Dynamical Behaviors of the Stochastic Hopfield Neural Networks with Mixed Time Delays

Abstract

This paper investigates dynamical behaviors of the stochastic Hopfield neural networks with mixed time delays. The mixed time delays under consideration comprise both the discrete time-varying delays and the distributed time-delays. By employing the theory of stochastic functional differential equations and linear matrix inequality (LMI) approach, some novel criteria on asymptotic stability, ultimate boundedness, and weak attractor are derived. Finally, a numerical example is given to illustrate the correctness and effectiveness of our theoretical results.

1. Introduction

The well-known Hopfield neural networks were firstly introduced by Hopfield [1, 2] in early 1980s. Since then, both the mathematical analysis and practical applications of Hopfield neural networks have gained considerable research attention. The Hopfield neural networks have already been successfully applied in many different areas such as combinatorial optimization, knowledge acquisition, and pattern recognition, see, for example, [3–5]. In both the biological and artificial neural networks, the interactions between neurons are generally asynchronous, which give rise to the inevitable signal transmission delays. Also, in electronic implementation of analog neural networks, time delay is usually time-varying due to the finite switching speed of amplifiers. Note that continuously distributed delays have gained particular attention, 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.

Recently, 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 and environmental fluctuations as well as different orders of ongoing events in the overall systems [6, 7]. Therefore, considerable attentions have been paid to investigate the dynamics of stochastic neural networks, and many results on stochastic neural networks with delays have been reported in the literature, see, for example, [8–30] and references therein. Among which, some sufficient criteria on the stability of uncertain stochastic neural networks were derived in [8–10]. Almost sure exponential stability of stochastic neural networks was discussed in [11–15]. In [16–22], mean square exponential stability and pth moment exponential stability of stochastic neural networks were investigated; Some sufficient criteria on the exponential stability for impulsive stochastic neural networks were established in [23–26]. In [27], the stability of discrete-time stochastic neural networks was analyzed, while exponential stability of stochastic neural networks with Markovian jump parameters is investigated in [28–30]. These references mainly considered the stability of equilibrium point of stochastic 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. They play an important role in investigating the uniqueness of equilibrium, global asymptotic stability, global exponential stability, the existence of periodic solution, its control and synchronization [31, 32], and so on. Recently, ultimate boundedness and attractor of several classes of neural networks with time delays have been reported. Some sufficient criteria were derived in [33, 34], but these results hold only under constant delays. Following, in [35], the globally robust ultimate boundedness of integrodifferential neural networks with uncertainties and varying delays was studied. After that, some sufficient criteria on the ultimate boundedness of neural networks with both varying and unbounded delays were derived in [36], but the concerned systems are deterministic ones. In [37, 38], a series of criteria on the boundedness, global exponential stability, and the existence of periodic solution for nonautonomous recurrent neural networks were established. In [39–41], the ultimate boundedness and attractor of the stochastic Hopfield 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 yet meaningful.

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

2. Preliminaries

Consider the following stochastic Hopfield neural networks with mixed time delays: 𝑑𝑥(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑔(𝑥(𝑠+𝜎))𝑑𝑠+𝐽𝑑𝑡₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))𝑑𝑤(𝑡),(1) where 𝑥=(𝑥₁,…,𝑥_𝑛)^𝑇 is the 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; 𝐽=(𝐽₁,…,𝐽_𝑛)^𝑇, 𝐽_𝑖 denotes the external bias on the 𝑖th unit; 𝑓_𝑗 and 𝑔_𝑗 denote activation functions, 𝑓(𝑥(𝑡))=(𝑓₁(𝑥₁(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡)))^𝑇, 𝑔(𝑥(𝑡))=(𝑔₁(𝑥₁(𝑡)),…,𝑔_𝑛(𝑥_𝑛(𝑡)))^𝑇; 𝜎₁,𝜎₂∈𝑅^𝑛×𝑛 are the diffusion coefficient matrices; 𝑤(𝑡) is one-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}; there exists a positive constant 𝜏 such that the transmission delay 𝜏(𝑡) satisfies 0≤𝜏(𝑡)≤𝜏.(2)

