Synchronization Control for Stochastic Neural Networks with Mixed Time-Varying Delays

Abstract

Synchronization control of stochastic neural networks with time-varying discrete and continuous delays has been investigated. A novel control scheme is proposed using the Lyapunov functional method and linear matrix inequality (LMI) approach. Sufficient conditions have been derived to ensure the global asymptotical mean-square stability for the error system, and thus the drive system synchronizes with the response system. Also, the control gain matrix can be obtained. With these effective methods, synchronization can be achieved. Simulation results are presented to show the effectiveness of the theoretical results.

1. Introduction

In recent years, stochastic neural networks (NNs) have gained particular research interests. These systems represent a class of stochastic systems that is popular in modeling practical systems which may experience random disturbances and parameters varying. Such a system can be found in biology systems, social systems, and wireless communication networks. Numerous results on stochastic neural network have been reported in the literature [1–5]. As one of the mostly investigated dynamical behaviors, the synchronization in stochastic NNs with or without time delays has drawn significant research interest; see, for example, [6–10] and the references therein.

On the other hand, time delays are frequently encountered in many practical control systems, such as aircraft, chemical, or process control systems. The existence of the time delays may be the source of instability of serious deterioration in the performance of the closed-loop systems. Thus, stability of delayed NNs has been a focal subject for research [11–15]. Most recently, significant and substantial progresses have been achieved in the synchronization of stochastic NNs. These include synchronization of randomly coupled neural networks with Markovian jumping and time-delay [15], adaptive synchronization for stochastic NNs with time-varying delays and distributed delays [16], passivity analysis for discrete time stochastic Markovian jump NNs with mixed time delays [17], adaptive synchronization for stochastic T-S fuzzy NNs with time-delay and Markovian jumping parameters [18], nonfragile synchronization of NNs with time-varying delay and randomly occurring controller gain fluctuation synchronization of biological NN systems with stochastic perturbations and time delays [19], global exponential adaptive synchronization of complex dynamical networks with neutral-type NN nodes and stochastic disturbances [20], adaptive synchronization for uncertain chaotic neural networks with mixed time delays using fuzzy disturbance observer [21], the effects of time delay on the stochastic resonance in feed-forward-loop NN motifs [22], state estimation for wireless network control system with stochastic uncertainty and time delay based on sliding mode observer [23], pinning synchronization in fixed and switching directed networks of Lorenz-type nodes [24], consensus tracking for higher-order multiagent systems with switching directed topologies and occasionally missing control inputs [25], and consensus tracking of multiagent systems with Lipschitz-type node dynamics and switching topologies [26].

Although substantial progresses have been made in the synchronization control of delayed stochastic neural network systems [16–32], there are still some problems which have not been fully studied. For example, many control schemes of time-varying delayed NNs assume that the derivative of the time-delay function is less than one. Also, few works have been done on the synchronization of mixed delayed stochastic NNs. In this paper, we propose sufficient conditions of the synchronization of the stochastic NN system, as well as the state feedback control design. At first, sufficient conditions are proposed in terms of LMIs to guarantee the stochastic asymptotical stability of the error system. The assumption that the derivative of the time-delay function is less than one is eliminated. Then, we give some corollaries to solve the problem in some special cases.

This paper is organized as follows. After the introduction in Section 1, the problem statement and preliminaries are presented in Section 2. Next, some sufficient conditions are presented for the synchronization of the delayed drive and response system. Also, some corollaries and remarks are given to show the advantages of this paper in Section 3. Then, the numerical simulation result is given in Section 4. A conclusion is drawn in Section 5.

2. Problem Statement and Preliminaries

Before proceeding, we introduce some notations which will be used later for derivations and discussions. ‖⋅‖ denotes the Euclidean norm of a vector or the Frobenius norm of a matrix. 𝑀>0 (<0) denotes that matrix 𝑀 is a positive (negative) definite matrix. 𝐸⋅ denotes the mathematical expectation.

