(𝑡)−𝑒^𝜆𝑡𝑔²_𝑗𝑥_𝑗(𝑡−𝜃)𝑑𝜃≤𝑒^𝑛𝜆𝑡_𝑗=1𝑢_3𝑗𝑘_𝑗(𝜈)𝑔²_𝑗𝑥_𝑗(𝑡)−𝑒^𝑛𝜆𝑡_𝑗=1𝑢_3𝑗^∞₀𝑘_𝑗(𝜃)𝑔_𝑗𝑥_𝑗(𝑡−𝜃)𝑑𝜃²=𝑒^𝜆𝑡𝑔^𝑇(𝑥(𝑡))𝑈₃𝐾(𝜈)𝑔(𝑥(𝑡))−𝑒^𝜆𝑡𝑡_−∞𝐾(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠^𝑇×𝑈₃^𝑡_−∞𝐾(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠,𝜕𝑉₃(𝑥(𝑡),𝑡)𝜕𝑡=𝑒^{𝜆(𝑡+𝜏)}𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)+𝑓^𝑇(𝑥(𝑡))𝑄₃𝑓(𝑥(𝑡−(𝜏̇(𝑡𝑒))1−))^{𝜆(𝑡−𝜏(𝑡)+𝜏)}×𝑥^𝑇(𝑡−𝜏(𝑡))𝑄₁𝑥(𝑡−𝜏(𝑡))+𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑄₃𝑓(𝑥(𝑡−𝜏(𝑡)))≤𝑒^{𝜆(𝑡+𝜏)}𝑥^𝑇(𝑡)𝑄₁𝑥(𝑡)+𝑓^𝑇(𝑥(𝑡))𝑄₃𝑓(𝑥(𝑡−𝑒))1−𝜇^𝜆𝑡×𝑥^𝑇(𝑡−𝜏(𝑡))𝑄₁𝑥(𝑡−𝜏(𝑡))+𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑄₃𝑓(𝑥(𝑡−𝜏(𝑡))),𝜕𝑉₄(𝑥(𝑡),𝑡)𝜕𝑡=𝑒^𝜆𝑡𝜏(𝑡)𝑥^𝑇(𝑡)𝑄₂𝑥(𝑡)+𝑓^𝑇(𝑥(𝑡))𝑄₄𝑓(𝑥(𝑡−(𝜏̇(𝑡×))1−))_{𝑡(𝑡)𝑡−𝜏}𝑒^𝜆𝑠𝑥^𝑇(𝑠)𝑄₂𝑥(𝑠)+𝑓^𝑇(𝑥(𝑠))𝑄₄𝑓(𝑥(𝑠))𝑑𝑠≤𝑒^𝜆𝑡𝜏𝑥^𝑇(𝑡)𝑄₂𝑥(𝑡)+𝑓^𝑇(𝑥(𝑡))𝑄₄𝑓(𝑥(𝑡)),(16)ℒ𝑉₁(𝑥(𝑡))=,𝑡𝜕𝑉₁(𝑥(𝑡),𝑡)+𝜕𝑡𝜕𝑉₁(𝑥(𝑡),𝑡)×(𝑡)(𝑥(𝑡(𝑥((𝑡𝜕𝑥−𝐶𝑥+𝐴𝑓))+𝐵𝑓𝑡−𝜏)))+𝐷^𝑡_−∞𝐾()𝑔(𝑥(𝑠𝜎𝑡−𝑠))𝑑𝑠+𝐽+trace₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))^𝑇𝜕²𝑉₁(𝑥(𝑡),𝑡)𝜕𝑥²×𝜎₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))=𝜆𝑒^𝜆𝑡𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑒^𝜆𝑡2𝑥^𝑇(𝑡)𝑃(𝑡)(𝑥(𝑡(𝑥((𝑡−𝐶𝑥+𝐴𝑓))+𝐵𝑓𝑡−𝜏)))+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠+𝐽+𝑒^𝜆𝑡𝜎₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))^𝑇×𝑃𝜎₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))≤𝜆𝑒^𝜆𝑡𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝑒^𝜆𝑡2𝑥^𝑇(𝑡)𝑃(𝑡)(𝑥(𝑡(𝑥((𝑡−𝐶𝑥+𝐴𝑓))+𝐵𝑓𝑡−𝜏)))+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠+𝑒^𝜆𝑡𝜆𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝜆⁻¹𝐽^𝑇𝑃𝐽+𝑒^𝜆𝑡𝜎₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))^𝑇×𝑃𝜎₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡)),(17) in which the following inequality is used: 2𝑥^𝑇𝑃𝐽≤𝜆𝑥^𝑇𝑃𝑥+𝜆⁻¹𝐽^𝑇𝑃^𝑇𝑃⁻¹𝑃𝐽=𝜆𝑥^𝑇𝑃𝑥+𝜆⁻¹𝐽^𝑇𝑃𝐽,for𝑃>0,𝜆>0.(18)