The initial conditions are given in the following form: 𝑥(𝑠)=𝜉(𝑠),−𝜏≤𝑠≤0,𝑗=1,…,𝑛,(3) where 𝜉(𝑠)=(𝜉₁(𝑠),…,𝜉_𝑛(𝑠))^𝑇 is 𝐶([−𝜏,0];𝑅^𝑛)-valued function, ℱ₀-measurable 𝑅^𝑛-valued random variable satisfying ‖𝜉‖²_𝜏=sup_{−𝜏≤𝑠≤0}𝐸‖𝜉(𝑠)‖²<∞,‖⋅‖ is the Euclidean norm, and 𝐶([−𝜏,0];𝑅^𝑛) is the space of all continuous 𝑅^𝑛-valued functions defined on [−𝜏,0].

Let 𝐹(𝑥_𝑡,𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐷∫^𝑡_{𝑡−𝜏(𝑡)}𝑔(𝑥(𝑠))𝑑𝑠+𝐽, 𝐺(𝑥_𝑡,𝑡)=𝜎₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡)), where 𝑥_𝑡={𝑥(𝑡+𝜃)∶−𝜏≤𝜃≤0,𝑡≥0}=𝜑(𝜃).(4) Then system (1) can be written by 𝑑𝑥(𝑡)=𝐹𝑥_𝑡,𝑡𝑑𝑡+𝐺𝑥_𝑡,𝑡𝑑𝑤(𝑡).(5) Throughout this paper, the following assumption will be considered.

(A1) There exist 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.(6)

Remark 1. It follows from [42] that under the assumption (A1), system (1) has a global solution on 𝑡≥0. Moreover, under assumption 1, it is not difficult to prove that 𝐹(𝑥_𝑡,𝑡) and 𝐺(𝑥_𝑡,𝑡) satisfy the local Lipschitz condition in [43].

Remark 2. We note that assumption (A1) is less conservative than that in [8, 9, 39], since the constants 𝑙⁻_𝑖, 𝑙⁺_𝑖, 𝑚⁻_𝑖 and 𝑚⁺_𝑖 are allowed to be positive, negative numbers, or zeros.

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. Denote by 𝐶(𝑅^𝑛×[−𝜏,∞);𝑅⁺) the family of continuous functions from 𝑅^𝑛×[−𝜏,∞) to 𝑅⁺=[0,∞). Let 𝐶^2,1(𝑅^𝑛×[−𝜏,∞);𝑅⁺) be the family of all continuous nonnegative functions 𝑉(𝑥,𝑡) defined on 𝑅^𝑛×[−𝜏,∞) such that they are continuously twice differentiable in 𝑥 and once in 𝑡. Given 𝑉∈𝐶^2,1(𝑅^𝑛×[−𝜏,∞);𝑅⁺), we define the functional 𝕃𝑉∶𝐶([−𝜏,0];𝑅^𝑛)×𝑅⁺→𝑅 by 𝕃𝑉𝜑,𝑡=𝑉_𝑡𝜑(0),𝑡+𝑉_𝑥𝜑(0)𝐹+1,𝑡𝜑,𝑡2trace𝐺^𝑇𝜑,𝑡𝑉_𝑥𝑥𝜑(0),𝑡𝐺𝜑,𝑡,(7) where 𝑉_𝑥(𝑥,𝑡)=(𝑉_𝑥1(𝑥,𝑡),…,𝑉_𝑥𝑛(𝑥,𝑡)) and 𝑉_𝑥𝑥(𝑥,𝑡)=(𝑉_{𝑥𝑖𝑥𝑗}(𝑥,𝑡))_𝑛×𝑛.

The following lemmas will be used in establishing our main results.

