Robust Synchronization Criterion for Coupled Stochastic Discrete-Time Neural Networks with Interval Time-Varying Delays, Leakage Delay, and Parameter Uncertainties

Abstract

The purpose of this paper is to investigate a delay-dependent robust synchronization analysis for coupled stochastic discrete-time neural networks with interval time-varying delays in networks coupling, a time delay in leakage term, and parameter uncertainties. Based on the Lyapunov method, a new delay-dependent criterion for the synchronization of the networks is derived in terms of linear matrix inequalities (LMIs) by constructing a suitable Lyapunov-Krasovskii’s functional and utilizing Finsler’s lemma without free-weighting matrices. Two numerical examples are given to illustrate the effectiveness of the proposed methods.

1. Introduction

In recent years, the problem of synchronization of coupled neural networks which is one of hot research fields of complex networks has been a challenging issue due to its potential applications such as physics, information sciences, biological systems, and so on. Here, complex networks, which are a set of interconnected nodes with specific dynamics, have been studied from various fields of science and engineering such as the World Wide Web, social networks, electrical power grids, global economic markets, and so on. Many mathematical models were proposed to describe various complex networks [1, 2]. Also, in the real applications of systems, there exists naturally time delay due to the finite information processing speed and the finite switching speed of amplifiers. It is well known that time delay often causes undesirable dynamic behaviors such as performance degradation and instability of the systems. So, some sufficient conditions for synchronization of coupled neural networks with time delay have been proposed in [3–5]. Moreover, the synchronization of delayed systems was applied in practical systems such as secure communication [6]. Furthermore, these days, most systems use digital computers (usually microprocessor or microcontrollers) with the necessary input/output hardware to implement the systems. The fundamental character of the digital computer is that it takes compute answers at discrete steps. Therefore, discrete-time modeling with time delay plays an important role in many fields of science and engineering applications. In this regard, various approaches to synchronization stability criterion for discrete-time complex networks with time delay have been investigated in the literature [7–9].

On the other hand, in implementation of many practical systems such as aircraft, chemical and biological systems, and electric circuits, there exist occasionally stochastic perturbations. It is not less important than the time delay as a considerable factor affecting dynamics in the fields of science and engineering applications. Therefore, the study on the problems for various forms of stochastic systems with time-delay has been addressed. For more details, see the literature [10–13] and references therein. Furthermore, on the problem of synchronization of coupled stochastic neural networks with time delay, various researches have been conducted [14–17]. Li and Yue [14] studied the synchronization stability problem for a class of complex networks with Markovian jumping parameters and mixed time delays. The model considered in [14] has stochastic coupling terms and stochastic disturbances to reflect more realistic dynamical behaviors of the complex networks that are affected by noisy environment. In [15], by utilizing novel Lyapunov-Krasovskii's functional with both lower and upper delay bounds, the synchronization criteria for coupled stochastic discrete-time neural networks with mixed delays were presented. Tang and Fang [16] derived several sufficient conditions for the synchronization of delayed stochastically coupled fuzzy cellular neural networks with mixed delays and uncertain hybrid coupling based on adaptive control technique and some stochastic analysis methods. In [17], by using Kronecker product as an effective tool, robust synchronization problem of coupled stochastic discrete-time neural networks with time-varying delay was investigated. Moreover, Song [18–20] addressed synchronization problem for the array of asymmetric, chaotic, and coupled connected neural networks with time-varying delay or nonlinear coupling. Also, in [21], robust exponential stability analysis of uncertain delayed neural networks with stochastic perturbation and impulse effects was investigated.

Very recently, a time delay in leakage term of the systems is being put to use in the problem of stability for neural networks as a considerable factor affecting dynamics for the worse in the systems [22, 23]. Li et al. [22] studied the existence and uniqueness of the equilibrium point of recurrent neural networks with time delays in the leakage term. By use of the topological degree theory, delay-dependent stability conditions of neural networks of neutral type with time delays in the leakage term were proposed in [23]. Unfortunately, to the best of authors’ knowledge, delay-dependent synchronization analysis of coupled stochastic discrete-time neural networks with time-varying delay in network coupling and leakage delay has not been investigated yet. Thus, by attempting the synchronization analysis for the model of coupled stochastic discrete-time neural networks with time delay in the leakage term, the model for coupled neural networks and its applications are closed to the practical networks. Here, delay-dependent analysis has been paid more attention than delay-independent one because the sufficient conditions for delay-dependent analysis make use of the information on the size of time delay [24]. That is, the former is generally less conservative than the latter.

Motivated by the above discussions, the problem of a new delay-dependent robust synchronization criterion for coupled stochastic discrete-time neural networks with interval time-varying delays in network coupling, the time delay in leakage term, and parameter uncertainties is considered for the first time. The coupled stochastic discrete-time neural networks are represented as a simple mathematical model by the use of Kronecker product technique. Then, by construction of a suitable Lyapunov-Krasovskii's functional and utilization of Finsler’s lemma without free-weighting matrices, a new synchronization criterion is derived in terms of LMIs. The LMIs can be formulated as convex optimization algorithms which are amenable to computer solution [25]. In order to utilize Finsler’s lemma as a tool of getting less conservative synchronization criteria on the number of decision variables, it should be noted that a new zero equality from the constructed mathematical model is devised. The concept of scaling transformation matrix will be utilized in deriving zero equality of the method. In [26], the effectiveness of Finsler’s lemma was illustrated by the improved passivity criteria of uncertain neural networks with time-varying delays. Finally, two numerical examples are included to show the effectiveness of the proposed method.