On the other hand, it follows from (A1) that for 𝑖=1,…,𝑛, 𝑓_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖(0)−𝑙⁺_𝑖𝑥_𝑖(𝑡×𝑓_𝑖)𝑥_𝑖(𝑡)−𝑓_𝑖(0)−𝑙⁻_𝑖𝑥_𝑖(𝑡𝑓)≤0,_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝑖(0)−𝑙⁺_𝑖𝑥_𝑖((𝑡×𝑓𝑡−𝜏))_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝑖(0)−𝑙⁻_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))≤0,(19)0≤𝑒^𝜆𝑡−2^𝑛_𝑖=1𝑢_1𝑖𝑓_𝑖𝑥_𝑖(𝑡)−𝑓_𝑖(0)−𝑙⁺_𝑖𝑥_𝑖(𝑡×𝑓_𝑖)𝑥_𝑖(𝑡)−𝑓_𝑖(0)−𝑙⁻_𝑖𝑥_𝑖(𝑡)−2^𝑛_𝑖=1𝑢_2𝑖𝑓_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝑖(0)−𝑙⁺_𝑖𝑥_𝑖((𝑡×𝑓𝑡−𝜏))_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑓_𝑖(0)−𝑙⁻_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))=𝑒^𝜆𝑡−2^𝑛_𝑖=1𝑢_1𝑖𝑓_𝑖𝑥_𝑖(𝑡)−𝑙⁺_𝑖𝑥_𝑖(𝑡×𝑓_𝑖)𝑥_𝑖(𝑡)−𝑙⁻_𝑖𝑥_𝑖(𝑡)−2^𝑛_𝑖=1𝑢_2𝑖𝑓_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑙⁺_𝑖𝑥_𝑖((𝑡×𝑓𝑡−𝜏))_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑙⁻_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−2^𝑛_𝑖=1𝑢_1𝑖𝑓²_𝑖(0)+2^𝑛_𝑖=1𝑢_1𝑖𝑓_𝑖(0)2𝑓_𝑖𝑥_𝑖(𝑡)−𝑙⁺_𝑖+𝑙⁻_𝑖𝑥_𝑖(𝑡)−2^𝑛_𝑖=1𝑢_2𝑖𝑓²_𝑖(0)+2^𝑛_𝑖=1𝑢_2𝑖𝑓_𝑖(0)2𝑓_𝑖𝑥_𝑖((𝑡−𝑙𝑡−𝜏))_+𝑖+𝑙⁻_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))≤𝑒^𝜆𝑡−2^𝑛_𝑖=1𝑢_1𝑖𝑓_𝑖𝑥_𝑖(𝑡)−𝑙⁺_𝑖𝑥_𝑖(𝑡×𝑓_𝑖)𝑥_𝑖(𝑡)−𝑙⁻_𝑖𝑥_𝑖(𝑡)−2^𝑛_𝑖=1𝑢_2𝑖𝑓_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑙⁺_𝑖𝑥_𝑖((𝑡×𝑓𝑡−𝜏))_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑙⁻_𝑖𝑥_𝑖((𝑡+𝑡−𝜏))_𝑛𝑖=14𝑢_1𝑖𝑓_𝑖(0)𝑓_𝑖𝑥