Lemma 3 (see [44]). For any positive definite matrix 𝑃>0, scalar 𝛾>0, vector function 𝑓∶[0,𝛾]→𝑅^𝑛 such that the integrations concerned are well defined, and the following inequality holds: ^𝛾₀𝑓(𝑠)𝑑𝑠^𝑇𝑃^𝛾₀𝑓(𝑠)𝑑𝑠≤𝛾^𝛾₀𝑓^𝑇(𝑠)𝑃𝑓(𝑠)𝑑𝑠.(8)

Lemma 4 (see [43]). Suppose that system (5) satisfies the local Lipschitz condition and the following assumptions hold. (A2) There are two functions 𝑉∈𝐶^2,1(𝑅^𝑛×[−𝜏,∞);𝑅⁺) and 𝑈∈𝐶(𝑅^𝑛×[−𝜏,∞);𝑅⁺) and two probability measures 𝜇(⋅) and 𝜇(⋅) on [−𝜏,0] such that lim_{‖𝑥‖→∞}inf_{0≤𝑡≤∞}𝑉(𝑥,𝑡)=∞,(9) while for all (𝜑,𝑡)∈𝐶([−𝜏,0];𝑅^𝑛)×𝑅⁺, 𝕃𝑉𝜑,𝑡≤𝛼₁−𝛼₂𝑉𝜑(0),𝑡+𝛼₃⁰_−𝜏𝑉𝜑(𝜃),𝑡+𝜃𝑑𝜇(𝜃)−𝑈𝜑(0),𝑡+𝛼⁰_−𝜏𝑈𝜑(𝜃),𝑡+𝜃𝑑𝜇(𝜃),(10) where 𝛼₁≥0, 𝛼₂>𝛼₃≥0 and 𝛼∈(0,1). (A3) If there is a pair of positive constants 𝑐 and 𝑝 such that 𝑐‖𝑥‖^𝑝≤𝑉(𝑥,𝑡),∀(𝑥,𝑡)∈𝑅^𝑛×[−𝜏,∞).(11) Then the unique global solution 𝑥(𝑡) to system (5) obeys limsup_𝑡→∞𝐸‖𝑥(𝑡)‖^𝑝≤𝛼₁,𝑐𝜀(12) where 𝜀=min{𝜀₁,𝜀₂} while 𝜀₂=−ln(𝛼)/𝜏 and 𝜀₁>0 is the unique root to the following equation: 𝛼₂=𝜀₁+𝛼₃𝑒^𝜀1𝜏.(13)
If, furthermore, 𝛼₁=0, then limsup_𝑡→∞1𝑡ln𝐸‖𝑥(𝑡)‖^𝑝≤−𝜀,limsup_𝑡→∞1𝑡𝑥(𝑡𝜀ln(‖)‖)≤−𝑝𝑎𝑙𝑚𝑜𝑠𝑡𝑠𝑢𝑟𝑒𝑙𝑦.(14)