The recurrent network under investigation is 𝑑𝑥(𝑡)=−𝐶𝑥(𝑡)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐷^𝑡_{(𝑡)𝑡−𝜎}𝑓(𝑥(𝑠))𝑑𝑠𝑑𝑡, (1) where 𝑥(𝑡)=[𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡)]^𝑇 is the system state associated with the neurons, 𝑛 denotes the number of neurons in the network, 𝑓(𝑥(𝑡))=[𝑓₁(𝑥₁(𝑡)),𝑓₂(𝑥₂(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡))] corresponds to the activation functions of neurons, and 𝜏(𝑡) and 𝜎(𝑡) are the time-varying discrete delay and continuous delay, respectively; the initial conditions are given by 𝑥(𝑡)=𝜙(𝑡)∈([−𝜏,0],𝑅^𝑛), where 𝜙(𝑡)∈([−𝜏,0],𝑅^𝑛) denotes the set of all continuous functions from [−𝜏,0] to 𝑅. 𝐶=diag(𝑐₁,𝑐₂,…,𝑐_𝑛) is a diagonal matrix, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛, 𝐵=(𝑏_𝑖𝑗)_𝑛×𝑛, and 𝐷=(𝑑_𝑖𝑗)_𝑛×𝑛 are the connection weight matrix, discrete time-delay, and continuous time-delay connection weight matrices, respectively.

In this paper, we consider model (1) as the master system. The response system is 𝑑𝑦(𝑡)=−𝐶𝑦(𝑡)+𝐴𝑓𝑦(𝑡)+𝐵𝑓𝑦(𝑡−𝜏(𝑡))+𝐷^𝑡_{𝑡−𝜎(𝑡)}𝑓𝑦(𝑠)(𝑡((𝑡)((𝑡(𝑡),𝑑𝑠+𝑢)𝑑𝑡+𝜎𝑡,𝑒,𝑒𝑡−𝜏)))𝑑𝜔 (2) where 𝐴,𝐵,𝐶, and 𝐷 are matrices which are the same as (1) and 𝑢(𝑡) is the controller. It has the same structure as the drive system. 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡) is the error state. 𝜔(𝑡)=[𝜔₁(𝑡),𝜔₂(𝑡),…,𝜔_𝑛(𝑡)]^𝑇 is a 𝑛 dimension Brownian motion defined on a complete probability space (Ω,𝐹,𝑃).

To propose our main results, it is necessary to make the following assumptions.(A1)Each function 𝑓_𝑖 is nondecreasing and globally Lipschitz with a constant 𝑘_𝑖>0: 𝑓_𝑖(𝑥)−𝑓_𝑖𝑦≤𝑘_𝑖𝑘𝑥−𝑦∀𝑥,𝑦∈𝑅,𝑖=1,2,…,𝑛,𝐾=diag₁,𝑘₂,…,𝑘_𝑛. (3)(A2)Matrix function 𝜎(𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡))) satisfies trace𝜎^𝑇((𝑡)((𝑡𝜎((𝑡)((𝑡≤𝑀𝑡,𝑒,𝑒𝑡−𝜏)))𝑡,𝑒,𝑒𝑡−𝜏)))₁𝑒(𝑡)²+𝑀₂𝑒(𝑡−𝜏(𝑡))², (4)where 𝑀₁ and 𝑀₂ are matrices with appropriate dimensions.(A3)Discrete delay 𝜏(𝑡) and continuous delay 𝜎(𝑡) are both differential functions of time, and the following conditions hold: 0≤𝜏(𝑡)≤𝜏,𝜏̇(𝑡)≤𝜏₁,0≤𝜎(𝑡)≤𝜎, (5)where 𝜏,𝜏₁, and 𝜎 are positive constants. It is worth to emphasis that the constraint, the derivative of time-delay function is less than one [1], is eliminated in this paper, while a more general upper boundary constraint is utilized instead of it. This improvement makes our results applicable for a wide range of time-delayed stochastic neural networks.

Let error state be 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡); subtracting (1) from (2) yields the synchronization error dynamical system as follows: 𝑑𝑒(𝑡)=−𝐶𝑒(𝑡)+𝐴𝑔(𝑡)+𝐵𝑔(𝑡−𝜏(𝑡))+𝐷^𝑡_{𝑡−𝜎(𝑡)}𝑔(𝑠)(𝑡((𝑡)((𝑡𝑑𝑠+𝑢)𝑑𝑡+𝜎𝑡,𝑒,𝑒𝑡−𝜏)))𝑑𝜔, (6) where 𝑔(𝑡)=𝑓(𝑦(𝑡))−𝑓(𝑥(𝑡)).

