Synchronization of Chaotic Neural Networks with Leakage Delay and Mixed Time-Varying Delays via Sampled-Data Control

Abstract

This paper investigates the synchronization problem for neural networks with leakage delay and both discrete and distributed time-varying delays under sampled-data control. By employing the Lyapunov functional method and using the matrix inequality techniques, a delay-dependent LMIs criterion is given to ensure that the master systems and the slave systems are synchronous. An example with simulations is given to show the effectiveness of the proposed criterion.

1. Introduction

Since the pioneering works of Pecora and Carroll [1], the synchronization of chaotic systems has received considerable attention due to its potential applications in biology, chemistry, secret communication, cryptography, nonlinear oscillation synchronization, and some other nonlinear fields [2]. It has been shown that the neural networks can exhibit chaotic behavior [3]. Therefore, it has a wider significance to study the problem on the synchronization of chaotic neural networks.

In the past decades, some works dealing with the synchronization of neural networks have also appeared; for example, see [4–22] and references therein. In [4], authors discussed the synchronization and computation in a chaotic neural network. In [7], the local synchronization and global exponential stability for an array of linearly coupled identical connected neural networks with delays were investigated without assuming that the coupling matrix is symmetric or irreducible; the linear matrix inequality approach was used to judge synchronization with global convergence property. In [8], authors presented an adaptive synchronization scheme between two different kinds of delayed chaotic neural networks with partly unknown parameters. An adaptive controller was designed to guarantee the global asymptotic synchronization of state trajectories for two different chaotic neural networks with time delay. In [9], the concept of 𝜇-synchronization was introduced; some sufficient conditions were derived for the global 𝜇-synchronization for the linearly coupled neural networks with delayed couplings, where the intrinsic systems are recurrently connected neural networks with unbounded time-varying delays, and the couplings include instant couplings and unbounded delayed couplings. In [10], authors proposed a general array model of coupled delayed neural networks with hybrid coupling, which is composed of constant coupling, discrete-delay coupling, and distributed-delay coupling. Based on the Lyapunov functional method and Kronecker product properties, several sufficient conditions were established to ensure global exponential synchronization based on the design of the coupling matrices, the inner linking matrices, and/or some free matrices representing the relationships between the system matrices. The conditions are expressed within the framework of linear matrix inequalities, which can be easily computed by the interior-point method. In addition, a typical chaotic cellular neural network was used as the node in the array to illustrate the effectiveness and advantages of the theoretical results. In [11], the globally robust synchronization problem was investigated for an array of coupled neural networks with uncertain parameters and time delays. Both the cases of linear coupling and nonlinear coupling were simultaneously taken into account. Several criteria for checking the robust exponential synchronization were given for the considered coupled neural networks. In [12], authors presented a new linear matrix inequality-based approach to an 𝐻^∞ output feedback control problem of master-slave synchronization of artificial neural networks with uncertain time delay. In [13], the problem of feedback controller design to achieve synchronization for neural network of neutral type with stochastic perturbation was considered. Based on Lyapunov method and LMI framework, a criterion for master-slave synchronization was obtained. In [16], authors investigated the globally exponential synchronization for linearly coupled neural networks with time-varying delay and impulsive disturbances. Since the impulsive effects discussed were regarded as disturbances, the impulses should not happen too frequently. The concept of average impulsive interval was used to formalize this phenomenon. By referring to an impulsive delay differential inequality, a criterion for the globally exponential synchronization of linearly coupled neural networks with impulsive disturbances was given. In [18], the projective synchronization between two continuous-time delayed neural systems with time-varying delay was investigated. A sufficient condition for synchronization of the coupled systems with modulated delay was presented analytically with the help of the Krasovskii-Lyapunov approach. In [19], the problem of guaranteed cost control for exponential synchronization of cellular neural networks with interval nondifferentiable and distributed time-varying delays via hybrid feedback control was considered. Several delay-dependent sufficient conditions for the exponential synchronization were obtained. In [20], authors studied the synchronization in an array of coupled neural networks with Markovian jumping and random coupling strength. By designing a novel Lyapunov function and using inequality techniques and the properties of random variables, several delay-dependent synchronization criteria were derived for the coupled networks of continuous-time version. Discrete-time analogues of the continuous-time networks were also formulated and studied. In [21], authors considered adaptive synchronization of chaotic Cohen-Grossberg neural networks with mixed time delays. In [22], 𝑝th moment exponential synchronization for stochastic delayed Cohen-Grossberg neural networks with Markovian switching was investigated.

On the other hand, with the development of networked control systems, sampled-data control in the presence of a constant input delay has been an important research area in recent years, because networked control systems are usually modeled as sampled-data systems under variable sampling with an additional network-induced delay [23]. There are some results dealing with the synchronization problem using sampled-data control; for example, see [23–38] and references therein. In [23], a new approach, the input delay approach, has been proposed that can deal well with the sampled-data control problems. The main idea of this approach is to convert the considered sampling period into a time-varying but bounded delay and then accomplish the sampled-data control or state estimation tasks by using the existing theory of time-delayed systems. In [24], the synchronization of chaotic system using a sampled-data fuzzy controller was studied. In [25–27], the synchronization for chaotic Lur’e systems using sampled-data control was investigated; several criteria were given to ensure that the master systems synchronize with the slave systems by using Lyapunov-Krasovskii functional and LMI approach. In [28–35], authors discussed the synchronization of chaotic system and complex networks by using sampled-data control. In [36–38], the synchronization of neural networks with time-varying delays was considered; by using sampled-data control method, several criteria for checking the synchronization were obtained. To the best of the authors knowledge, there is no results on the problem of the synchronization for neural networks with leakage delay and both discrete and distributed time-varying delays [38]. Therefore, there is a need to further extend the synchronization results reported in [38].