3. Main Results

Theorem 5. Suppose that there exist some matrices 𝑃>0, 𝑈_𝑖=diag{𝑢_𝑖1,…,𝑢_𝑖𝑛}≥0(𝑖=1,2,3) and positive constants 𝛾₁, 𝛾₂, 𝜆 such that 𝜆⁻¹𝜏𝛾₂⁻¹∈(0,1) and ΔΣ=¹0𝑃𝐴+𝐿₂𝑈₁𝑃𝐵𝑈₃𝑀₂∗Δ₂0𝐿₂𝑈₂0∗∗Δ₃00∗∗∗Δ₄0∗∗∗∗Δ₅<0,(15) where Δ₁=(𝛾₁+2𝜆)𝑃+2𝜎^𝑇₁𝑃𝜎₁−𝑃𝐶−𝐶𝑃+𝑈₁(𝜆𝐼−2𝐿₁)+𝑈₃(𝜆𝐼−2𝑀₁), Δ₂=2𝜎^𝑇₂𝑃𝜎₂+(𝜆𝐼−2𝐿₁)𝑈₂, Δ₃=2(𝜆−1)𝑈₁, Δ₄=2(𝜆−1)𝑈₂, Δ₅=2(𝜆−1)𝑈₃+𝛾₂𝐷^𝑇𝑃𝐷, 𝐿₁=diag{𝑙⁻₁𝑙⁺₁,…,𝑙⁻_𝑛𝑙⁺_𝑛}, 𝐿₂=diag{𝑙⁻₁+𝑙⁺₁,…,𝑙⁻_𝑛+𝑙⁺_𝑛}, 𝑀₁=diag{𝑚⁻₁𝑚⁺₁,…,𝑚⁻_𝑛𝑚⁺_𝑛}, 𝑀₂=diag{𝑚⁻₁+𝑚⁺₁,…,𝑚⁻_𝑛+𝑚⁺_𝑛}, ∗ means the symmetric terms.
Then, the following results hold.(i) System (1) is stochastically ultimately bounded; that is, for any 𝛿∈(0,1), there exists a positive constant 𝐶=𝐶(𝛿) such that the solution 𝑥(𝑡) of system (1) satisfies limsup_𝑡→∞𝑃{‖𝑥(𝑡)‖≤𝐶}≥1−𝛿.(16)(ii)If 𝛼₁=0, where 𝛼₁=max(𝛾₃,0),𝜀>0 is the same as defined in Lemma 4, 𝛾₃=𝜆⁻¹𝐽^𝑇+𝑃𝐽_𝑛𝑖=1𝑢_1𝑖+𝑢_2𝑖×−2𝑓²_𝑖(0)+2𝜆⁻¹𝑓²_𝑖(0)+𝜆⁻¹𝑙⁺_𝑖+𝑙⁻_𝑖²𝑓²_𝑖(0+_𝑛𝑖=1)𝑢_3𝑖−2𝑔²_𝑖(0)+2𝜆⁻¹𝑔²_𝑖(0)+𝜆⁻¹𝑚⁺_𝑖+𝑚⁻_𝑖²𝑔²_𝑖(0),(17)
then limsup_𝑡→∞1𝑡ln𝐸‖𝑥(𝑡)‖²≤−𝜀,limsup_𝑡→∞1𝑡𝑥(𝑡𝜀ln(‖)‖)≤−2𝑎𝑙𝑚𝑜𝑠𝑡𝑠𝑢𝑟𝑒𝑙𝑦.(18)