In this paper, we design a memoryless state feedback controller: 𝑢(𝑡)=𝐺𝑒(𝑡), (7) where 𝐺∈𝑅^𝑛×𝑛 is a constant gain matrix.

Substituting the controller into the error system (6), we get 𝑑𝑒(𝑡)=(−𝐶+𝐺)𝑒(𝑡)+𝐴𝑔(𝑡)+𝐵𝑔(𝑡−𝜏(𝑡))+𝐷^𝑡_{𝑡−𝜎(𝑡)}𝑔(𝑠)((𝑡)((𝑡𝑑𝑠𝑑𝑡+𝜎𝑡,𝑒,𝑒𝑡−𝜏)))𝑑𝜔. (8) It is well known that system (8) has a unique solution [33].

Definition 1. The system (8) is called globally asymptotically mean-square stable, if the following holds: lim_𝑡→∞𝐸‖𝑒(𝑡)‖²=0,forany𝑒𝑡₀, (9) where 𝐸⋅ is the mathematical expectation. The following Lemma is given.

Lemma 2 (Schur complement lemma). For matrices 𝐴,𝐵, and 𝐶 with compatible dimensions, the following three conditions are equivalent: (𝑎)𝐶𝐴𝐶_𝑇𝐵<0;(𝑏)𝐴−𝐶𝐵⁻¹𝐶^𝑇<0,𝐵<0,𝑖𝑓𝐵𝑖𝑠𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑏𝑙𝑒;(𝑐)𝐵−𝐶^𝑇𝐴⁻¹𝐶<0,𝐴<0,𝑖𝑓𝐴𝑖𝑠𝑖𝑛𝑣𝑒𝑟𝑡𝑖𝑏𝑙𝑒. (10)(11)(12)

Lemma 3. For vector function 𝑓(𝑡)∈𝑅^𝑛 and symmetric matrix 𝑃>0, the following holds: ^𝑏_𝑎𝑓^𝑇(𝑡)𝑑𝑡𝑃^𝑏_𝑎𝑓(𝑡)𝑑𝑡≤(𝑏−𝑎)^𝑏_𝑎𝑓^𝑇(𝑡)𝑃𝑓(𝑡)𝑑𝑡. (13)

3. Criteria of Synchronization

In this section, new criteria are presented for the global asymptotical stability of the equilibrium point of the neural network defined by (8), and thus the drive system (1) synchronizes with the response system (2). Its proof is based on a new Lyapunov functional method and linear matrix inequality (LMI) approach.

Theorem 4. Under the assumptions A1–A3, the equilibrium point of model (8) is called globally asymptotically stable in mean-square, if there exist symmetric positive-definite matrices 𝑃_𝑗(𝑗=1,…,4), diagonal matrices 𝐷₁,𝐷₂, and general matrix 𝑄₁ such that the following inequalities hold:ΠΠ=¹¹𝑄₁Π₁₃Π₁₄0𝑃₁𝐷+𝜏²𝐷𝜏(−𝐶+𝐺)^𝑇𝑃₁∗Π₂₂0𝐾^𝑇𝐷₂−𝑄₁00∗∗Π₃₃𝜏²𝐴^𝑇𝑃₁𝐵0𝜏²𝐴^𝑇𝑃₁𝐷0∗∗∗Π₄₄0𝜏²𝐵^𝑇𝑃₁𝐷0∗∗∗∗−𝑃₁00∗∗∗∗∗−𝑃₄+𝜏²𝐷^𝑇𝑃₁𝐷0∗∗∗∗∗∗−𝑃₁<0,𝑃₁<𝜌𝐼, (14)(15)where Π₁₁=𝑃₁(−𝐶+𝐺)+(−𝐶+𝐺)^𝑇𝑃₁+𝑃₂+𝜌𝑀^𝑇₁𝑀₁,Π₁₃=𝑃₁𝐴+𝐾^𝑇𝐷₁+𝜏²(−𝐶+𝐺)^𝑇𝑃₁Π𝐴,₁₄=𝑃₁𝐵+𝜏²(−𝐶+𝐺)^𝑇𝑃₁Π𝐵,₂₂=−1−𝜏₁𝑃₂−𝑄₁−𝑄^𝑇₁+𝜌𝑀^𝑇₂𝑀₂,Π₃₃=𝑃₃−2𝐷₁+𝜎²𝑃₄+𝜏²𝐴^𝑇𝑃₁Π𝐴,₄₄=−1−𝜏₁𝑃₃−2𝐷₂+𝜏²𝐵^𝑇𝑃₁𝐵. (16)