_𝑖(𝑡)+2𝑢_1𝑖𝑓_𝑖(0)𝑙⁺_𝑖+𝑙⁻_𝑖𝑥_𝑖(𝑡+_𝑛𝑖=1)4𝑢_2𝑖𝑓_𝑖(0)𝑓_𝑖𝑥_𝑖((𝑡+𝑡−𝜏))2𝑢_2𝑖𝑓_𝑖(0)𝑙⁺_𝑖+𝑙⁻_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))≤𝑒^𝜆𝑡−2^𝑛_𝑖=1𝑢_1𝑖𝑓_𝑖𝑥_𝑖(𝑡)−𝑙⁺_𝑖𝑥_𝑖(𝑡×𝑓_𝑖)𝑥_𝑖(𝑡)−𝑙⁻_𝑖𝑥_𝑖(𝑡)−2^𝑛_𝑖=1𝑢_2𝑖𝑓_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑙⁺_𝑖𝑥_𝑖((𝑡×𝑓𝑡−𝜏))_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))−𝑙⁻_𝑖𝑥_𝑖((𝑡+𝑡−𝜏))_𝑛𝑖=1𝜆𝑓²_𝑖𝑥_𝑖(𝑡)+4𝜆⁻¹𝑓²_𝑖(0)𝑢²_1𝑖+𝜆𝑥²_𝑖(𝑡)+𝜆⁻¹𝑓²_𝑖(0)𝑢²_1𝑖𝑙⁺_𝑖+𝑙⁻_𝑖²+^𝑛_𝑖=1𝜆𝑓²_𝑖𝑥_𝑖(𝑡−𝜏(𝑡))+4𝜆⁻¹𝑓²_𝑖(0)𝑢²_2𝑖+𝜆𝑥²_𝑖(𝑡−𝜏(𝑡))+𝜆⁻¹𝑓²_𝑖(0)𝑢²_2𝑖𝑙⁺_𝑖+𝑙⁻_𝑖².(20)
Similarly, one may obtain 0≤𝑒^𝜆𝑡−2^𝑛_𝑖=1𝑢_4𝑖𝑔_𝑖𝑥_𝑖(𝑡)−𝑔_𝑖(0)−𝑚⁺_𝑖𝑥_𝑖(𝑡×𝑔_𝑖)𝑥_𝑖(𝑡)−𝑔_𝑖(0)−𝑚⁻_𝑖𝑥_𝑖(𝑡)≤𝑒^𝜆𝑡−2^𝑛_𝑖=1𝑢_4𝑖𝑔_𝑖𝑥_𝑖(𝑡)−𝑚⁺_𝑖𝑥_𝑖(𝑡×𝑔_𝑖)𝑥_𝑖(𝑡)−𝑚⁻_𝑖𝑥_𝑖(𝑡+_𝑛𝑖=1)𝜆𝑔²_𝑖𝑥_𝑖(𝑡)+4𝜆⁻¹𝑔²_𝑖(0)𝑢²_4𝑖+𝜆𝑥²_𝑖(𝑡)+𝜆⁻¹𝑔²_𝑖(0)𝑢²_4𝑖𝑚⁺_𝑖+𝑚⁻_𝑖².(21)
Therefore, from (15)–(21), it follows that 𝑑𝑉(𝑥(𝑡),𝑡)≤𝑒^𝜆𝑡2𝑥^𝑇(𝑡)𝑃𝜎₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))𝑑𝑤(𝑡)+𝑒^𝜆𝑡𝜂^𝑇(𝑡)Γ𝜂(𝑡)𝑑𝑡+𝑒^𝜆𝑡𝐶₁𝑑𝑡≤𝑒^𝜆𝑡2𝑥^𝑇(𝑡)𝑃𝜎₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))𝑑𝑤(𝑡)+𝑒^𝜆𝑡𝐶₁𝑑𝑡,(22) where 𝜂(𝑡)=𝑥^𝑇(𝑡),𝑥^𝑇(𝑡−𝜏(𝑡)),𝑓^𝑇(𝑥(𝑡)),𝑓^𝑇(𝑥((𝑡,𝑔𝑡−𝜏)))_𝑇(𝑥(𝑡)),^𝑡_−∞𝐾(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠^𝑇𝑇,𝐶(23)₁=𝜆⁻¹𝐽^𝑇+𝑃𝐽_𝑛𝑖=14𝜆⁻¹𝑓²_𝑖(0)𝑢²_1𝑖+𝜆⁻¹𝑓²_𝑖(0)𝑢²_1𝑖𝑙⁺_𝑖+𝑙⁻_𝑖²+4𝜆⁻¹𝑓²_𝑖(0)𝑢²_2𝑖+𝜆⁻¹𝑓²_𝑖(0)𝑢²_2𝑖𝑙⁺_𝑖+𝑙⁻_𝑖²+4𝜆⁻¹𝑔²_𝑖(0)𝑢²_4𝑖+𝜆⁻¹𝑔²_𝑖(0)𝑢²_4𝑖𝑚⁺_𝑖+𝑚⁻_𝑖².(24)
Thus, one may obtain that 𝑉(𝑥(𝑡),𝑡)≤𝑉(𝑥(0),0)+𝑒^𝜆𝑡𝜆⁻¹𝐶₁+^𝑡₀2𝑥^𝑇(𝑠)𝑃𝜎₁𝑥(𝑠)+𝜎₂𝑥((𝑠(𝑠),𝐸‖‖𝑠−𝜏))𝑑𝑤(25)𝑥(𝑡)²≤𝑒^−𝜆𝑡𝐸𝑉(𝑥(0),0)+𝜆⁻¹𝐶₁𝜆_min(𝑃)=𝑒^−𝜆𝑡𝐸𝑉(𝑥(0),0)𝜆_min(𝑃)+𝐶_∗,(26) where 𝐶_∗=𝜆⁻¹𝐶₁/𝜆_min(𝑃). Equation (26) implies that (8) holds. The proof is completed.