Motivated by the previous discussions, the objective of this paper is to study the synchronization for neural networks with leakage delay and both discrete and distributed time-varying delays by using sampled-data control approach. The obtained sufficient conditions do not require the differentiability of time-varying delays and are expressed in terms of linear matrix inequalities, which can be checked numerically using the effective LMI toolbox in MATLAB. An example is given to show the effectiveness and less conservatism of the proposed criterion.

Notations. The notations are quite standard. Throughout this paper, ℝ^𝑛 and ℝ^𝑛×𝑚 denote, respectively, the 𝑛-dimensional Euclidean space and the set of all 𝑛×𝑚 real matrices. ‖⋅‖ refers to the Euclidean vector norm. 𝐴^𝑇 represents the transpose of matrix 𝐴 and the asterisk “∗” in a matrix is used to represent the term which is induced by symmetry. 𝐼 is the identity matrix with compatible dimension. 𝑋>𝑌 means that 𝑋 and 𝑌 are symmetric matrices and that 𝑋−𝑌 is positive definite. Matrices, if not explicitly specified, are assumed to have compatible dimensions.

2. Model Description and Preliminaries

Consider the following neural networks with leakage delay and mixed time-varying delays: 𝑥̇(𝑡)=−𝐷𝑥(𝑡−𝛿)+𝐴𝑓(𝑥(𝑡))+𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))+𝐶^𝑡_{𝑡−𝜎(𝑡)}𝑓(𝑥(𝑠))𝑑𝑠+𝐽(𝑡), (1) for 𝑡≥0, where 𝑥(𝑡)=(𝑥₁(𝑡),𝑥₂(𝑡),…,𝑥_𝑛(𝑡))^𝑇∈ℝ^𝑛 is the state vector of the network at time 𝑡, 𝑛 corresponds to the number of neurons, 𝐷∈ℝ^𝑛×𝑛 is a positive diagonal matrix, 𝐴∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑛, and 𝐶∈ℝ^𝑛×𝑛 are the connection weight matrix, the discretely delayed connection weight matrix, and the distributively delayed connection weight matrix, respectively, 𝑓(𝑥(𝑡))=(𝑓₁(𝑥₁(𝑡)),𝑓₂(𝑥₂(𝑡)),…,𝑓_𝑛(𝑥_𝑛(𝑡)))^𝑇∈ℝ^𝑛 denotes the neuron activation at time 𝑡, 𝐽(𝑡)=(𝐽₁(𝑡),𝐽₂(𝑡),…,𝐽_𝑛(𝑡))^𝑇∈ℝ^𝑛 is an external input vector, and 𝛿, 𝜏(𝑡), and 𝜎(𝑡) denote the leakage delay, discrete time-varying delay and the distributed time-varying delay, respectively.

In this paper, system (1) is regarded as the master system and a slave system for (1) can be described by the following equation: 𝑦̇(𝑡)=−𝐷𝑦(𝑡−𝛿)+𝐴𝑓𝑦(𝑡)+𝐵𝑓𝑦(𝑡−𝜏(𝑡))+𝐶^𝑡_{𝑡−𝜎(𝑡)}𝑓𝑦(𝑠)𝑑𝑠+𝑢(𝑡)+𝐽(𝑡), (2) where 𝑢(𝑡)∈ℝ^𝑛 is the appropriate control input that will be designed in order to obtain a certain control objective.

Throughout this paper, we make the following assumptions.(H1)For any 𝑗∈{1,2,…,𝑛}, there exist constants 𝐹⁻_𝑗 and 𝐹⁺_𝑗 such that 𝐹⁻_𝑗≤𝑓_𝑗𝛼₁−𝑓_𝑗𝛼₂𝛼₁−𝛼₂≤𝐹⁺_𝑗, (3) for all 𝛼₁≠𝛼₂.(H2)The leakage delay 𝛿, the discrete time-varying delays 𝜏(𝑡), and the distributed time-varying delay 𝜎(𝑡) satisfy the following conditions: 0≤𝛿,0≤𝜏(𝑡)≤𝜏,0≤𝜎(𝑡)≤𝜎, (4) where 𝛿, 𝜏, and 𝜎 are constants.

By defining the error signal as 𝑒(𝑡)=𝑦(𝑡)−𝑥(𝑡), the error system for (1) and (2) can be represented as follows: 𝑒̇(𝑡)=−𝐷𝑒(𝑡−𝛿)+𝐴𝑔(𝑒(𝑡))+𝐵𝑔(𝑒(𝑡−𝜏(𝑡)))+𝐶^𝑡_{𝑡−𝜎(𝑡)}𝑔(𝑒(𝑠))𝑑𝑠+𝑢(𝑡), (5) where 𝑔(𝑒(𝑡))=𝑓(𝑦(𝑡))−𝑓(𝑥(𝑡)).