Proof. The Lyapunov function candidate is given as 𝑉(𝑡)=⁵_𝑖=1𝑉_𝑖(𝑡), (17) where 𝑉₁(𝑡)=𝑒^𝑇(𝑡)𝑃₁𝑒(𝑡),𝑉₂(𝑡)=^𝑡_{(𝑡)𝑡−𝜏}𝑒^𝑇(𝑠)𝑃₂𝑒(𝑠)𝑉𝑑𝑠,₃(𝑡)=^𝑡_{𝑡−𝜏(𝑡)}𝑔^𝑇(𝑠)𝑃₃𝑔(𝑠)𝑉𝑑𝑠,₄(𝑡)=𝜏^𝜏₀^𝑡_𝑡−𝑠𝑒^𝑇̇(𝜃)𝑃₁𝑒̇(𝜃)𝑉𝑑𝜃𝑑𝑠,₅(𝑡)=𝜎⁰_−𝜎^𝑡_𝑡+𝑠𝑔^𝑇(𝜃)𝑃₄𝑔(𝜃)𝑑𝜃𝑑𝑠. (18) The weak infinitesimal operator 𝐿 of the process {𝑥_𝑡=𝑥(𝑡+𝑠),𝑡≥0,−𝜏≤𝑡≤0} is given by 𝐿𝑉₁=2𝑒^𝑇(𝑡)𝑃₁)𝑒(𝑡)(𝑡)((𝑡(−𝐶+𝐺+𝐴𝑔+𝐵𝑔𝑡−𝜏))+𝐷^𝑡_{(𝑡)𝑡−𝜎}𝑔(𝑠)𝜎𝑑𝑠+trace^𝑇(𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)))×𝑃₁𝜎(𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡))). (19) By (5) and (14), we have trace𝜎^𝑇(𝑡,𝑒(𝑡),𝑒(𝑡−𝜏(𝑡)))×𝑃₁𝜎((𝑡)((𝑡𝜎𝑡,𝑒,𝑒𝑡−𝜏)))≤𝜌trace^𝑇((𝑡)((𝑡((𝑡)((𝑡𝑒𝑡,𝑒,𝑒𝑡−𝜏)))×𝜎𝑡,𝑒,𝑒𝑡−𝜏)))=𝜌^𝑇(𝑡)𝑀^𝑇₁𝑀₁𝑒(𝑡)+𝑒^𝑇(𝑡−𝜏(𝑡))𝑀^𝑇₂𝑀₂𝑒(𝑡−𝜏(𝑡)),𝐿𝑉₂(𝑡)=𝑒^𝑇(𝑡)𝑃₂𝑒(𝑡)−(1−𝜏̇(𝑡))𝑒^𝑇×(𝑡−𝜏(𝑡))𝑃₂𝑒(𝑡−𝜏(𝑡)),𝐿𝑉₃(𝑡)=𝑔^𝑇(𝑡)𝑃₃𝑔(𝑡)−(1−𝜏̇(𝑡))𝑔^𝑇×(𝑡−𝜏(𝑡))𝑃₃𝑔(𝑡−𝜏(𝑡)),𝐿𝑉₄(𝑡)=𝜏²𝑒^𝑇̇(𝑡)𝑃₁𝑒̇(𝑡)−𝜏^𝑡_𝑡−𝜏𝑒^𝑇̇(𝑠)𝑃₁𝑒̇(𝑠)𝑑𝑠,𝐿𝑉₅(𝑡)=𝜎²𝑔^𝑇(𝑡)𝑃₄𝑔(𝑡)−𝜎^𝑡_𝑡−𝜎𝑔^𝑇(𝑠)𝑃₄𝑔(𝑠)𝑑𝑠. (20)(21)(22)(23)(24) By Lemma 3, we obtain −𝜏^𝑡_𝑡−𝜏𝑒^𝑇̇(𝑠)𝑃₁𝑒̇(𝑠)(𝑡)𝑑𝑠≤−𝜏^𝑡_{(𝑡)𝑡−𝜏}𝑒^𝑇̇(𝑠)𝑃₁𝑒̇(𝑠)𝑑𝑠≤−^𝑡_{𝑡−𝜏(𝑡)}𝑒^𝑇̇(𝑠)𝑑𝑠𝑃₁^𝑡_{𝑡−𝜏(𝑡)}𝑒̇(𝑠)𝑑𝑠,−𝜎^𝑡_𝑡−𝜎𝑔^𝑇(𝑠)𝑃₄𝑔(𝑠)(𝑡)𝑑𝑠≤−𝜎^𝑡_{(𝑡)𝑡−𝜎}𝑔^𝑇(𝑠)𝑃₄𝑔(𝑠)𝑑𝑠≤−^𝑡_{𝑡−𝜎(𝑡)}𝑔^𝑇(𝑠)𝑑𝑠𝑃₄^𝑡_{𝑡−𝜎(𝑡)}𝑔(𝑠)𝑑𝑠. (25)(26) By Newton-Leibniz formula, we have 𝑒(𝑡)−𝑒(𝑡−𝜏(𝑡))=^𝑡_{𝑡−𝜏(𝑡)}𝑒̇(𝑠)𝑑𝑠. (27) From assumption A1, we get 𝑔^𝑇(𝑡)𝐷₁𝑔(𝑡)≤𝑔^𝑇(𝑡)𝐷₁(𝑡),𝑔𝐾𝑒_𝑇(𝑡−𝜏(𝑡))𝐷₂𝑔(𝑡−𝜏(𝑡))≤𝑔^𝑇(𝑡−𝜏(𝑡))𝐷₂𝐾𝑒(𝑡−𝜏(𝑡)), (28) where 𝐾,𝐷₁, and 𝐷₂ are positive-definite diagonal matrices. By (27) and (28), we define several nonnegative expressions as follows: 𝐿1=2𝑒(𝑡−𝜏(𝑡))𝑄₁𝑒(𝑡)−𝑒(𝑡−𝜏(𝑡))−^𝑡_{𝑡−𝜏(𝑡)}𝑒̇(𝑠)𝑑𝑠=0,𝐿2=2𝑔^𝑇(𝑡)𝐷₁𝐾𝑒(𝑡)−𝑔^𝑇(𝑡)𝐷₁𝑔(𝑡)≥0,𝐿3=2𝑔^𝑇(𝑡−𝜏(𝑡))𝐷₂𝐾𝑒(𝑡−𝜏(𝑡))−𝑔^𝑇(𝑡−𝜏(𝑡))×𝐷₂𝑔(𝑡−𝜏(𝑡))≥0. (29)(30)(31) Then expressions 𝐿1,𝐿2, and 𝐿3 are to be added to the inequality of 𝐿𝑉(𝑡) to facilitate the proof.