Theorem 3 shows that there exists 𝑡₀>0 such that for any 𝑡≥𝑡₀, 𝑃{‖𝑥(𝑡)‖≤𝐶}≥1−𝛿. Let 𝐵_𝐶 be denoted by 𝐵_𝐶=𝑥∈𝑅^𝑛∣‖𝑥(𝑡)‖≤𝐶,𝑡≥𝑡₀.(27)

Clearly, 𝐵_𝐶 is closed, bounded, and invariant. Moreover, limsup_𝑡→∞inf_{𝑦∈𝐵𝐶}𝑥(𝑡)−𝑦=0(28) with no less than probability 1−𝛿, which means that 𝐵_𝐶 attracts the solutions infinitely many times with no less than probability 1−𝛿; so we may say that 𝐵_𝐶 is a weak attractor for the solutions.

Theorem 4. Suppose that all conditions of Theorem 3 hold, then there exists a weak attractor 𝐵_𝐶 for the solutions of system (1).

Theorem 5. Suppose that all conditions of Theorem 3 hold and 𝑓(0)=𝑔(0)=𝐽=0; then, zero solution of system (1) is mean square exponential stability and almost sure exponential stability.

Proof. If 𝑓(0)=𝑔(0)=𝐽=0, then 𝐶₁=𝐶_∗=0. By (25) and the semimartingale convergence theorem used in [35], zero solution of system (1) is almost sure exponential stability. It follows from (26) that zero solution of system (1) is mean square exponential stability.