Notation. ℝ^𝑛 is the 𝑛-dimensional Euclidean space, and ℝ^𝑚×𝑛 denotes the set of all 𝑚×𝑛 real matrices. For symmetric matrices 𝑋 and 𝑌, 𝑋>𝑌 (resp., 𝑋≥𝑌) means that the matrix 𝑋−𝑌 is positive definite (resp., nonnegative). 𝑋^⟂ denotes a basis for the null-space of 𝑋. 𝐼_𝑛 and 0_𝑛 and 0_𝑚×𝑛 denote 𝑛×𝑛 identity matrix and 𝑛×𝑛 and 𝑚×𝑛 zero matrices, respectively. ‖⋅‖ refers to the Euclidean vector norm or the induced matrix norm. 𝜆_max(⋅) means the maximum eigenvalue of a given square matrix. diag{⋯} denotes the block diagonal matrix. ⋆ represents the elements below the main diagonal of a symmetric matrix. Let (Ω,ℱ,{𝐹_𝑡}_𝑡≥0,𝒫) be complete probability space with a filtration {𝐹_𝑡}_𝑡≥0 satisfying the usual conditions (i.e., it is right continuous and ℱ₀ contains all 𝒫-pull sets). 𝔼{⋅} stands for the mathematical expectation operator with respect to the given probability measure 𝒫.

2. Problem Statements

Consider the following discrete-time delayed neural networks: 𝑦(𝑘+1)=(𝐴+Δ𝐴)𝑦(𝑘−𝜏)+𝑊₁+Δ𝑊₁𝑔𝑦(𝑘)+𝑊₂+Δ𝑊₂𝑔𝑦(𝑘−ℎ(𝑘))+𝑏, (1) where 𝑛 denotes the number of neurons in a neural network, 𝑦(⋅)=[𝑦₁(⋅),…,𝑦_𝑛(⋅)]^𝑇∈ℝ^𝑛 is the neuron state vector, 𝑔(⋅)=[𝑔₁(⋅),…,𝑔_𝑛(⋅)]^𝑇∈ℝ^𝑛 denotes the neuron activation function vector, 𝑏=[𝑏₁,…,𝑏_𝑛]^𝑇∈ℝ^𝑛 means a constant external input vector, 𝐴=diag{𝑎₁,…,𝑎_𝑛}∈ℝ^𝑛×𝑛(0<𝑎_𝑞<1,𝑞=1,…,𝑛) is the state feedback matrix, 𝑊_𝑞∈ℝ^𝑛×𝑛  (𝑞=1,2) are the connection weight matrices, and Δ𝐴 and Δ𝑊_𝑞(𝑞=1,2) are the parameter uncertainties of the form Δ𝐴,Δ𝑊₁,Δ𝑊₂=𝐷𝐹(𝑘)𝐸_𝑎,𝐸₁,𝐸₂, (2) where 𝐹(𝑘) is a real uncertain matrix function with Lebesgue measurable elements satisfying 𝐹^𝑇(𝑘)𝐹(𝑘)≤𝐼. (3) The delays ℎ(𝑘) and 𝜏 are interval time-varying delays and leakage delay, respectively, satisfying 0<ℎ_𝑚≤ℎ(𝑘)≤ℎ_𝑀,0<𝜏, (4) where ℎ_𝑚 and ℎ_𝑀 are positive integers.

The neuron activation functions, 𝑔_𝑝(𝑦_𝑝(⋅))(𝑝=1,…,𝑛), are assumed to be nondecreasing, bounded, and globally Lipschitz; that is, 𝑙⁻_𝑝≤𝑔_𝑝𝜉_𝑝−𝑔_𝑝𝜉_𝑞𝜉_𝑝−𝜉_𝑞≤𝑙⁺_𝑝,∀𝜉_𝑝,𝜉_𝑞∈ℝ,𝜉_𝑝≠𝜉_𝑞, (5) where 𝑙⁻_𝑝 and 𝑙⁺_𝑝 are constant values.

For simplicity, in stability analysis of the network (1), the equilibrium point 𝑦^∗=[𝑦^∗₁,…,𝑦^∗_𝑛]^𝑇 is shifted to the origin by the utilization of the transformation 𝑦(⋅)=𝑦(⋅)−𝑦^∗, which leads the network (1) to the following form: 𝑦(𝑘+1)=(𝐴+Δ𝐴)𝑦(𝑘−𝜏)+𝑊₁+Δ𝑊₁𝑔𝑦(𝑘)+𝑊₂+Δ𝑊₂𝑔𝑦(𝑘−ℎ(𝑘)), (6) where 𝑦(⋅)=[𝑦₁(⋅),…,𝑦_𝑛(⋅)]^𝑇∈ℝ^𝑛 is the state vector of the transformed network, and 𝑔(𝑦(⋅))=[𝑔₁(𝑦₁(⋅)),…,𝑔_𝑛(𝑦_𝑛(⋅))]^𝑇 is the transformed neuron activation function vector with 𝑔_𝑞(𝑦_𝑞(⋅))=𝑔_𝑞(𝑦_𝑞(⋅)+𝑦^∗_𝑞)−𝑔_𝑞(𝑦^∗_𝑞)  (𝑞=1,…,𝑛) satisfies, from (5), 𝑙⁻_𝑝≤𝑔_𝑝(𝜉_𝑝)/𝜉_𝑝≤𝑙⁺_𝑝,∀𝜉_𝑝≠0, which is equivalent to 𝑔_𝑝𝑦_𝑝(𝑘)−𝑙⁻_𝑝𝑦_𝑝(𝑘)𝑔_𝑝𝑦_𝑝(𝑘)−𝑙⁺_𝑝𝑦_𝑝(𝑘)≤0. (7)