In many real-world applications, it is difficult to guarantee that the state variables transmitted to controllers are continuous. In addition, in order to make full use of modern computer technique, the sampled-data feedback control is applied to synchronize delayed neural networks [36]. In this paper, the following sampled-data controller is adopted [38]: 𝑢(𝑡)=𝐾𝑒𝑡_𝑘,𝑡_𝑘≤𝑡<𝑡_𝑘+1, (6) where 𝐾 is sampled-data feedback controller gain matrix to be determined, 𝑡_𝑘 denotes the sampling instant and satisfies 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯, lim_𝑘→+∞𝑡_𝑘=+∞. Moreover, the sampling period under consideration is assumed to be bounded by a known constant ℎ; that is, 𝑡_𝑘+1−𝑡_𝑘≤ℎ, (7) for any integer 𝑘≥0, where ℎ is a positive scalar and represents the largest sampling interval.

Substituting control law (6) into the error system (5) yields 𝑒̇(𝑡)=−𝐷𝑒(𝑡−𝛿)+𝐴𝑔(𝑒(𝑡))+𝐵𝑔(𝑒(𝑡−𝜏(𝑡)))+𝐶^𝑡_{𝑡−𝜎(𝑡)}𝑔(𝑒(𝑠))𝑑𝑠+𝐾𝑒𝑡_𝑘. (8)

Clearly, it is difficult to analyze the synchronization of neural networks based on error system (8) because of the discrete term 𝑒(𝑡_𝑘). Therefore, the input delay approach [23] is applied; that is, a sawtooth function is defined as follows: 𝛾(𝑡)=𝑡−𝑡_𝑘,𝑡_𝑘≤𝑡<𝑡_𝑘+1. (9) It can be found from (7) and (9) that 0≤𝛾(𝑡)<ℎ and 𝛾̇(𝑡)=1 for 𝑡≠𝑡_𝑘.

By substituting (9) into (8), we get that 𝑒̇(𝑡)=−𝐷𝑒(𝑡−𝛿)+𝐴𝑔(𝑒(𝑡))+𝐵𝑔(𝑒(𝑡−𝜏(𝑡)))+𝐶^𝑡_{𝑡−𝜎(𝑡)}𝑔(𝑒(𝑠))𝑑𝑠+𝐾𝑒𝑡−𝛾(𝑡). (10)

The main purpose of this paper is to design controller with the form (6) to ensure that master system (1) synchronizes with slave system (2). In other words, we are interested in finding a feedback gain matrix 𝐾 such that error system (10) is stable.

To prove our result, the following lemmas that can be found in [39] are necessary.

Lemma 1 (see [39]). For any constant matrix 𝑊∈ℝ^𝑚×𝑚, 𝑊>0, scalar 0<ℎ(𝑡)<ℎ, and vector function 𝜔(⋅)∶[0,ℎ]→ℝ^𝑚 such that the integrations concerned are well defined; then ₀^ℎ(𝑡)𝜔(𝑠)𝑑𝑠^𝑇𝑊₀^ℎ(𝑡)𝜔(𝑠)(𝑡)𝑑𝑠≤ℎ₀^ℎ(𝑡)𝜔^𝑇(𝑠)𝑊𝜔(𝑠)𝑑𝑠. (11)

Lemma 2 (see [39]). Given constant matrices 𝑃, 𝑄, and 𝑅, where 𝑃^𝑇=𝑃, 𝑄^𝑇=𝑄, then 𝑅𝑃𝑅_𝑇−𝑄<0 (12) is equivalent to the following conditions: 𝑄>0,𝑃+𝑅𝑄⁻¹𝑅^𝑇<0. (13)

3. Main Results

Theorem 3. Suppose that (H1) and (H2) hold. If there exist seven symmetric positive definite matrices 𝑃_𝑖 (𝑖=1,2,3,4,5,6,7), four positive diagonal matrices 𝑊₁, 𝑊₂, 𝑅₁, and 𝑅₂, and nine matrices 𝑋₁₁, 𝑋₁₂, 𝑋₂₂, 𝑄₁, 𝑄₂, 𝑄₃, 𝑄₄, 𝑄₅, and 𝑍 such that the following LMIs hold: 𝑋𝑋=¹¹𝑋₁₂∗𝑋₂₂Ω>0,Ω=¹Ω₂∗Ω₃<0, (14)(15) whereΩ₁=Ω₁₁Ω₁₂Ω₁₃𝑋₂₂𝑄₃0Ω₁₇0Ω₁₉𝑄₁𝐵𝑄₁𝐶∗Ω₂₂−𝑄₁𝐷𝑋₁₂00𝑍0Ω₂₉𝑄₁𝐵𝑄₁𝐶∗∗−𝑃₁−𝑋₁₂0000000∗∗∗−𝑃₂0000000∗∗∗∗Ω₅₅𝑄₂000𝐹₄𝑅₂0∗∗∗∗∗−𝑃₃00000∗∗∗∗∗∗Ω₇₇𝑄₂000∗∗∗∗∗∗∗−𝑃₅000∗∗∗∗∗∗∗∗Ω₉₉00∗∗∗∗∗∗∗∗∗−𝑅₂0∗∗∗∗∗∗∗∗∗∗−𝑃₇,Ω₂=𝑄₃𝑄₅0000000000000000𝑄₂00000000𝑄₄0000000000000000,Ω₃=−1𝜏𝑃₄10000−ℎ𝑃₆10000−𝜏𝑃₄01000−ℎ𝑃₆, (16)
in which Ω₁₁=𝑋₁₂+𝑋^𝑇₁₂+𝑃₁+𝛿²𝑃₂+𝑃₃+𝑃₅−𝑄₃−𝑄^𝑇₃−𝑄₅−𝑄^𝑇₅−𝐹₃𝑅₁, Ω₁₂=𝑋₁₁−𝐹₁𝑊₁+𝐹₂𝑊₂−𝑄₁, Ω₁₃=−𝑋₁₂−𝑄₁𝐷, Ω₁₇=𝑍+𝑄₅, Ω₁₉=𝑄₁𝐴+𝐹₄𝑅₁, Ω₂₂=𝜏𝑃₄+ℎ𝑃₆−𝑄₁−𝑄^𝑇₁, Ω₂₉=𝑊₁−𝑊₂+𝑄₁𝐴, Ω₅₅=−𝑄₂−𝑄^𝑇₂−𝐹₃𝑅₂, Ω₇₇=−𝑄₄−𝑄^𝑇₄, and Ω₉₉=𝜎²𝑃₇−𝑅₁, then master system (1) and slave system (2) are synchronous. Moreover, the desired controller gain matrix 𝐾 in (6) can be given by 𝐾=𝑄₁⁻¹𝑍. (17)