Remark 6. If one takes 𝑄₂=𝑄₄=0 in Theorem 3, then it is not required that 𝜇≤1. Furthermore, If one takes 𝑄₁=𝑄₂=𝑄₃=𝑄₄=0, then 𝜏(𝑡) may be nondifferentiable or the boundedness of 𝜏̇(𝑡) is unknown.

Remark 7. Assumption (A1) is less conservative than that in [32] since the constants 𝑙⁻_𝑖,𝑙⁺_𝑖,𝑚⁻_𝑖, and 𝑚⁺_𝑖 are allowed to be positive, negative, or zero. System (1) includes mixed time-delays, which is more complex than that in [33]. The systems concerned in [26–31] are deterministic, so the stochastic system studied in this paper is more complex and realistic.
When 𝜎₁=𝜎₂=0, system (1) becomes the following determined system: 𝑑𝑥(𝑡)(𝑡)(𝑥(𝑡(𝑥((𝑡𝑑𝑡=−𝐶𝑥+𝐴𝑓))+𝐵𝑓𝑡−𝜏)))+𝐷^𝑡_−∞𝐾(𝑡−𝑠)𝑔(𝑥(𝑠))𝑑𝑠+𝐽.(29)

Definition 8. System (29) is said to be uniformly bounded, if, for each 𝐻>0, there exists a constant 𝑀=𝑀(𝐻)>0 such that [𝑡₀∈𝑅, 𝜙∈𝐶(−∞,0], ‖𝜙‖≤𝐻, 𝑡>𝑡₀] imply ‖𝑥(𝑡,𝑡₀,𝜙)‖≤𝑀, where ‖𝜙‖=sup_𝑡≤0‖𝜙(𝑡)‖.

Theorem 9. System (29) is uniformly bounded provided that 𝜏(𝑡) satisfies 0≤𝜏(𝑡)≤𝜏, 𝜏̇(𝑡)≤𝜇≤1, and there exist some matrices 𝑃>0, 𝑄_𝑖≥0, and 𝑈_𝑖=diag(𝑢_𝑖1,…,𝑢_𝑖𝑛)≥0(𝑖=1,2,3,4) such that the following linear matrix inequality holds: ΣΣ=¹¹0𝑃𝐴+𝐿₂𝑈₁𝑃𝐵𝑀₂𝑈₄𝑃𝐷∗Σ₂₂0𝐿₂𝑈₂00∗∗Σ₃₃000∗∗∗Σ₄₄00∗∗∗∗Σ₅₅0∗∗∗∗∗−𝑈₃<0,(30) where ∗ denotes the corresponding symmetric terms, Σ_𝑖𝑖 (𝑖=1,2,3,4,5),𝐿₁,𝐿₂,𝑀₁,𝑀₂ are the same as in Theorem 3.