Therefore, combining (19)–(29) we have 𝐿𝑉(𝑡)≤⁵_𝑖=1𝐿𝑉_𝑖(𝑡)+𝐿1+𝐿2+𝐿3≤𝑒^𝑇(𝑡)2𝑃₁(−𝐶+𝐺)+𝜌𝑀^𝑇₁𝑀₁+𝑃₂𝑒(𝑡)+𝑒^𝑇(𝑡−𝜏(𝑡))−1−𝜏₁𝑃₂+𝜌𝑀^𝑇₂𝑀₂−2𝑄₁×𝑒(𝑡−𝜏(𝑡))+𝑔^𝑇(𝑡)𝑃₃−2𝐷₁+𝜎²𝑃₄𝑔(𝑡)+𝑔^𝑇(𝑡−𝜏(𝑡))−1−𝜏₁𝑃₃−2𝐷₂𝑔((𝑡−𝑡−𝜏))_{𝑡𝑡−𝜏}𝑒^𝑇̇(𝑠)𝑑𝑠𝑃₁^𝑡_𝑡−𝜏𝑒̇(𝑠)−𝑑𝑠_{𝑡𝑡−𝜎}𝑔^𝑇(𝜃)𝑑𝜃𝑃₄^𝑡_𝑡−𝜎𝑔(𝜃)𝑑𝜃+2𝑒^𝑇(𝑡)𝑄₁𝑒(𝑡−𝜏(𝑡))+𝑒^𝑇(𝑡)2𝑃₁𝐴+2𝐾^𝑇𝐷₁𝑔(𝑡)+2𝑒^𝑇(𝑡)𝑃₁𝐵𝑔(𝑡−𝜏(𝑡))+2𝑒^𝑇(𝑡)𝑃₁𝐷^𝑡_{𝑡−𝜎(𝑡)}𝑔(𝜃)𝑑𝜃+2𝑒^𝑇(𝑡−𝜏(𝑡))𝐾^𝑇𝐷₂𝑔(𝑡−𝜏(𝑡))+2𝑒^𝑇(𝑡−𝜏(𝑡))𝑄₁^𝑡_𝑡−𝜏𝑒̇(𝑠)+)𝑒(𝑡)(𝑡)((𝑡𝑑𝑠(−𝐶+𝐺+𝐴𝑔+𝐵𝑔𝑡−𝜏))+𝐷^𝑡_{(𝑡)𝑡−𝜎}𝑔(𝑠)𝑑𝑠^𝑇×𝜏²𝑃₁)𝑒(𝑡)(𝑡)((𝑡(−𝐶+𝐺+𝐴𝑔+𝐵𝑔𝑡−𝜏))+𝐷^𝑡_{(𝑡)𝑡−𝜎}𝑔(𝑠)𝑑𝑠≤𝜉^𝑇𝑁𝜉, (32) where 𝜉=𝑒^𝑇(𝑡),𝑒^𝑇(𝑡−𝜏(𝑡)),𝑔^𝑇(𝑡),𝑔^𝑇(𝑡−𝜏(𝑡))_{𝑡(𝑡)𝑡−𝜏},𝑒^𝑇̇(𝑠)𝑑𝑠,^𝑡_{(𝑡)𝑡−𝜎}𝑔^𝑇(𝑠)𝑑𝑠^𝑇,𝑁𝑁=¹¹𝑄₁Π₁₃Π₁₄0𝑃₁𝐷+𝜏²𝐷∗Π₂₂0𝐾^𝑇𝐷₂−𝑄₁0∗∗Π₃₃𝜏²𝐴^𝑇𝑃₁𝐵0𝜏²𝐴^𝑇𝑃₁𝐷∗∗∗Π₄₄0𝜏²𝐵^𝑇𝑃₁𝐷∗∗∗∗−𝑃₁0∗∗∗∗∗−𝑃₄+𝜏²𝐷^𝑇𝑃₁𝐷,𝑁₁₁=𝑃₁(−𝐶+𝐺)+(−𝐶+𝐺)^𝑇𝑃₁+𝑃₂+𝜌𝑀^𝑇₁𝑀₁+𝜏²(−𝐶+𝐺)^𝑇𝑃₁(−𝐶+𝐺). (33) We introduce matrices 𝑀 and 𝐻 which satisfy 𝐻Π=𝑀𝐻_𝑇−𝑃₁,𝑁=𝑀−𝐻−𝑃₁⁻¹𝐻^𝑇, (34) where Π𝑀=¹¹𝑄₁Π₁₃Π₁₄0𝑃₁𝐷+𝜏²𝐷∗Π₂₂0𝐾^𝑇𝐷₂−𝑄₁0∗∗Π₃₃𝜏²𝐴^𝑇𝑃₁𝐵0𝜏²𝐴^𝑇𝑃₁𝐷∗∗∗Π₄₄0𝜏²𝐵^𝑇𝑃₁𝐷∗∗∗∗−𝑃₁0∗∗∗∗∗−𝑃₄+𝜏²𝐷^𝑇𝑃₁𝐷,𝐻=𝜏(−𝐶+𝐺)^𝑇𝑃₁00000. (35) By Lemma 2, we get that Π<0⇔𝑁<0,−𝑃₁<0. Thus 𝐿𝑉<0. From (24) and the Ito formula, it is obvious that 𝐸𝑉(𝑡)−𝐸𝑉𝑡₀=𝐸^𝑡_𝑡0𝐿𝑉(𝑠)𝑑𝑠. (36) From the definition of 𝑉(𝑡) in (15), there exists positive constant 𝜆₁ such that 𝜆₁𝐸‖𝑒(𝑡)‖²≤𝐸𝑉(𝑡)≤𝐸𝑉𝑡₀+𝐸^𝑡_𝑡0(𝑠)𝑡𝐿𝑉𝑑𝑠≤𝐸𝑉₀+𝜆_max𝐸^𝑡_𝑡0‖𝑒(𝑠)‖²𝑑𝑠, (37) where 𝜆_max is the maximal eigenvalue of 𝑁 and it is negative.
Therefore, from (26) and the discussion in [34], we know that the equilibrium of (11) is globally asymptotically stable in mean-square. This completes the proof.