Proof. From assumption (H1), we know that ^𝑒𝑖₀^(𝑡)𝑔_𝑖(𝑠)−𝐹⁻_𝑖𝑠𝑑𝑠≥0,_{𝑒𝑖0(𝑡)}𝐹⁺_𝑖𝑠−𝑔_𝑖(𝑠)𝑑𝑠≥0,𝑖=1,2,…,𝑛. (18) Let 𝑊₁=diag(𝑤₁₁,𝑤₁₂,…,𝑤_1𝑛) and 𝑊₂=diag(𝑤₂₁,𝑤₂₂,…,𝑤_2𝑛), and consider the following Lyapunov-Krasovskii functional: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)+𝑉₄(𝑡)+𝑉₅(𝑡)+𝑉₆(𝑡), (19) where 𝑉₁(𝑡)=∫𝑒(𝑡)_{𝑡𝑡−𝛿}𝑒(𝑠)𝑑𝑠^𝑇𝑋₁₁𝑋₁₂∗𝑋₂₂𝑒(𝑡)∫^𝑡_𝑡−𝛿𝑒(𝑠),𝑉𝑑𝑠₂(𝑡)=2^𝑛_𝑖=1𝑤_1𝑖^𝑒𝑖₀^(𝑡)𝑔_𝑖(𝑠)−𝐹⁻_𝑖𝑠𝑑𝑠+2^𝑛_𝑖=1𝑤_2𝑖^𝑒𝑖₀^(𝑡)𝐹⁺_𝑖𝑠−𝑔_𝑖(𝑠)𝑉𝑑𝑠,₃(𝑡)=^𝑡_𝑡−𝛿𝑒^𝑇(𝑠)𝑃₁𝑒(𝑠)𝑑𝑠+𝛿⁰_−𝛿^𝑡_𝑡+𝜉𝑒^𝑇(𝑠)𝑃₂𝑒(𝑠)𝑉𝑑𝑠𝑑𝜉,₄(𝑡)=^𝑡_𝑡−𝜏𝑒^𝑇(𝑠)𝑃₃𝑒(𝑠)+𝑑𝑠_{0−𝜏𝑡𝑡+𝜉}𝑒^𝑇̇(𝑠)𝑃₄𝑒̇(𝑠)𝑉𝑑𝑠𝑑𝜉,₅(𝑡)=^𝑡_𝑡−ℎ𝑒^𝑇(𝑠)𝑃₅𝑒(𝑠)+𝑑𝑠_{0−ℎ𝑡𝑡+𝜉}𝑒^𝑇̇(𝑠)𝑃₆𝑒̇(𝑠)𝑉𝑑𝑠𝑑𝜉,₆(𝑡)=𝜎⁰_−𝜎^𝑡_𝑡+𝜉𝑔^𝑇(𝑒(𝑠))𝑃₇𝑔(𝑒(𝑠))𝑑𝑠𝑑𝜉. (20) Calculating the time derivative of 𝑉_𝑖(𝑡) (𝑖=1,2,3,4,5,6), we obtain 𝑉̇₁(𝑡)𝑒(𝑡)∫=2_{𝑡𝑡−𝛿}𝑒(𝑠)𝑋𝑑𝑠¹¹𝑋₁₂∗𝑋₂₂𝑒̇(𝑡)𝑒(𝑡)−𝑒(𝑡−𝛿)^𝑇=𝑒^𝑇(𝑡)𝑋₁₂+𝑋^𝑇₁₂𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑋₁₁𝑒̇(𝑡)−2𝑒^𝑇(𝑡)𝑋₁₂𝑒(𝑡−𝛿)+2𝑒^𝑇(𝑡)𝑋₂₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑒̇𝑑𝑠+2^𝑇(𝑡)𝑋₁₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠−2𝑒^𝑇(𝑡−𝛿)𝑋₂₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑉̇𝑑𝑠,₂(𝑡)=2𝑒^𝑇̇(𝑡)𝑊₁𝑔(𝑒(𝑡))−𝐹₁𝑒(𝑡)+2𝑒^𝑇̇(𝑡)𝑊₂𝐹₂𝑒(𝑡)−𝑔(𝑒(𝑡))=2𝑒^𝑇(𝑡)−𝐹₁𝑊₁+𝐹₂𝑊₂𝑒̇(𝑡)+2𝑒^𝑇̇(𝑡)𝑊₁−𝑊₂𝑔(𝑒(𝑡,𝑉̇))₃(𝑡)=𝑒^𝑇(𝑡)𝑃₁+𝛿²𝑃₂𝑒(𝑡)−𝑒^𝑇(𝑡−𝛿)𝑃₁𝑒(𝑡−𝛿)−𝛿^𝑡_𝑡−𝛿𝑒^𝑇(𝑠)𝑃₂𝑒(𝑠)𝑑𝑠≤𝑒^𝑇(𝑡)𝑃₁+𝛿²𝑃₂𝑒(𝑡)−𝑒^𝑇(𝑡−𝛿)𝑃₁𝑒()−𝑡−𝛿_{𝑡𝑡−𝛿}𝑒(𝑠)𝑑𝑠^𝑇𝑃₂^𝑡_𝑡−𝛿𝑒(𝑠),𝑉̇𝑑𝑠₄(𝑡)=𝑒^𝑇(𝑡)𝑃₃𝑒(𝑡)−𝑒^𝑇(𝑡−𝜏)𝑃₃𝑒()𝑒̇𝑡−𝜏+𝜏^𝑇(𝑡)𝑃₄𝑒̇(𝑡)−^𝑡_𝑡−𝜏𝑒^𝑇̇(𝑠)𝑃₄𝑒̇(𝑠)𝑉̇𝑑𝑠,₅(𝑡)=𝑒^𝑇(𝑡)𝑃₅𝑒(𝑡)−𝑒^𝑇(𝑡−ℎ)𝑃₅𝑒()𝑒̇𝑡−ℎ+ℎ^𝑇(𝑡)𝑃₆𝑒̇(𝑡)−^𝑡_𝑡−ℎ𝑒^𝑇̇(𝑠)𝑃₆𝑒̇(𝑠)𝑉̇𝑑𝑠.