In this paper, a model of coupled stochastic discrete-time neural networks with interval time-varying delays in network coupling, leakage delay, and parameter uncertainties is considered as 𝑦_𝑖(𝑘+1)=(𝐴+Δ𝐴)𝑦_𝑖(𝑘−𝜏)+𝑊₁+Δ𝑊₁𝑔𝑦_𝑖(𝑘)+𝑊₂+Δ𝑊₂𝑔𝑦_𝑖((𝑘+𝑘−ℎ))_𝑁𝑗=1𝑔_𝑖𝑗Γ𝑦_𝑗(𝑘−ℎ(𝑘))1+𝜔₁(𝑘)+𝜎_𝑖𝑘,𝑦_𝑖(𝑘),𝑦_𝑖(𝑘−ℎ(𝑘))𝜔₂(𝑘),𝑖=1,2,…,𝑁, (8) where 𝑁 is the number of couple nodes, 𝑦_𝑖(𝑘)=[𝑦_𝑖1(𝑘),…,𝑦_𝑖𝑛(𝑘)]^𝑇∈ℝ^𝑛 is the state vector of the 𝑖th node, Γ∈ℝ^𝑛×𝑛 is the constant inner-coupling matrix of nodes, which describe the individual coupling between the subnetworks, 𝐺=[𝑔_𝑖𝑗]_𝑁×𝑁 is the outer-coupling matrix representing the coupling strength and the topological structure of the network satisfies the diffusive coupling connections 𝑔_𝑖𝑗=𝑔_𝑗𝑖,𝑔≥0𝑖≠𝑗_𝑖𝑖=−^𝑁_{𝑗=1,𝑖≠𝑗}𝑔_𝑖𝑗𝑖,𝑗=1,2,…,𝑁, (9) and 𝜔_𝑞(𝑘)(𝑞=1,2) are 𝑚-dimensional Wiener processes (Brownian Motion) on (Ω,ℱ,{𝐹_𝑡}_𝑡≥0,𝒫) which satisfy 𝔼𝜔_𝑞(𝑘𝔼𝜔)=0,_2𝑞(𝑘𝔼𝜔)=1,_𝑞(𝑖)𝜔_𝑞𝑗=0𝑖≠𝑗. (10) Here, 𝜔₁(𝑘) and 𝜔₂(𝑘), which are mutually independent, are the coupling strength disturbance and the system noise, respectively. And the nonlinear uncertainties 𝜎_𝑖(⋅,⋅,⋅)∈ℝ^𝑛×𝑚(𝑖=1,…,𝑁) are the noise intensity functions satisfying the Lipschitz condition and the following assumption: 𝜎^𝑇_𝑖𝑘,𝑦_𝑖(𝑘),𝑦_𝑖(𝑘−ℎ(𝑘))𝜎_𝑖𝑘,𝑦_𝑖(𝑘),𝑦_𝑖((𝑘≤𝐻𝑘−ℎ))₁𝑦_𝑖(𝑘)²+𝐻₂𝑦_𝑖(𝑘−ℎ(𝑘))², (11) where 𝐻_𝑞(𝑞=1,2) are constant matrices with appropriate dimensions.

Remark 1. According to the graph theory [27], the outer-coupling matrix 𝐺 is called the negative Laplacian matrix of undirected graph. A physical meaning of the matrix 𝐺 is the bilateral connection between node 𝑖 and 𝑗. If the matrix 𝐺 cannot satisfy symmetric, the unidirectional connection between nodes 𝑖 and 𝑗 is expressed. At this time, the matrix 𝐺 is called the negative Laplacian matrix of directed graph. Therefore, new numerical model and strong sufficient condition guaranteed to the stability for networks are needed. Moreover, in order to analyze the consensus problem for multiagent systems, the Laplacian matrix of directed graph was used [28].
For the convenience of stability analysis for the network (8), the following Kronecker product and its properties are used.

Lemma 2 (see [29]). Let ⊗ denote the notation of Kronecker product. Then, the following properties of Kronecker product are easily established:(i)(𝛼𝐴)⊗𝐵=𝐴⊗(𝛼𝐵),(ii)(𝐴+𝐵)⊗𝐶=𝐴⊗𝐶+𝐵⊗𝐶,(iii)(𝐴⊗𝐵)(𝐶⊗𝐷)=(𝐴𝐶)⊗(𝐵𝐷),(iv)(𝐴⊗𝐵)^𝑇=𝐴^𝑇⊗𝐵^𝑇.

Let us define 𝑥(𝑘)=𝑦₁(𝑘),…,𝑦_𝑁(𝑘)^𝑇,𝑓(𝑥(𝑘))=𝑔𝑦₁(𝑘),…,𝑔𝑦_𝑁(𝑘)^𝑇,𝜎(𝑡)=𝜎₁(⋅,⋅,⋅),…,𝜎_𝑁(⋅,⋅,⋅)^𝑇. (12) Then, with Kronecker product in Lemma 2, the network (8) can be represented as 𝑥(𝑘+1)=𝐼_𝑁⊗𝐴(𝑘)𝑥(𝑘−𝜏)+𝐼_𝑁⊗𝑊₁(𝑘)𝑓(𝑥(𝑘+𝐼))_𝑁⊗𝑊₂(𝑘)𝑓(𝑥((𝑘+()𝑥((𝑘𝑘−ℎ)))𝐺⊗Γ𝑘−ℎ))1+𝜔₁(𝑘)+𝜎(𝑡)𝜔₂(𝑡), (13) where 𝐴(𝑘)=𝐴+𝐷𝐹(𝑘)𝐸_𝑎, 𝑊₁(𝑘)=𝑊₁+𝐷𝐹(𝑘)𝐸₁, and 𝑊₂(𝑘)=𝑊₂+𝐷𝐹(𝑘)𝐸₂.