Proof. From Σ<0, there exists a sufficiently small 𝜆∈(0,𝜈) such that ΓΓ=¹¹0𝑃𝐴+𝐿₂𝑈₁𝑃𝐵𝑀₂𝑈₄𝑃𝐷∗Γ₂₂0𝐿₂𝑈₂00∗∗Γ₃₃000∗∗∗Γ₄₄00∗∗∗∗Γ₅₅0∗∗∗∗∗−𝑈₃<0,(31) where 𝐼 is identity matrix, Γ₁₁=𝑒^𝜆𝜏𝑄₁+𝜏𝑄₂−𝑃𝐶−𝐶𝑃−2𝐿₁𝑈₁−2𝑀₁𝑈₄+2𝜆𝑃+2𝜆𝐼, Γ₂₂=Σ₂₂+𝜆𝐼, Γ₃₃=𝜆𝐼+𝑒^𝜆𝜏𝑄₃+𝜏𝑄₄−2𝑈₁, Γ₄₄=Σ₄₄+𝜆𝐼, and Γ₅₅=Σ₅₅+𝜆𝐼.
We still consider the Lyapunov-Krasovskii functional 𝑉(𝑥(𝑡)) in (13). From (16)–(21), one may obtain ‖𝑥(𝑡)‖²≤𝜆⁻¹_min(𝑃)𝑒^−𝜆𝑡𝑉(𝑥(0),0)+𝜆⁻¹𝐶₁≤𝜆⁻¹_min(𝑃)𝑉(𝑥(0),0)+𝜆⁻¹𝐶₁≤𝜆⁻¹_min(𝑃)𝜆_max(𝑃)𝜉²+^𝑛_𝑗=1𝑢_3𝑗^∞₀𝑘_𝑗(𝜃)×⁰_−𝜃𝑒^{𝜆(𝑠+𝜃)}𝑔²_𝑗𝑥_𝑗(𝑠)𝑑𝑠𝑑𝜃+⁰_−𝜏𝑒^{𝜆(𝑠+𝜏)}×𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+𝑓^𝑇(𝑥(𝑠))𝑄₃𝑓(𝑥(𝑠))𝑑𝑠+⁰_−𝜏⁰_𝑠𝑒^𝜆𝜃×𝑥^𝑇(𝜃)𝑄₂𝑥(𝜃)+𝑓^𝑇(𝑥(𝜃))𝑄₄𝑓(𝑥(𝜃))𝑑𝜃𝑑𝑠+𝜆⁻¹𝐶₁,(32) where ‖𝜉‖²=sup_𝑡≤0‖𝑥(𝑡)‖², 𝐶₁ is the same as in (24). Note that ^𝑛_𝑗=1𝑢_3𝑗^∞₀𝑘_𝑗(𝜃)⁰_−𝜃𝑒^{𝜆(𝑠+𝜃)}𝑔²_𝑗𝑥_𝑗(𝑠≤)𝑑𝑠𝑑𝜃_𝑛𝑗=12𝑢_3𝑗^∞₀𝑘_𝑗(𝜃)⁰_−𝜃𝑒^{𝜆(𝑠+𝜃)}max_{1≤𝑗≤𝑛}𝑚⁻_𝑗²,𝑚⁺_𝑗²𝑥²_𝑗(𝑠)+𝑔²_𝑗(0≤)𝑑𝑠𝑑𝜃_𝑛𝑗=12𝑢_3𝑗^∞₀𝑘_𝑗(𝜃)⁰_−𝜃𝑒^{𝜆(𝑠+𝜃)}max_{1≤𝑗≤𝑛}𝑚⁻_𝑗²,𝑚⁺_𝑗²×‖𝑥(𝑠)‖²+𝑔(0)²≤𝑑𝑠𝑑𝜃_𝑛𝑗=12𝑢_3𝑗𝜆⁻¹𝑘_𝑗(𝜈)×max_{1≤𝑗≤𝑛}𝑚⁻_𝑗²,𝑚⁺_𝑗²𝜉²+𝑔(0)²,_0−𝜏𝑒^{𝜆(𝑠+𝜏)}𝑥^𝑇(𝑠)𝑄₁𝑥(𝑠)+𝑓^𝑇(𝑥(𝑠))𝑄₃𝑓(𝑥(𝑠≤))𝑑𝑠_0−𝜏𝑒^{𝜆(𝑠+𝜏)}𝜆_max𝑄₁‖𝑥(𝑠)‖²+𝜆_max𝑄₃𝑓(𝑥(𝑠))²𝑑𝑠≤⁰_−𝜏𝑒^{𝜆(𝑠+𝜏)}𝜆_max𝑄₁𝜉²+2𝜆_max𝑄₃×𝑓(0)²+max_{1≤𝑖≤𝑛}𝑙⁻_𝑖²,𝑙⁺_𝑖²𝜉²𝑑𝑠≤𝜆⁻¹𝜆_max𝑄₁𝜉²+2𝜆_max𝑄₃×𝑓(0)²+max_{1≤𝑖≤𝑛}𝑙⁻_𝑖²,𝑙⁺_𝑖²𝜉²𝑑𝑠,_{0−𝜏0𝑠}𝑒^𝜆𝜃𝑥^𝑇(𝜃)𝑄₂𝑥(𝜃)+𝑓^𝑇(𝑥(𝜃))𝑄₄𝑓(𝑥(𝜃≤))𝑑𝜃𝑑𝑠_{0−𝜏0𝑠}𝑒^𝜆𝜃𝜆_max𝑄₂𝜉²+2𝜆_max𝑄₄×𝑓(0)²+max_{1≤𝑖≤𝑛}𝑙⁻_𝑖²,𝑙⁺_𝑖²𝜉²𝑑𝜃𝑑𝑠≤𝜏𝜆⁻¹𝜆_max𝑄₂𝜉²+2𝜆_max𝑄₄×𝑓(0)²+max_{1≤𝑖≤𝑛}𝑙⁻_𝑖²,𝑙⁺_𝑖²𝜉².(33) So, system (29) is uniformly bounded.