₆(𝑡)=𝜎²𝑔^𝑇(𝑒(𝑡))𝑃₇𝑔(𝑒(𝑡))−𝜎^𝑡_𝑡−𝜎𝑔^𝑇(𝑒(𝑠))𝑃₇𝑔(𝑒(𝑠))𝑑𝑠≤𝜎²𝑔^𝑇(𝑒(𝑡))𝑃₇𝑔(𝑒(𝑡(𝑡)))−𝜎^𝑡_{𝑡−𝜎(𝑡)}𝑔^𝑇(𝑒(𝑠))𝑃₇𝑔(𝑒(𝑠))𝑑𝑠≤𝜎²𝑔^𝑇(𝑒(𝑡))𝑃₇𝑔(𝑒(𝑡−))_{𝑡(𝑡)𝑡−𝜎}𝑔(𝑒(𝑠))𝑑𝑠^𝑇𝑃₇^𝑡_{(𝑡)𝑡−𝜎}𝑔(𝑒(𝑠))𝑑𝑠. (21)(22)(23)(24)(25)(26) In deriving inequalities (23) and (26), we have made use of Lemma 1. It follows from inequalities (21)–(26) that 𝑉̇(𝑡)≤𝑒^𝑇(𝑡)𝑋₁₂+𝑋^𝑇₁₂+𝑃₁+𝛿²𝑃₂+𝑃₃+𝑃₅𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑋₁₁−𝐹₁𝑊₁+𝐹₂𝑊₂𝑒̇(𝑡)−2𝑒^𝑇(𝑡)𝑋₁₂𝑒(𝑡−𝛿)+2𝑒^𝑇(𝑡)𝑋₂₂×^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠+𝑒^𝑇̇(𝑡)𝜏𝑃₄+ℎ𝑃₆𝑒̇(𝑡)+2𝑒^𝑇̇(𝑡)×𝑋₁₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠+2𝑒^𝑇̇(𝑡)𝑊₁−𝑊₂𝑔(𝑒(𝑡))−𝑒^𝑇(𝑡−𝛿)𝑃₁𝑒(𝑡−𝛿)−2𝑒^𝑇(𝑡−𝛿)𝑋₂₂×^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠−𝑒^𝑇(𝑡−𝜏)𝑃₃𝑒()−𝑡−𝜏_{𝑡𝑡−𝛿}𝑒(𝑠)𝑑𝑠^𝑇𝑃₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠−𝑒^𝑇(𝑡−ℎ)𝑃₅𝑒()−𝑡−ℎ_{𝑡𝑡−𝜏}𝑒^𝑇̇(𝑠)𝑃₄𝑒̇(𝑠)𝑑𝑠−^𝑡_𝑡−ℎ𝑒^𝑇̇(𝑠)𝑃₆𝑒̇(𝑠)𝑑𝑠+𝜎²𝑔^𝑇(𝑒(𝑡))𝑃₇𝑔(𝑒(𝑡−))_{𝑡(𝑡)𝑡−𝜎}𝑔(𝑒(𝑠))𝑑𝑠^𝑇𝑃₇^𝑡_{(𝑡)𝑡−𝜎}𝑔(𝑒(𝑠))𝑑𝑠. (27)
From model (10), we have 0=2𝑒^𝑇(𝑡)+𝑒^𝑇̇(𝑡)𝑄₁×−𝑒̇(𝑡)−𝐷𝑒(𝑡−𝛿)+𝐴𝑔(𝑒(𝑡))+𝐵𝑔(𝑒(𝑡−𝜏(𝑡)))+𝐶^𝑡_{(𝑡)𝑡−𝜎}𝑔(𝑒(𝑠))𝑑𝑠+𝐾𝑒𝑡−𝛾(𝑡). (28) From Newton-Leibniz formulation and assumption (H2), we have 