In addition, for stability analysis, (13) can be rewritten as follows: 𝑥(𝑘+1)=𝜂(𝑘)+𝜚(𝑘)𝜔(𝑘), (14) where 𝜂(𝑘)=𝐼_𝑁⊗𝐴𝑥(𝑘−𝜏)+𝐼_𝑁⊗𝑊₁𝑓(𝑥(𝑘+𝐼))_𝑁⊗𝑊₂𝑓(𝑥((𝑘+()𝑥((𝑘+𝐼𝑘−ℎ)))𝐺⊗Γ𝑘−ℎ))_𝑁𝑝(𝑘),𝑝(𝑘)=𝐼⊗𝐷_𝑁(𝑘)𝑞(𝑘),𝑞(𝑘)=𝐼⊗𝐹_𝑁⊗𝐸_𝑎𝑥(𝑘−𝜏)+𝐼_𝑁⊗𝐸₁𝑓(𝑥(𝑘+𝐼))_𝑁⊗𝐸₂𝑓(𝑥((𝑘,𝜚(𝑘)=)𝑥((𝑘(𝑘,𝜔𝑡−ℎ)))[(𝐺⊗Γ𝑘−ℎ)),𝜎)]_𝑇(𝑘)=𝜔^𝑇₁(𝑘),𝜔^𝑇₂(𝑘). (15)

The aim of this paper is to investigate the delay-dependent synchronization stability analysis of the network (14) with interval time-varying delays in network coupling, leakage delay, and parameter uncertainties. In order to do this, the following definition and lemmas are needed.

Definition 3 (see [7]). The network (8) is said to be asymptotically synchronized if the following condition holds: lim_𝑡→∞𝑥_𝑖(𝑘)−𝑥_𝑗(𝑘)=0,𝑖,𝑗=1,2,…,𝑁. (16)

Lemma 4 (see [3]). Let 𝑈=[𝑢_𝑖𝑗]_𝑁×𝑁, 𝑃∈ℝ^𝑛×𝑛, 𝑥^𝑇=[𝑥₁,𝑥₂,…,𝑥_𝑛]^𝑇, and 𝑦^𝑇=[𝑦₁,𝑦₂,…,𝑦_𝑛]^𝑇. If 𝑈=𝑈^𝑇 and each row sum of  𝑈 is zero, then 𝑥^𝑇(𝑈⊗𝑃)𝑦=−_{1≤𝑖<𝑗≤𝑁}𝑢_𝑖𝑗𝑥_𝑖−𝑥_𝑗^𝑇𝑃𝑦_𝑖−𝑦_𝑗. (17)

Lemma 5 (see [30]). For any constant matrix 0<𝑀=𝑀^𝑇∈ℝ^𝑛×𝑛, integers ℎ_𝑚 and ℎ_𝑀 satisfying 1≤ℎ_𝑚≤ℎ_𝑀, and vector function 𝑥(𝑘)∈ℝ^𝑛, the following inequality holds: −ℎ_𝑀−ℎ_𝑚+1^ℎ𝑀_𝑘=ℎ𝑚𝑥^𝑇(𝑘)𝑀𝑥(𝑘)≤−^ℎ𝑀_𝑘=ℎ𝑚𝑥(𝑘)^𝑇𝑀^ℎ𝑀_𝑘=ℎ𝑚𝑥(𝑘). (18)

Lemma 6 (see [31] (Finsler's lemma)). Let 𝜁∈ℝ^𝑛, Φ=Φ^𝑇∈ℝ^𝑛×𝑛, and Υ∈ℝ^𝑚×𝑛 such that 𝑟𝑎𝑛𝑘(Υ)<𝑛. The following statements are equivalent: (i)𝜁^𝑇Φ𝜁<0, ∀Υ𝜁=0, 𝜁≠0, (ii)Υ^⟂𝑇ΦΥ^⟂<0.