Corollary 5. Under the assumptions A1–A3, the equilibrium point of model (8) is called globally asymptotically stable in mean-square, if there exist symmetric positive-definite matrices 𝑃_𝑗(𝑗=1,…,4), diagonal matrices 𝐷₁,𝐷₂, and general matrices 𝑄₁ and 𝑊, such that the following inequalities hold:ΠΠ=¹¹𝑄₁Π₁₃Π₁₄0𝑃₁𝐷+𝜏²𝐷𝜏−𝐶^𝑇𝑃₁+𝑊^𝑇∗Π₂₂0𝐾^𝑇𝐷₂−𝑄₁00∗∗Π₃₃𝜏²𝐴^𝑇𝑃₁𝐵0𝜏²𝐴^𝑇𝑃₁𝐷0∗∗∗Π₄₄0𝜏²𝐵^𝑇𝑃₁𝐷0∗∗∗∗−𝑃₁00∗∗∗∗∗−𝑃₄+𝜏²𝐷^𝑇𝑃₁𝐷0∗∗∗∗∗∗−𝑃₁𝑃<0,₁<𝜌𝐼, (38)where Π₁₁=−𝑃₁𝐶−𝐶^𝑇𝑃₁+𝑊+𝑊^𝑇+𝑃₂+𝜌𝑀^𝑇₁𝑀₁,Π₁₃=𝑃₁𝐴+𝐾^𝑇𝐷₁+𝜏²−𝐶^𝑇𝑃₁+𝑊^𝑇Π𝐴,₁₄=𝑃₁𝐵+𝜏²−𝐶^𝑇𝑃₁+𝑊^𝑇Π𝐵,₂₂=−1−𝜏₁𝑃₂−𝑄₁−𝑄^𝑇₁+𝜌𝑀^𝑇₂𝑀₂,Π₃₃=𝑃₃−2𝐷₁+𝜎²𝑃₄+𝜏²𝐴^𝑇𝑃₁Π𝐴,₄₄=−1−𝜏₁𝑃₃−2𝐷₂+𝜏²𝐵^𝑇𝑃₁𝐵. (39) Furthermore, the control gain matrix 𝐺 is given as 𝐺=𝑃₁⁻¹𝑊. (40)