0=−2𝑒^𝑇(𝑡−𝜏(𝑡))×𝑄₂𝑒(𝑡−𝜏(𝑡))−𝑒(𝑡−𝜏)−^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑒̇(𝑠)𝑑𝑠≤−2𝑒^𝑇(𝑡−𝜏(𝑡))𝑄₂𝑒(𝑡−𝜏(𝑡))+2𝑒^𝑇(𝑡−𝜏(𝑡))𝑄₂𝑒(𝑡−𝜏)+𝜏𝑒^𝑇(𝑡−𝜏(𝑡))×𝑄₂𝑃₄⁻¹𝑄^𝑇₂𝑒(𝑡−𝜏(𝑡))+^{𝑡−𝜏(𝑡)}_𝑡−𝜏𝑒^𝑇̇(𝑠)𝑃₄𝑒̇(𝑠)𝑑𝑠,0=−2𝑒^𝑇(𝑡)𝑄₃𝑒(𝑡)−𝑒(𝑡−𝜏(𝑡))−^𝑡_{(𝑡)𝑡−𝜏}𝑒̇(𝑠)𝑑𝑠≤−2𝑒^𝑇(𝑡)𝑄₃𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑄₃𝑒(𝑡−𝜏(𝑡))+𝜏𝑒^𝑇(𝑡)𝑄₃𝑃₄⁻¹𝑄^𝑇₃𝑒(𝑡)+^𝑡_{𝑡−𝜏(𝑡)}𝑒^𝑇̇(𝑠)𝑃₄𝑒̇(𝑠)𝑑𝑠,0=−2𝑒^𝑇𝑡−𝛾(𝑡)×𝑄₄𝑒𝑡−𝛾(𝑡)−𝑒(𝑡−ℎ)−^{𝑡−𝛾(𝑡)}_𝑡−ℎ𝑒̇(𝑠)𝑑𝑠≤−2𝑒^𝑇𝑡−𝛾(𝑡)𝑄₄𝑒𝑡−𝛾(𝑡)+2𝑒^𝑇𝑡−𝛾(𝑡)𝑄₂𝑒(𝑡−ℎ)+ℎ𝑒^𝑇𝑡−𝛾(𝑡)𝑄₄𝑃₆⁻¹𝑄^𝑇₄𝑒(𝑡)+𝑡−𝛾_{𝑡−𝛾(𝑡)𝑡−ℎ}𝑒^𝑇̇(𝑠)𝑃₆𝑒̇(𝑠)𝑑𝑠,0=−2𝑒^𝑇(𝑡)𝑄₅𝑒(𝑡)−𝑒𝑡−𝛾(𝑡)−^𝑡_{(𝑡)𝑡−𝛾}𝑒̇(𝑠)𝑑𝑠≤−2𝑒^𝑇(𝑡)𝑄₅𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑄₅𝑒𝑡−𝛾(𝑡)+ℎ𝑒^𝑇(𝑡)𝑄₅𝑃₆⁻¹𝑄^𝑇₅𝑒(𝑡)+^𝑡_{(𝑡)𝑡−𝛾}𝑒^𝑇̇(𝑠)𝑃₆𝑒̇(𝑠)𝑑𝑠. (29) In addition, for positive diagonal matrices 𝑅₁>0 and 𝑅₂>0, we can get from assumption (H1) that [40] 𝑒(𝑡)𝑔(𝑒(𝑡))^𝑇𝐹₃𝑅₁−𝐹₄𝑅₁−𝐹₄𝑅₁𝑅₁𝑒(𝑡)𝑔(𝑒(𝑡𝑒((𝑡𝑔(𝑒((𝑡))≤0,𝑡−𝜏))𝑡−𝜏)))^𝑇𝐹₃𝑅₂−𝐹₄𝑅₂−𝐹₄𝑅₂𝑅₂×𝑒((𝑡𝑔(𝑒((𝑡𝑡−𝜏))𝑡−𝜏)))≤0. (30)(31) It follows from (27)–(31) that 𝑉̇(𝑡)≤𝑒^𝑇(𝑡)2𝑋₁₂+𝑃₁+𝛿²𝑃₂+𝑃₃+𝑃₅−2𝑄₃+𝜏𝑄₃𝑃₄⁻¹𝑄^𝑇₃−2𝑄₅+ℎ𝑄₅𝑃₆⁻¹𝑄^𝑇₅−𝐹₃𝑅₁𝑒(𝑡)+2𝑒^𝑇(𝑡)𝑋₁₁−𝐹₁𝑊₁+𝐹₂𝑊₂−𝑄₁𝑒̇(𝑡)−2𝑒^𝑇(𝑡)𝑋₁₂+𝑄₁𝐷𝑒(𝑡−𝛿)+2𝑒^𝑇(𝑡)𝑋₂₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠+2𝑒^𝑇(𝑡)𝑄₃𝑒(𝑡−𝜏(𝑡))+2𝑒^𝑇(𝑡)𝑄₁𝐾+𝑄₅𝑒𝑡−𝛾(𝑡)+2𝑒^𝑇(𝑡)𝑄₁𝐴+𝐹₄𝑅₁𝑔(𝑒(𝑡))+2𝑒^𝑇(𝑡)𝑄₁𝐵𝑔(𝑒(𝑡−𝜏(𝑡)))+2𝑒^𝑇(𝑡)𝑄₁𝐶^𝑡_{𝑡−𝜎(𝑡)}𝑔(𝑒(𝑠+𝑒̇))𝑑𝑠_𝑇(𝑡)𝜏𝑃₄+ℎ𝑃₆−2𝑄₁𝑒̇(𝑡)−2𝑒^𝑇̇(𝑡)𝑄₁𝐷𝑒(𝑡−𝛿)+2𝑒^𝑇̇(𝑡)𝑋₁₂×^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠+2𝑒^𝑇̇(𝑡)𝑄₁(𝑡)𝑒̇𝐾𝑒𝑡−𝛾+2^𝑇(𝑡)𝑊