3. Main Results

In this section, a new synchronization criterion for the network (14) will be proposed. For the sake of simplicity on matrix representation, 𝑒_𝑖(𝑖=1,…,9)∈ℝ^9𝑛×𝑛 are defined as block entry matrices (e.g., 𝑒₂=[0_𝑛,𝐼_𝑛,0_𝑛,0_𝑛,0_𝑛,0_𝑛,0_𝑛,0_𝑛,0_𝑛]^𝑇). The notations of several matrices are defined as follows: 𝜁^𝑇(𝑘)=𝑥^𝑇(𝑘),𝑥^𝑇(𝑘−𝜏),𝑥^𝑇𝑘−ℎ_𝑚,𝑥^𝑇((𝑘,𝑥𝑘−ℎ))_𝑇𝑘−ℎ_𝑀,𝜂(𝑘)−𝑥(𝑘)^𝑇,𝑓^𝑇(𝑥(𝑘,𝑓))_𝑇(𝑥(𝑘−ℎ(𝑘))),𝑝^𝑇(𝑘,𝑧_𝑖𝑗)(𝑘)=𝑥_𝑖(𝑘)−𝑥_𝑗(𝑘),𝑓𝑧_𝑖𝑗(𝑘)=𝑓𝑥_𝑖(𝑘)−𝑓𝑥_𝑗(𝑘,𝜂_𝑖𝑗)(𝑘)=𝜂_𝑖(𝑘)−𝜂_𝑗(𝑘),𝑝_𝑖𝑗(𝑘)=𝑝_𝑖(𝑘)−𝑝_𝑗(𝑘),𝜁^𝑇_𝑖𝑗(𝑘)=𝑧^𝑇_𝑖𝑗(𝑘),𝑧^𝑇_𝑖𝑗(𝑘−𝜏),𝑧^𝑇_𝑖𝑗𝑘−ℎ_𝑚,𝑧^𝑇_𝑖𝑗((𝑘,𝑧𝑘−ℎ))_𝑇𝑖𝑗𝑘−ℎ_𝑀,𝜂_𝑖𝑗(𝑘)−𝑧_𝑖𝑗(𝑘)^𝑇,𝑓^𝑇𝑧_𝑖𝑗(𝑘),𝑓^𝑇𝑧_𝑖𝑗(𝑘−ℎ(𝑘)),𝑝^𝑇_𝑖𝑗(𝑘,Υ_𝑖𝑗)=−𝐼_𝑛,𝐴,0_𝑛,−𝑁𝑔_𝑖𝑗Γ,0_𝑛,−𝐼_𝑛,𝑊₁,𝑊₂,𝐷,Σ=𝑃+ℎ²_𝑚𝑅₁+ℎ_𝑀−ℎ_𝑚²𝑅₂+𝜏²𝑆₂,Ξ₁=𝑒₁𝑃𝑒^𝑇₆+𝑒₆𝑃𝑒^𝑇₁+𝑒₆𝑃𝑒^𝑇₆,Ξ₂=𝑒₁𝑄₁𝑒^𝑇₁−𝑒₃𝑄₁−𝑄₂𝑒^𝑇₃−𝑒₅𝑄₂𝑒^𝑇₅,Ξ₃=𝑒₆ℎ²_𝑚𝑅₁+ℎ_𝑀−ℎ_𝑚²𝑅₂𝑒^𝑇₆−𝑒₁−𝑒₃𝑅₁𝑒₁−𝑒₃^𝑇−𝑒₃−𝑒₄𝑅₂𝑒₃−𝑒₄^𝑇−𝑒₄−𝑒₅𝑅₂𝑒₄−𝑒₅^𝑇−𝑒₃−𝑒₄𝑇^𝑇𝑒₄−𝑒₅^𝑇−𝑒₄−𝑒₅𝑇𝑒₃−𝑒₄^𝑇,Ξ₄=𝑒₁𝑆₁𝑒^𝑇₁−𝑒₂𝑆₁𝑒^𝑇₂+𝑒₆𝜏²𝑆₂𝑒^𝑇₆−𝑒₁−𝑒₂𝑆₂𝑒₁−𝑒₂^𝑇,Ξ₅=𝑒₄𝑁^𝑁_𝑙=1𝑔_𝑖𝑙𝑔_𝑙𝑗Γ^𝑇ΣΓ𝑒^𝑇₄+𝑒₁𝜌𝐻^𝑇₁𝐻₁𝑒^𝑇₁+𝑒₄𝜌𝐻^𝑇₂𝐻₂𝑒^𝑇₄,Ξ₆=−𝑒₁2𝐿_𝑚𝐷₁𝐿_𝑝𝑒^𝑇₁+𝑒₁𝐿_𝑚+𝐿_𝑝𝐷₁𝑒^𝑇₇+𝑒₁𝐿_𝑚+𝐿_𝑝𝐷₁𝑒^𝑇₇^𝑇−𝑒₇2𝐷₁𝑒^𝑇₇−𝑒₄2𝐿_𝑚𝐷₂𝐿_𝑝𝑒^𝑇₄+𝑒₄𝐿_𝑚+𝐿_𝑝𝐷₂𝑒^𝑇₈+𝑒₄𝐿_𝑚+𝐿_𝑝𝐷₂𝑒^𝑇₈^𝑇−𝑒₈2𝐷₂𝑒^𝑇₈,Ξ₇=−𝑒₉𝜖𝐼_𝑛𝑒^𝑇₉,Ψ=0_𝑛,𝐸_𝑎,0_𝑛,0_𝑛,0_𝑛,0_𝑛,𝐸₁,𝐸₂,0_𝑛. (19)

Then, the main result of this paper is presented as follows.

Theorem 7. For given positive integers ℎ_𝑚, ℎ_𝑀 and 𝜏, diagonal matrices 𝐿_𝑚=diag{𝑙⁻₁,…,𝑙⁻_𝑛} and 𝐿_𝑝=diag{𝑙⁺₁,…,𝑙⁺_𝑛}, the network (14) is asymptotically synchronized for ℎ_𝑚≤ℎ(𝑘)≤ℎ_𝑀, if there exist positive scalars 𝜌, 𝜖, positive definite matrices 𝑃, 𝑄₁, 𝑄₂, 𝑅₁, 𝑅₂, 𝑆₁, 𝑆₂, positive diagonal matrices 𝐷₁, 𝐷₂, and any matrix 𝑇 satisfying the following LMIs for 1≤𝑖<𝑗≤𝑁: Σ−𝜌𝐼_𝑛𝑅≤0,₂𝑇⋆𝑅₂Υ≥0,𝑗−𝑖_𝑖𝑗^⟂0_9𝑛×𝑛0_𝑛×8𝑛𝐼_𝑛^𝑇7_𝑙∑₌₁Ξ_𝑙𝜖Ψ^𝑇⋆−𝜖𝐼_𝑛×𝑗−𝑖Υ_𝑖𝑗^⟂0_9𝑛×𝑛0_𝑛×8𝑛𝐼_𝑛<0, (20)(21)(22) where Σ, Υ_𝑖𝑗, Ξ_𝑙  (𝑙=1,…,7), and Ψ are defined in (19).