Proof. Let the Lyapunov function 𝑉(𝑥,𝑡)=𝑥^𝑇(𝑡)𝑃𝑥(𝑡). Applying Itô’s formula in [42] to 𝑉(𝑡) along with system (1), one may obtain the following: 𝑑𝑉(𝑥,𝑡)=2𝑥^𝑇(𝑡)𝑃𝜎₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))𝑑𝑤(𝑡)+𝕃𝑉𝜑,𝑡𝑑𝑡,(19) where 𝕃𝑉𝜑,𝑡=2𝑥^𝑇(𝑡)𝑃−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑔(𝑥(𝑠+𝜎))𝑑𝑠+𝐽₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))^𝑇×𝑃𝜎₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))≤2𝑥^𝑇(𝑡)𝑃(𝑡)(𝑥(𝑡(𝑥((𝑡−𝐶𝑥+𝐴𝑓))+𝐵𝑓𝑡−𝜏)))+2𝜆𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝜆⁻¹𝐽^𝑇𝑃𝐽+𝜆^−1𝑡_{(𝑡)𝑡−𝜏}𝐷𝑔(𝑥(𝑠))𝑑𝑠^𝑇×𝑃^𝑡_{(𝑡)𝑡−𝜏}(𝑥(𝑠𝑥𝐷𝑔))𝑑𝑠+2^𝑇(𝑡)𝜎^𝑇₁𝑃𝜎₁𝑥(𝑡)+𝑥^𝑇(𝑡−𝜏(𝑡))𝜎^𝑇₂𝑃𝜎₂𝑥(𝑡−𝜏(𝑡)).(20) From Lemma 3, it follows that 𝜆^−1𝑡_{(𝑡)𝑡−𝜏}𝐷𝑔(𝑥(𝑠))𝑑𝑠^𝑇𝑃^𝑡_{(𝑡)𝑡−𝜏}𝐷𝑔(𝑥(𝑠))𝑑𝑠≤𝜆^−1𝑡_𝑡−𝜏𝐷𝑔(𝑥(𝑠))𝑑𝑠^𝑇𝑃^𝑡_𝑡−𝜏𝐷𝑔(𝑥(𝑠))𝑑𝑠≤𝜆⁻¹𝜏^𝑡_𝑡−𝜏𝑔^𝑇(𝑥(𝑠))𝐷^𝑇𝑃𝐷𝑔(𝑥(𝑠))𝑑𝑠=𝜆⁻¹𝜏⁰_−𝜏𝑔^𝑇(𝑥(𝑡+𝜃))𝐷^𝑇𝑃𝐷𝑔(𝑥(𝑡+𝜃))𝑑𝜃=𝜆⁻¹𝜏⁰_−𝜏𝑔^𝑇𝜑(𝜃)𝐷^𝑇𝑃𝐷𝑔𝜑(𝜃)𝑑𝜃.(21) From (A1), it follows that for 𝑖=1,…,𝑛, 0≤−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_1𝑖𝑓²_𝑖𝑥_𝑖(𝑡)−𝑙⁺_𝑖+𝑙⁻_𝑖𝑥_𝑖(𝑡)𝑓_𝑖𝑥_𝑖(𝑡)+𝑙⁺_𝑖𝑙⁻_𝑖𝑥²_𝑖(𝑡)+𝑓²_𝑖(0)−2𝑓_𝑖(0)𝑓_𝑖𝑥_𝑖(𝑡)+𝑙⁺_𝑖+𝑙⁻_𝑖𝑥_𝑖(𝑡)𝑓_𝑖(0)=−2^𝑛_𝑖=1𝑢_1𝑖𝑓²_𝑖𝑥_𝑖(𝑡)−𝑙⁺_𝑖+𝑙⁻_𝑖𝑥_𝑖(𝑡)𝑓_𝑖𝑥_𝑖(𝑡)+𝑙⁺_𝑖𝑙⁻_𝑖𝑥²