4. One Example

Example 1. Consider system (1) with 𝐽=(0,1)^𝑇, 𝐾(𝑡)=diag(𝑒^−𝑡,2𝑒^−2𝑡),𝜈=0.5, and ,,𝐴=−0.10.40.2−0.5,𝐵=0.1−1−1.40.4𝐶=1.4001.65𝐷=−0.60.711.15,𝜎₁=,𝜎0.230.10.30.2₂=.0.1−0.20.20.3(34)

The activation functions 𝑓_𝑖(𝑥_𝑖)=𝑥_𝑖+sin(𝑥_𝑖), 𝑔_𝑖(𝑥_𝑖)=tanh(𝑥_𝑖)(𝑖=1,2) satisfy, 𝑙⁻_𝑖=𝑚⁻_𝑖=0, 𝑙⁺_𝑖=𝑚⁺_𝑖=1. Then, one computes 𝐿₁=𝑀₁=0, 𝐿₂=𝑀₂=diag(1,1), and 𝐾(0.5)=diag(2,4/3). By using MATLAB’s LMI Control Toolbox [34], for 𝜇=0.0035 and 𝜏=1, based on Theorem 3, such system is stochastically ultimately bounded when 𝑃, 𝑈_𝑖, and 𝑄_𝑖 (𝑖=1,2,3,4) satisfy 𝑃=1.77480.23420.23421.9398,𝑈₁=,𝑈1.3091002.1602₂=218.321500274.1607,𝑈₃=,𝑈1.0825001.4021₄=2.1620001.8493,𝑄₁=,𝑄275.3435104.0781104.0781397.1452₂=31.312931.437731.437774.4756,𝑄₃=,𝑄1.0857−1.2164−1.21642.9090₄=.189.366116.395116.395192.8987(35)

From Figure 1, it is easy to see that 𝑥(𝑡) is stochastically ultimately bounded.

(a)

(b)

Figure 1 

Time trajectories (a) as well as the phase portrait (b) for the system in Example 1.

5. Conclusions

A proper Lyapunov functional and linear matrix inequalities are employed to investigate the ultimate boundedness, stability, and weak attractor of stochastic Hopfield neural networks with both time-varying and continuously distributed delays. New results and sufficient criteria are derived after extensive deductions. From the proposed sufficient conditions, one can easily prove that zero solution of such network is mean square exponentially stable and almost surely exponentially stable by applying the semimartingale convergence theorem.

Acknowledgments

The authors thank the editor and the reviewers for their insightful comments and valuable suggestions. This work was supported by the National Natural Science Foundation of China (nos. 10801109, 11271295, and 11047114), Science and Technology Research Projects of Hubei Provincial Department of Education (D20131602), and Young Talent Cultivation Projects of Guangdong (LYM09134).