Proof. Define a matrix 𝑈 as 𝑈=𝑢_{𝑖𝑗𝑁×𝑁}=𝑁−1−1⋯−1−1𝑁−1−1⋮⋮−1⋱−1−1⋯−1𝑁−1 (23) and the forward difference of 𝑥(𝑘) and 𝑉(𝑘) as (𝑘)()(𝑘)(𝑘)(𝑘)(𝑘)𝜔(𝑡),(𝑘)()(𝑘).Δ𝑥=𝑥𝑘+1−𝑥=𝜂−𝑥+𝜚Δ𝑉=𝑉𝑘+1−𝑉 (24) Let us consider the following Lyapunov-Krasovskii's functional candidate as 𝑉(𝑘)=𝑉₁(𝑘)+𝑉₂(𝑘)+𝑉₃(𝑘)+𝑉₄(𝑘), (25) where 𝑉₁(𝑘)=𝑥^𝑇(𝑘)𝑥(𝑘),𝑉)(𝑈⊗𝑃₂(𝑘)=^𝑘−1_{𝑠=𝑘−ℎ𝑚}𝑥^𝑇(𝑠)𝑈⊗𝑄₁𝑥(𝑠)+^{𝑘−ℎ𝑚−1}_{𝑠=𝑘−ℎ𝑀}𝑥^𝑇(𝑠)𝑈⊗𝑄₂𝑥(𝑠),𝑉₃(𝑘)=ℎ_𝑚⁻¹_{𝑠=−ℎ𝑚}^𝑘−1_{𝑢=𝑘+𝑠}Δ𝑥^𝑇(𝑢)𝑈⊗𝑅₁(𝑢)+ℎΔ𝑥_𝑀−ℎ_𝑚^{−ℎ𝑚−1}_{𝑠=−ℎ𝑀}^𝑘−1_{𝑢=𝑘+𝑠}Δ𝑥^𝑇(𝑢)𝑈⊗𝑅₂(𝑢),𝑉Δ𝑥₄(𝑘)=^𝑘−1_{𝑠=𝑘−𝜏}𝑥^𝑇(𝑠)𝑈⊗𝑆₁𝑥(𝑠)+𝜏⁻¹_𝑠=−𝜏^𝑘−1_{𝑢=𝑘+𝑠}Δ𝑥^𝑇(𝑢)𝑈⊗𝑆₂Δ𝑥(𝑢). (26) The mathematical expectation of Δ𝑉(𝑘) is calculated as follows: 𝔼Δ𝑉₁(𝑘𝑥)=𝔼^𝑇(𝑘+1)(𝑈⊗𝑃)𝑥(𝑘+1)−𝑥^𝑇(𝑘)𝑥(𝑘(𝑘)(𝑘)(𝑈⊗𝑃)=𝔼(Δ𝑥+𝑥))^𝑇(𝑈⊗𝑃)(Δ𝑥(𝑘)+𝑥(𝑘))−𝑥^𝑇(𝑘)(𝑈⊗𝑃)𝑥(𝑘)=𝔼Δ𝑥^𝑇(𝑘)(𝑈⊗𝑃)Δ𝑥(𝑘)+2Δ𝑥^𝑇(𝑘)𝑥(𝑘𝜂(𝑘)(𝑘))(𝑈⊗𝑃)=𝔼−𝑥^𝑇()𝜂(𝑘)(𝑘)+𝜚(𝑘)𝜔(𝑘𝑈⊗𝑃−𝑥)^𝑇(𝜚(𝑘)𝜔(𝑘𝜂(𝑘)(𝑘)𝑈⊗𝑃))+2−𝑥^𝑇()𝑥(𝑘𝜂(𝑘)(𝑘)𝑈⊗𝑃)=𝔼−𝑥^𝑇(𝑈⊗𝑃)𝜂(𝑘)−𝑥(𝑘)+𝑥^𝑇(𝑡−ℎ(𝑘))(𝐺⊗Γ)^𝑇(𝑈⊗𝑃)(𝐺⊗Γ)𝑥(𝑡−ℎ(𝑘))_Θ1+𝜎^𝑇(𝑘)(𝑈⊗𝑃_Ω1)𝜎(𝑘)+2𝜂(𝑘)−𝑥(𝑘)^𝑇()𝑥(𝑘,𝔼𝑈⊗𝑃)Δ𝑉₂(𝑘𝑥)=𝔼^𝑇(𝑘)𝑈⊗𝑄₁𝑥(𝑘)−𝑥^𝑇𝑘−ℎ_𝑚𝑈⊗𝑄₁−𝑄₂𝑥𝑘−ℎ_𝑚−𝑥^𝑇𝑘−ℎ_𝑀𝑈⊗𝑄₂𝑥𝑘−ℎ_𝑀,𝔼Δ𝑉₃(𝑘)=𝔼Δ𝑥^𝑇(𝑘)𝑈⊗ℎ²_𝑚𝑅₁+ℎ_𝑀−ℎ_𝑚²𝑅₂Δ𝑥(𝑘)−ℎ_𝑚^𝑘−1_{𝑠=𝑘−ℎ𝑚}Δ𝑥^𝑇(𝑠)𝑈⊗𝑅₁(𝑠)−ℎΔ𝑥_𝑀−ℎ_𝑚^{𝑘−ℎ𝑚−1}_{𝑠=𝑘−ℎ𝑀}Δ𝑥^𝑇(𝑠)𝑈⊗𝑅₂(𝑠)𝜂(𝑘)(𝑘)Δ𝑥=𝔼−𝑥^𝑇𝑈⊗ℎ²_𝑚𝑅₁+ℎ_𝑀−ℎ_𝑚²𝑅₂×𝜂(𝑘)(𝑘)+𝑥−𝑥_𝑇(𝑡−ℎ(𝑘))(𝐺⊗Γ)^𝑇×𝑈⊗ℎ²_𝑚𝑅₁+ℎ_𝑀−ℎ_𝑚²𝑅₂×(𝐺⊗Γ)𝑥(𝑡−ℎ(𝑘))_Θ2+𝜎^𝑇(𝑘)𝑈⊗ℎ²_𝑚𝑅₁+ℎ_𝑀−ℎ_𝑚²𝑅₂𝜎(𝑘)_Ω2−ℎ_𝑚^𝑘−1_{𝑠=𝑘−ℎ𝑚}Δ𝑥^𝑇(𝑠)𝑈⊗𝑅₁(𝑠)−ℎΔ𝑥_𝑀−ℎ_𝑚^{𝑘−ℎ𝑚−1}_{𝑠=𝑘−ℎ𝑀}Δ𝑥^𝑇(𝑠)𝑈⊗𝑅₂(𝑠),𝔼Δ𝑥Δ𝑉₄(𝑘𝑥)=𝔼^𝑇(𝑘)𝑈⊗𝑆₁𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏)𝑈⊗𝑆₁𝑥(𝑘−𝜏)+Δ𝑥^𝑇(𝑘)𝑈⊗𝜏²𝑆₂Δ𝑥(𝑘)−𝜏^𝑘−1_{𝑠=𝑘−𝜏}Δ𝑥^𝑇(𝑠)𝑈⊗𝑆₂(𝑠𝑥Δ𝑥)=𝔼^𝑇(𝑘)𝑈⊗𝑆₁𝑥(𝑘)−𝑥^𝑇(𝑘−𝜏)𝑈⊗𝑆₁𝑥()+𝜂(𝑘)(𝑘)𝑘−𝜏−𝑥^𝑇𝑈⊗𝜏²𝑆₂𝜂(𝑘)−𝑥(𝑘)+𝑥^𝑇(𝑡−ℎ(𝑘))(𝐺⊗Γ)^𝑇𝑈⊗𝜏²𝑆₂(𝐺⊗Γ)𝑥(𝑡−ℎ(𝑘))_Θ3+𝜎^𝑇(𝑘)𝑈⊗𝜏²𝑆₂𝜎(𝑘)_Ω3−𝜏^𝑘−1_{𝑠=𝑘−𝜏}Δ𝑥^𝑇(𝑠)𝑈⊗𝑆₂Δ𝑥(𝑠). (27) By Lemmas 4 and 5, the sum terms of 𝔼{Δ𝑉₃(𝑘)} are bounded as