Proof. Let 𝑊=𝑃₁𝐺 in Theorem 4; it is easy to get the result.

Corollary 6. If the systems (1) and (2) have no integral time-delay items (e.g., 𝐷=0), the result can be simplified as follows. Under assumptions A1–A3 and 𝐷=0, the equilibrium point of model (8) is called globally asymptotically stable in mean-square, if there exist symmetric positive-definite matrices 𝑃_𝑗(𝑗=1,…,4), diagonal matrices 𝐷₁,𝐷₂, and general matrices 𝑄₁ and 𝑊, such that the following inequalities hold:ΠΠ=¹¹𝑄₁Π₁₃Π₁₄0𝜏−𝐶^𝑇𝑃₁+𝑊^𝑇∗Π₂₂0𝐾^𝑇𝐷₂−𝑄₁0∗∗Π₃₃𝜏²𝐴^𝑇𝑃₁𝐵00∗∗∗Π₄₄00∗∗∗∗−𝑃₁0∗∗∗∗∗0∗∗∗∗∗−𝑃₁𝑃<0,₁<𝜌𝐼, (41)where Π₁₁=−𝑃₁𝐶−𝐶^𝑇𝑃₁+𝑊+𝑊^𝑇+𝑃₂+𝜌𝑀^𝑇₁𝑀₁,Π₁₃=𝑃₁𝐴+𝐾^𝑇𝐷₁+𝜏²−𝐶^𝑇𝑃₁+𝑊^𝑇Π𝐴,₁₄=𝑃₁𝐵+𝜏²−𝐶^𝑇𝑃₁+𝑊^𝑇Π𝐵,₂₂=−1−𝜏₁𝑃₂−𝑄₁−𝑄^𝑇₁+𝜌𝑀^𝑇₂𝑀₂,Π₃₃=𝑃₃−2𝐷₁+𝜎²𝑃₄+𝜏²𝐴^𝑇𝑃₁Π𝐴,₄₄=−1−𝜏₁𝑃₃−2𝐷₂+𝜏²𝐵^𝑇𝑃₁𝐵. (42) Furthermore, the control gain matrix 𝐺 is given as 𝐺=𝑃₁⁻¹𝑊. (43)