_𝑖(𝑡+_𝑛𝑖=1)𝑢_1𝑖−2𝑓²_𝑖(0)+4𝑓_𝑖(0)𝑓_𝑖𝑥_𝑖(𝑡)−2𝑙⁺_𝑖+𝑙⁻_𝑖𝑥_𝑖(𝑡)𝑓_𝑖(0)≤−2^𝑛_𝑖=1𝑢_1𝑖𝑓²_𝑖𝑥_𝑖(𝑡)−𝑙⁺_𝑖+𝑙⁻_𝑖𝑥_𝑖(𝑡)𝑓_𝑖𝑥_𝑖(𝑡)+𝑙⁺_𝑖𝑙⁻_𝑖𝑥²_𝑖(𝑡+_𝑛𝑖=1)𝑢_1𝑖−2𝑓²_𝑖(0)+2𝜆𝑓²_𝑖𝑥_𝑖(𝑡)+𝜆⁻¹𝑓²_𝑖(0+)𝜆𝑥²_𝑖(𝑡)+𝜆⁻¹𝑙⁺_𝑖+𝑙⁻_𝑖²𝑓²_𝑖(0()𝑓)=2𝜆−1^𝑇(𝑥(𝑡))𝑈₁𝑓(𝑥(𝑡))+2𝑓^𝑇(𝑥(𝑡))𝑈₁𝐿₂𝑥(𝑡)+𝑥^𝑇(𝑡)𝑈₁𝜆𝐼−2𝐿₁𝑥(𝑡)+^𝑛_𝑖=1𝑢_1𝑖−2𝑓²_𝑖(0)+2𝜆⁻¹𝑓²_𝑖(0)+𝜆⁻¹𝑙⁺_𝑖+𝑙⁻_𝑖²𝑓²_𝑖(0).(22) Similarly, one derives that 0≤−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()𝑓𝑡−𝜏))≤2𝜆−1^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑈₂𝑓(𝑥(𝑡−𝜏(𝑡)))+2𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑈₂𝐿₂𝑥(𝑡−𝜏(𝑡))+𝑥^𝑇(𝑡−𝜏(𝑡))𝑈₂𝜆𝐼−2𝐿₁𝑥((𝑡+𝑡−𝜏))_𝑛𝑖=1𝑢_2𝑖−2𝑓²_𝑖(0)+2𝜆⁻¹𝑓²_𝑖(0)+𝜆⁻¹𝑙⁺_𝑖+𝑙⁻_𝑖²𝑓²_𝑖(0),0≤−2^𝑛_𝑖=1𝑢_3𝑖𝑔_𝑖𝑥_𝑖(𝑡)−𝑔_𝑖(0)−𝑚⁺_𝑖𝑥_𝑖(𝑡×𝑔_𝑖)𝑥_𝑖(𝑡)−𝑔_𝑖(0)−𝑚⁻_𝑖𝑥_𝑖(𝑡()𝑔)≤2𝜆−1^𝑇(𝑥(𝑡))𝑈₃𝑔(𝑥(𝑡))+2𝑔^𝑇(𝑥(𝑡))𝑈₃𝑀₂𝑥(𝑡)+𝑥^𝑇(𝑡)𝑈₃𝜆𝐼−2𝑀₁𝑥(𝑡)+^𝑛_𝑖=1𝑢_3𝑖−2𝑔²_𝑖(0)+2𝜆⁻¹𝑔²_𝑖(0)+𝜆⁻¹𝑚⁺_𝑖+𝑚⁻_𝑖²𝑔²_𝑖(0).(23)
Further from (20)–(23), one derives 𝕃𝑉𝜑,𝑡≤2𝑥^𝑇(𝑡)𝑃−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+2𝜆𝑥^𝑇(𝑡)𝑃𝑥(𝑡)+𝜆⁻¹𝜏⁰_−𝜏𝑔^𝑇𝜑(𝜃)𝐷^𝑇𝜑(𝜃)𝑥𝑃𝐷𝑔𝑑𝜃+2^𝑇(𝑡)𝜎^𝑇₁𝑃𝜎₁𝑥(𝑡)+𝑥^𝑇(𝑡−𝜏(𝑡))𝜎^𝑇₂𝑃𝜎₂𝑥((𝑡()𝑓𝑡−𝜏))+2𝜆−1^𝑇(𝑥(𝑡))𝑈₁𝑓(𝑥(𝑡))+2𝑓^𝑇(𝑥(𝑡))𝑈₁𝐿₂𝑥(𝑡)+𝑥^𝑇(𝑡)𝑈₁𝜆𝐼−2𝐿₁𝑥(𝑡)+2(𝜆−1)𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑈₂𝑓(𝑥(𝑡−𝜏(𝑡)))+2𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑈₂𝐿₂𝑥(𝑡−𝜏(𝑡))+𝑥^𝑇(𝑡−𝜏(𝑡))𝑈₂𝜆𝐼−2𝐿₁𝑥((𝑡()𝑔𝑡−𝜏))+2𝜆−1^𝑇(𝑥(𝑡))𝑈₃𝑔(𝑥(𝑡))+2𝑔^𝑇(𝑥(𝑡))𝑈₃𝑀₂𝑥(𝑡)+𝑥^𝑇(𝑡)𝑈₃𝜆𝐼−2𝑀₁𝑥(𝑡)+𝛾₃+𝛾₁𝑥^𝑇(𝑡)𝑃𝑥(𝑡)−𝛾₁𝑉𝜑(0),𝑡+𝛾₂𝑔^𝑇(𝑥(𝑡))𝐷^𝑇𝑃𝐷𝑔(𝑥(𝑡))−𝛾₂𝑔^𝑇𝜑(0)𝐷^𝑇𝑃𝐷𝑔𝜑(0)≤𝜂^𝑇(𝑡)Σ𝜂(𝑡)−𝛾₁𝑉𝜑(0),𝑡−𝛾₂𝑔^𝑇𝜑(0)𝐷^𝑇𝑃𝐷𝑔𝜑(0)+𝛾₃+𝜆⁻¹𝜏⁰_−𝜏𝑔^𝑇𝜑(𝜃)𝐷^𝑇𝑃𝐷𝑔𝜑(𝜃)𝑑𝜃≤𝛾₃−𝛾₁𝑉𝜑(0),𝑡−𝑈𝜑(0),𝑡+𝜆⁻¹𝜏𝛾₂⁻¹⁰_−𝜏𝑈𝜑(𝜃),𝑡+𝜃𝑑𝜃,(24) where 𝜂(𝑡)=(𝑥^𝑇(𝑡), 𝑥^𝑇(𝑡−𝜏(𝑡)), 𝑓^𝑇(𝑥(𝑡)), 𝑓^𝑇(𝑥(𝑡−𝜏(𝑡))), 𝑔^𝑇(𝑥(𝑡)))^𝑇, 𝜑(0)=𝑥(𝑡), 𝑈(𝑥,𝑡)=𝛾₂𝑔^𝑇(𝑥)𝐷^𝑇𝑃𝐷𝑔(𝑥), 𝛾₃=𝜆⁻¹𝐽^𝑇+𝑃𝐽_𝑛𝑖=1𝑢_1𝑖+𝑢_2𝑖×−2𝑓²_𝑖(0)+2𝜆⁻¹𝑓²_𝑖(0)+𝜆⁻¹𝑙⁺_𝑖+𝑙⁻_𝑖²𝑓²_𝑖(0+_𝑛𝑖=1)𝑢_3𝑖−2𝑔²_𝑖(0)+2𝜆⁻¹𝑔²_𝑖(0)+𝜆⁻¹𝑚⁺_𝑖+𝑚⁻_𝑖²𝑔²_𝑖(0).(25) Let 𝛼₁=max(𝛾₃,0), 𝛼₂=𝛾₁, 𝛼₃=0, 𝛼=𝜆⁻¹𝜏𝛾₂⁻¹, 𝑐=𝜆_min(𝑃). Then it follows from Lemma 4 that limsup_𝑡→∞𝐸‖𝑥(𝑡)‖²≤𝛼₁𝜆_min(𝑃)𝜀≤𝛾₄𝜆_min(𝑃)𝜀,(26) where 𝜀>0 is the same as defined in Lemma 4, 𝛾₄=𝜆⁻¹𝐽^𝑇+𝑃𝐽_𝑛𝑖=1𝑢_1𝑖+𝑢_2𝑖2𝜆⁻¹𝑓²_𝑖(0)+𝜆⁻¹𝑙⁺_𝑖+𝑙⁻_𝑖²𝑓²_𝑖(0+_𝑛𝑖=1)𝑢_3𝑖2𝜆⁻¹𝑔²_𝑖(0)+𝜆⁻¹𝑚⁺_𝑖+𝑚⁻_𝑖²𝑔²_𝑖(0).(27) Therefore, for any 𝛿>0, it follows from Chebyshev’s inequality that limsup_𝑡→∞𝑃‖𝑥(𝑡)‖>𝛾₄𝛿𝜆_min(𝑃)𝜀≤limsup_𝑡→∞𝐸‖𝑥(𝑡)‖²𝛾₄/𝛿𝜆_min(𝑃)𝜀≤𝛿.(28) If, furthermore, 𝛼₁=0, then it follows from Lemma 4 that (ii) holds.