follows: −ℎ_𝑚^𝑘−1_{𝑠=𝑘−ℎ𝑚}Δ𝑥^𝑇(𝑠)𝑈⊗𝑅₁Δ𝑥(𝑠)≤−^𝑘−1_{𝑠=𝑘−ℎ𝑚}Δ𝑥(𝑠)^𝑇𝑈⊗𝑅₁^𝑘−1_{𝑠=𝑘−ℎ𝑚}Δ𝑥(𝑠)=−_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑘)𝑒^𝑇₁−𝑒^𝑇₃^𝑇𝑅₁𝑒^𝑇₁−𝑒^𝑇₃𝜁_𝑖𝑗(𝑘),−ℎ_𝑀−ℎ_𝑚^{𝑘−ℎ𝑚−1}_{𝑠=𝑘−ℎ𝑀}Δ𝑥^𝑇(𝑠)𝑈⊗𝑅₂Δ𝑥(𝑠)≤−^{𝑘−ℎ(𝑘)−1}_𝑠=𝑘∑_−ℎ𝑀Δ𝑥_{𝑘−ℎ𝑚−1}(𝑠)_(𝑘)∑_{𝑠=𝑘−ℎ}Δ𝑥(𝑠)^𝑇1𝛼_𝑘𝑈⊗𝑅₂0_𝑁𝑛0_𝑁𝑛11−𝛼_𝑘𝑈⊗𝑅₂×^{𝑘−ℎ(𝑘)−1}_𝑠=𝑘∑_−ℎ𝑀Δ𝑥_{𝑘−ℎ𝑚−1}(𝑠)_(𝑘)∑_{𝑠=𝑘−ℎ}Δ𝑥(𝑠)=−_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑘)𝑒^𝑇₄−𝑒^𝑇₅𝑒^𝑇₃−𝑒^𝑇₄^𝑇×1𝛼_𝑘𝑅₂0_𝑛0_𝑛11−𝛼_𝑘𝑅₂𝑒^𝑇₄−𝑒^𝑇₅𝑒^𝑇₃−𝑒^𝑇₄𝜁_𝑖𝑗(𝑘), (28)(29) where 𝛼_𝑘=(ℎ_𝑀−ℎ(𝑘))(ℎ_𝑀−ℎ_𝑚)⁻¹, which satisfies 0<𝛼_𝑘<1.
Also, by Theorem 7 in [32], the following inequality for any matrix 𝑇 holds 1−𝛼_𝑘𝛼_𝑘𝐼_𝑛0_𝑛0_𝑛−𝛼_𝑘1−𝛼_𝑘𝐼_𝑛𝑅₂𝑇⋆𝑅₂×1−𝛼_𝑘𝛼_𝑘𝐼_𝑛0_𝑛0_𝑛−𝛼_𝑘1−𝛼_𝑘𝐼_𝑛≥0, (30) which implies 1𝛼_𝑘𝑅₂0_𝑛0_𝑛11−𝛼_𝑘𝑅₂≥𝑅₂𝑇⋆𝑅₂, (31) then, an upper bound of the sum term (29) of 𝔼{Δ𝑉₃(𝑘)} can be rebounded as −ℎ_𝑀−ℎ_𝑚^{𝑘−ℎ𝑚−1}_{𝑠=𝑘−ℎ𝑀}Δ𝑥^𝑇(𝑠)𝑈⊗𝑅₂Δ𝑥(𝑠)≤−_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑘)𝑒^𝑇₄−𝑒^𝑇₅𝑒^𝑇₃−𝑒^𝑇₄^𝑇𝑅₂𝑇⋆𝑅₂×𝑒^𝑇₄−𝑒^𝑇₅𝑒^𝑇₃−𝑒^𝑇₄𝜁_𝑖𝑗(𝑘). (32) Similarly, the sum term of 𝔼{Δ𝑉₄(𝑘)} is bounded as −𝜏^𝑘−1_{𝑠=𝑘−𝜏}Δ𝑥^𝑇(𝑠)𝑈⊗𝑆₂Δ𝑥(𝑠)≤−_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑘)𝑒₁−𝑒₂𝑆₂𝑒₁−𝑒₂^𝑇𝜁_𝑖𝑗(𝑘). (33) Also, by properties of Kronecker product in Lemma 2 and 𝑈𝐺=𝐺𝑈=𝑁𝐺, the terms Θ_𝑞(𝑞=1,2,3) in (27) are calculated as follows: ³_𝑙=1Θ_𝑙=𝑥^𝑇(𝑡−ℎ(𝑘))(𝐺⊗Γ)^𝑇(𝑈⊗Σ)(𝐺⊗Γ)𝑥(𝑡−ℎ(𝑘))=𝑥^𝑇(𝑡−ℎ(𝑘))𝑁𝐺^𝑇𝐺⊗Γ^𝑇ΣΓ𝑥(𝑡−ℎ(𝑘)), (34) where Σ is defined in (19), and, if Σ≤𝜌𝐼_𝑛, then, from (11), the upper bound of terms Ω_𝑞(𝑞=1,2,3) in (27) is calculated as follows: ³_𝑙=1Ω_𝑙=𝜎^𝑇(𝑘)𝜎(𝑘)𝑥)(𝑈⊗Σ≤𝜌^𝑇(𝑘)𝑈⊗𝐻^𝑇₁𝐻₁𝑥(𝑘)+𝑥^𝑇(𝑡−ℎ(𝑘))𝑈⊗𝐻^𝑇₂𝐻₂𝑥(𝑡−ℎ(𝑘)). (35) Then, by utilizing Lemma 4, an upper bound of 𝔼{Δ𝑉(𝑘)=∑⁴_𝑙=1Δ𝑉_𝑙(𝑘)} can be written as follows: 