₁−𝑊₂+𝑄₁𝐴𝑔(𝑒(𝑡𝑒̇))+2^𝑇(𝑡)𝑄₁𝐵𝑔(𝑒(𝑡−𝜏(𝑡)))+2𝑒^𝑇̇(𝑡)𝑄₁𝐶×^𝑡_{(𝑡)𝑡−𝜎}𝑔(𝑒(𝑠))𝑑𝑠−𝑒^𝑇(𝑡−𝛿)𝑃₁𝑒(𝑡−𝛿)−2𝑒^𝑇(𝑡−𝛿)𝑋₂₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠−^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠^𝑇×𝑃₂^𝑡_𝑡−𝛿𝑒(𝑠)𝑑𝑠+𝑒^𝑇((𝑡×𝑡−𝜏))−2𝑄₂+𝜏𝑄₂𝑃₄⁻¹𝑄^𝑇₂−𝐹₃𝑅₂𝑒(𝑡−𝜏(𝑡))+2𝑒^𝑇(𝑡−𝜏(𝑡))𝑄₂𝑒(𝑡−𝜏)+2𝑒^𝑇(𝑡−𝜏(𝑡))×𝐹₄𝑅₂𝑔(𝑒(𝑡−𝜏(𝑡)))−𝑒^𝑇(𝑡−𝜏)𝑃₃𝑒(𝑡−𝜏)+𝑒^𝑇𝑡−𝛾(𝑡)−2𝑄₄+ℎ𝑄₄𝑃₆⁻¹𝑄^𝑇₄𝑒𝑡−𝛾(𝑡)+2𝑒^𝑇𝑡−𝛾(𝑡)𝑄₂𝑒(𝑡−ℎ)−𝑒^𝑇(𝑡−ℎ)𝑃₅𝑒(𝑡−ℎ)+𝑔^𝑇(𝑒(𝑡))𝜎²𝑃₇−𝑅₁𝑔(𝑒(𝑡))−𝑔^𝑇(𝑒(𝑡−𝜏(𝑡)))𝑅₂𝑔(𝑒((𝑡−𝑡−𝜏)))_{𝑡(𝑡)𝑡−𝜎}𝑔(𝑒(𝑠))𝑑𝑠^𝑇𝑃₇^𝑡_{(𝑡)𝑡−𝜎}𝑔(𝑒(𝑠))𝑑𝑠=𝜉^𝑇(𝑡)Π𝜉(𝑡), (32) where 𝜉(𝑡)=(𝑒^𝑇(𝑡),𝑒^𝑇̇(𝑡),𝑒^𝑇(𝑡−𝛿),∫^𝑡_𝑡−𝛿𝑒^𝑇(𝑠)𝑑𝑠,𝑒^𝑇(𝑡−𝜏(𝑡)),𝑒^𝑇(𝑡−𝜏),𝑒^𝑇(𝑡−𝛾(𝑡)),𝑒^𝑇(𝑡−ℎ), 𝑔^𝑇(𝑒(𝑡)),𝑔^𝑇(𝑒(𝑡−𝜏(𝑡))),∫^𝑡_{𝑡−𝜎(𝑡)}𝑔^𝑇(𝑒(𝑠))𝑑𝑠)^𝑇 and ΠΠ=¹¹Ω₁₂Ω₁₃𝑋₂₂𝑄₃0Ω₁₇0Ω₁₉𝑄₁𝐵𝑄₁𝐶∗Ω₂₂−𝑄₁𝐷𝑋₁₂00𝑄₁𝐾0Ω₂₉𝑄₁𝐵𝑄₁𝐶∗∗−𝑃₁−𝑋₁₂0000000∗∗∗−𝑃₂0000000∗∗∗∗Π₅₅𝑄₂000𝐹₄𝑅₂0∗∗∗∗∗−𝑃₃00000∗∗∗∗∗∗Π₇₇𝑄₂000∗∗∗∗∗∗∗−𝑃₅000∗∗∗∗∗∗∗∗Ω₉₉00∗∗∗∗∗∗∗∗∗−𝑅₂0∗∗∗∗∗∗∗∗∗∗−𝑃₇, (33)
with Π₁₁=𝑋₁₂+𝑋^𝑇₁₂+𝑃₁+𝛿²𝑃₂+𝑃₃+𝑃₅−𝑄₃−𝑄^𝑇₃+𝜏𝑄₃𝑃₄⁻¹𝑄^𝑇₃−𝑄₅−𝑄^𝑇₅+ℎ𝑄₅𝑃₆⁻¹𝑄^𝑇₅−𝐹₃𝑅₁, Π₅₅=−𝑄₂−𝑄^𝑇₂+𝜏𝑄₂𝑃₄⁻¹𝑄^𝑇₂−𝐹₃𝑅₂, Π₇₇=−𝑄₄−𝑄^𝑇₄+ℎ𝑄₄𝑃₆⁻¹𝑄^𝑇₄.
By using Lemma 2 and noting 𝐾=𝑄₁⁻¹𝑍, it is easy to verify the equivalence of Π<0 and Ω<0. Thus, one can derive from (15) and (32) that 𝑉̇(𝑡)≤0, (34) which implies that error-state system (10) is global and asymptotically stable; that is, master system (1) and slave system (2) are synchronous. The proof is completed.