Proof. Let 𝐷=0 in system (8). It is much similar to the proof of Theorem 4 and is then omitted.

Remark 7. In this paper, we use a memoryless state feedback control to achieve the synchronization of the stochastic delayed neural network system. Compared with the delayed feedback control [1], it is easy to implement. The biggest advantage of this approach is that we only need to know the boundaries of time delays, instead of exact values of time delays.

Remark 8. In Corollary 5, we give an approach to choose the control gain matrix 𝐺 and it is helpful for the design of the controller to let the drive system synchronize with the response system.

4. Numerical Simulations

In this section, we present numerical simulations to show the effectiveness of the theoretical results.

Consider the drive system (1) of a delayed Hopfield neural network with coefficient matrices as ,,𝜏(𝑡)()(𝑡)(𝑡),𝑓(𝑡)=𝑥𝐴=1−0.5−0.21,𝐵=−0.5−0.1−0.3−0.6𝐶=1001,𝐷=1001=0.3+0.3sin4𝑡,𝜎=0.4−0.2sintanh₁tanh𝑥₂. (44) The corresponding response system refers to (2), where 𝜎((𝑡)((𝑡=‖𝑒(𝑡0‖𝑒((𝑡𝑡,𝑒,𝑒𝑡−𝜏))).)‖00.6𝑡−𝜏))‖ (45) It is easy to see that 𝐾=𝑀=𝑀₁=𝐼, where 𝐼 is the identity matrix. By Theorem 4 and Corollary 5, we can get the following feasible solutions: ,𝑄𝑊=−1918.212.477.5−1984.4₁=,𝐷545.55683.694317.0650557.7364₁=,𝐷1046.5001207.4₂=,,𝑃226.325900242.7899𝜌=1084.8,𝐺=−1.83520.04250.1038−1.9002₁=1.046.20.016.90.016.91.044.7,𝑃₂=,𝑃52.1765−18.3810−18.381048.2007₃=94.2132−39.7350−39.735086.2442,𝑃₄=.2626.1148.4148.42760.3 (46) The initial states of drive system and response system are 𝑥(0)=[0.750]^𝑇 and 𝑦(0)=[0.50.1]^𝑇.

The results are shown in Figures 1, 2, 3, and 4.

(a)

(b)

Figure 1 

Phase trajectories of drive (a) and response (b) system.

(a)

(b)

Figure 2 

State trajectories of drive system.

(a)

(b)

Figure 3 

State trajectories of response system.

(a)

(b)

Figure 4 

State trajectories of error system.

5. Conclusion

In this paper, we considered synchronization control of stochastic neural networks with time-varying delays. We use Lyapunov functional method and linear matrix inequality (LMI) technique to solve this problem. Several sufficient conditions have been derived to ensure the global asymptotical stability for the error system, and thus the drive system synchronizes with the response system. Also, the control gain can be obtained. The results are novel since there are few works about the synchronization of mixed delayed system and the constraint of the derivative of the time-delay function is relaxed. It is easy to apply these sufficient conditions to the real networks. Finally, a numerical simulation is presented to verify the theoretical results.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editor and the reviewers for their helpful and valuable comments and suggestions, which have improved the presentation. This work is supported by National Natural Science Foundation of China under Grant nos. 61273352, 61175111, 61174046, 60904030, 60874045, and 60874030, Natural Science Foundation of the Jiangsu Higher Education Institutions (nos. 10KJB510027 and 09KJB510019), Postdoctoral Science Research Project of Jiangsu Province (no. 1102167C).