Theorem 5 shows that there exists 𝑡₀>0 such that for any 𝑡≥𝑡₀, 𝑃{‖𝑥(𝑡)‖≤𝐶}≥1−𝛿. Let 𝐵_𝐶 denote by 𝐵_𝐶=𝑥∈𝑅^𝑛∣‖𝑥(𝑡)‖≤𝐶,𝑡≥𝑡₀.(29) Clearly, 𝐵_𝐶 is closed, bounded, and invariant. Moreover, limsup_𝑡→∞inf_{𝑦∈𝐵𝐶}𝑥(𝑡)−𝑦=0,(30) 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 6. Suppose that all conditions of Theorem 5 hold, then there exists a weak attractor 𝐵_𝐶  for the solutions of system (1).

Remark 7. Compared with [39–41], assumption (A1) is less conservative than that in [39] and system (1) includes mixed time delays, which is more complex than that in [39–41]. In addition, Lemma 4 is firstly used to investigate the dynamical behaviors of stochastic neural networks with mixed time delays. The bound for 𝕃𝑉 may be in a much weaker form. Our results do not only deal with the asymptotic moment estimation but also the path wise (almost sure) estimation.

4. Numerical Example

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

Example 8. Consider the following stochastic Hopfield neural networks with mixed time delays: 𝑑𝑥(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐷^𝑡_{(𝑡)𝑡−𝜏}𝑔(𝑥(𝑠+𝜎))𝑑𝑠+𝐽𝑑𝑡₁𝑥(𝑡)+𝜎₂𝑥(𝑡−𝜏(𝑡))𝑑𝑤(𝑡),(31) where 𝐽=(0.01,0.01)^𝑇, 𝜏(𝑡)=0.2|sin𝑡|, ,,𝜎𝐴=−0.20.30.2−0.5,𝐵=0.5−1−1.40.8𝐶=1.2001.2,𝐷=0.3−0.10.10.4₁=0.20.10.30.2,𝜎₂=,0.1−0.2−0.20.3(32) and 𝑓(𝑥)=𝑔(𝑥)=tanh(𝑥), 𝑤(𝑡) is one-dimensional Brownian motion. Then 𝐿₁=𝑀₁=0, 𝐿₂=𝑀₂=diag{1,1}. By using the Matlab LMI Control Toolbox [45], for 𝛾₁=𝛾₂=𝜆=0.5, 𝜏=0.2, based on Theorem 5, such system is stochastically ultimately bounded when 𝑃, 𝑈₁, 𝑈₂ and 𝑈₃ are chosen as follows: 𝑃=0.32780.27440.27440.3567,𝑈₁=,𝑈0.0668000.2255₂=0.1263000.1218,𝑈₃=.0.0901000.1031(33)
For 𝛿=0.01, 𝛾₁=𝛾₂=𝜆=0.5, 𝜏=0.2, we obtain 𝜀=0.5 and constant 𝐶=𝐶(𝛿)=𝐽^𝑇𝑃𝐽/𝜆_min(𝑃)𝛿𝜆𝜀=0.1141/0.0675=1.3001. Then 𝐵_𝐶={𝑥∈𝑅²∣‖𝑥(𝑡)‖≤1.3001,𝑡≥𝑡₀}, 𝑃(𝑥∈𝐵_𝐶)≥0.99. For the system in Example 8 (color online), Figure 1(a) shows time trajectories, and Figure 1(b) shows the set 𝐵_𝐶 and several typical phase portraits, where initial value for 𝑡<0 is chosen as 𝑥(𝑡)=(50,80). In Figure 1(b), only phase portraits for 𝑡≥0 are shown. From Figure 1, one can easily find that these trajectories are almost all attracted by the set 𝐵_𝐶.

(a)

(b)

Figure 1 

Time trajectories (a) as well as the set 𝐵𝐶 and several typical phase portraits (b) for the system in Example 8 (color online). Where initial values for 𝑡<0 are chosen as 𝑥(𝑡)=(50,80). For (b), only phase portraits for 𝑡≥0 are shown.

5. Conclusion

In this paper, by using the theory of stochastic functional differential equations and linear matrix inequality, new results and sufficient criteria on the asymptotic stability, ultimate boundedness, and attractor of stochastic Hopfield neural networks with mixed time delays are established. A numerical example is also presented to demonstrate the correctness of our theoretical results.

Acknowledgments

The authors thank the Editor and the Reviewers for their detailed comments and valuable suggestions. This work was supported by the National Natural Science Foundation of China (no. 11271295, 10926128, and 11047114), Science and Technology Research Projects of Hubei Provincial Department of Education (no. Q20111607, Q20111611, D20131602, and D20131604), and Young Talent Cultivation Projects of Guangdong (LYM09134).