4. Numerical Example

To verify the effectiveness of the theoretical result of this paper, consider the following example.

Example 1. Consider master system (1) and slave system (2) with the following parameters:,,𝑓𝐷=1001,𝐴=1.80.1−4.32.9𝐵=−1.6−0.1−0.2−2.7,𝐶=−0.30.10.1−0.2₁(𝛼)=𝑓₂(𝛼)=tanh(𝛼),𝐽₁(𝑡)=𝐽₂(𝑡)(𝑡)||(𝑡)|(.=0,𝛿=0.3,𝜏=0.5sin𝑡,𝜎=0.2cos2𝑡)| (35)

The chaotic behaviors of master system (1) and slave system (2) with 𝑢(𝑡)=0 are given in Figure 1 and Figure 2, respectively, with the initial states chosen as 𝑥(𝑠)=(−0.1,0.1)^𝑇, 𝑦(𝑠)=(−0.5sin(23𝑡),−0.6cos(5𝑡))^𝑇, and 𝑠∈[−0.5,0].

Figure 1 

The chaotic behavior of master system (1).

Figure 2 

The chaotic behavior of slave system (2) with 𝑢(𝑡)=0.

It can be verified that assumptions (H1) and (H2) are satisfied, and 𝐹₁=0, 𝐹₂=𝐼, 𝐹₃=0, 𝐹₄=diag{0.5,0.5}, 𝜏=0.5, and 𝜎=0.2.

Further, the sampling period is taken as ℎ=0.7; by using the MATLAB LMI Control Toolbox, a solution to the LMIs in (14)-(15) is found as follows: 𝑃₁=10⁻⁸,𝑃0.5575−0.0458−0.04580.0813₂=10⁻⁷,𝑃0.1589−0.0521−0.05210.1440₃=10⁻⁸,𝑃0.7818−0.0275−0.02750.0804₄=10⁻⁹,𝑃0.5382−0.0061−0.00610.0069₅=10⁻⁸,𝑃0.1046−0.0257−0.02570.0731₆=10⁻⁸,𝑃0.2393−0.0215−0.02150.0690₇=10⁻⁷,𝑄0.3430−0.0452−0.04520.3334₁=10⁻⁹,𝑄0.3710−0.0167−0.00640.0142₂=10⁻⁹,𝑄0.9695−0.0121−0.01210.0465₃=10⁻⁸,𝑄0.1136−0.0020−0.00130.0066₄=10⁻⁷,𝑄0.4169−0.0372−0.04030.1167₅=10⁻⁷,0.2032−0.0197−0.01990.0622𝑍=10⁻⁹,𝑊−0.37650.04560.0973−0.1631₁=10⁻¹⁰,𝑊0.3365000.3390₂=10⁻⁹,𝑅0.7363000.0811₁=10⁻⁸,𝑅0.6542000.4249₂=10⁻⁸,𝑋0.2055000.0173₁₁=10⁻¹⁰,𝑋0.9837−0.2378−0.23780.5765₁₂=10⁻⁹,𝑋−0.08130.03400.0424−0.1056₂₂=10⁻⁸.0.2687−0.0933−0.09330.2625 (36) Subsequently, we can obtain from 𝐾=𝑄₁⁻¹𝑍 that 𝐾=−0.7228−0.39946.5033−11.6367. (37) According to Theorem 3, master system (1) and slave system (2) are synchronous under sampled-data controller (6). Figure 3 depicts the synchronization errors of state variables between master system (1) and slave system (2). The numerical simulations clearly verify the effectiveness of the developed sampled-data control approach in the synchronization of two chaotic neural networks with discrete and distributed time-varying delays as well as leakage delay.

(a)

(b)

Figure 3 

State trajectory of 𝑒1(𝑡) and 𝑒2(𝑡) of error system (8).

5. Conclusions

In this paper, we have dealt with the synchronization problems for chaotic neural networks with leakage delay and both discrete and distributed time-varying delays. Based on the sampled-data control techniques, Lyapunov stability theory, and the matrix inequality techniques, a delay-dependent criterion sufficient condition has been developed to guarantee synchronization of the considered coupled neural networks. An example has been provided to demonstrate the effectiveness of the proposed criterion since the feasible solutions to the given LMIs criterion in this paper have been found.

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 anonymous reviewers for their valuable suggestions and comments which have led to a much improved paper. This work was supported by the National Natural Science Foundation of China under Grants 61273021, 60974132, 11172247, and 51208538 and in part by the Natural Science Foundation Project of CQ cstc2013jjB40008.