𝔼{Δ𝑉(𝑘)}≤𝔼_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑘)⁵_𝑙=1Ξ_𝑙𝜁_𝑖𝑗(𝑘). (36) From (7), for any positive diagonal matrices 𝐷_𝑞(𝑞=1,2), the following inequalities hold. 0≤_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑘)Ξ₆𝜁_𝑖𝑗(𝑘). (37) Since the relational expression between 𝑝(𝑘) and 𝑞(𝑘), 𝑝^𝑇(𝑘)𝑝(𝑘)≤𝑞^𝑇(𝑘)𝑞(𝑘), holds from the second equality of the system (14), there exists a positive scalar 𝜖 satisfying the following inequality: 0≤_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑘)𝜖Ψ^𝑇Ψ+Ξ₇𝜁_𝑖𝑗(𝑘). (38) From (36)–(38), by S-procedure [25], the 𝔼{Δ𝑉(𝑘)} has a new upper bound as follows: 𝔼{Δ𝑉(𝑘)}≤𝔼_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑘)⁷_𝑙=1Ξ_𝑙+𝜖Ψ^𝑇Ψ𝜁_𝑖𝑗(𝑘). (39) Also, the network (14) with the augmented matrix 𝜁_𝑖𝑗(𝑘) can be rewritten as follows: 𝔼_{1≤𝑖<𝑗≤𝑁}𝑗−𝑖Υ_𝑖𝑗𝜁_𝑖𝑗(𝑘)=0_𝑛×1. (40) Here, in order to illustrate the process of obtaining (40), let us define the following: Λ=Λ₁,Λ₂,…,Λ_𝑁=[𝑁,𝑁−1,…,1]⊗𝐼_𝑛∈ℝ^{𝑛×𝑁𝑛}. (41) By (14), (23), and properties of Kronecker product in Lemma 2, we have the following zero equality: 0_𝑛×1=𝔼{Λ(𝑈⊗𝐴)𝑥(𝑘−𝜏)+Λ(𝑁𝐺⊗Γ)𝑥(𝑘−ℎ(𝑘))−Λ𝑈⊗𝐼_𝑛𝜂(𝑘)−𝑥(𝑘)+Λ𝑈⊗𝑊₁𝑓(𝑥(𝑘))+Λ𝑈⊗𝑊₂𝑓(𝑥(𝑘−ℎ(𝑘)))+Λ(𝑈⊗𝐷)𝑝(𝑘). (42) By Lemma 4, the first term of (42) can be obtained as follows: Λ()𝑥()=𝑈⊗𝐴𝑘−𝜏𝑁𝐼_𝑛,…,𝐼_{𝑛𝑛×𝑁𝑛}(𝑈⊗𝐴_{𝑁𝑛×𝑁𝑛})𝑥₁(𝑘−𝜏),…,𝑥_𝑁(𝑘−𝜏)^𝑇_𝑁𝑛×1=−_{1≤𝑖<𝑗≤𝑁}𝑢_𝑖𝑗Λ_𝑖−Λ_𝑗𝐴𝑥_𝑖(𝑘−𝜏)−𝑥_𝑗(𝑘−𝜏)=_{1≤𝑖<𝑗≤𝑁}Λ_𝑖−Λ_𝑗𝐴𝑧_𝑖𝑗(𝑘−𝜏)=_{1≤𝑖<𝑗≤𝑁}(𝑁+1−𝑖)𝐼_𝑛−𝑁+1−𝑗𝐼_𝑛𝐴𝑧_𝑖𝑗(𝑘−𝜏)=_{1≤𝑖<𝑗≤𝑁}𝑗−𝑖𝐴𝑧_𝑖𝑗(𝑘−𝜏). (43) Similarly, the other terms of (42) are calculated as follows: Λ(𝑁𝐺⊗Γ)𝑥(𝑘−ℎ(𝑘))=−_{1≤𝑖<𝑗≤𝑁}𝑁𝑔_𝑖𝑗Λ_𝑖−Λ_𝑗Γ×𝑥_𝑖(𝑡−ℎ(𝑘))−𝑥_𝑗(𝑡−ℎ(𝑘))=−_{1≤𝑖<𝑗≤𝑁}𝑗−𝑖𝑁𝑔_𝑖𝑗Γ𝑧_𝑖𝑗(𝑘−ℎ(𝑘)),−Λ𝑈⊗𝐼_𝑛𝜂(𝑘)−𝑥(𝑘)=_{1≤𝑖<𝑗≤𝑁}𝑢_𝑖𝑗Λ_𝑖−Λ_𝑗𝐼_𝑛×𝜂_𝑖(𝑘)−𝑥_𝑖(𝑘)−𝜂_𝑗(𝑘)−𝑥_𝑗(𝑘)=−_{1≤𝑖<𝑗≤𝑁}𝑗−𝑖𝜂_𝑖𝑗(𝑘)−𝑧_𝑖𝑗(𝑘,Λ)𝑈⊗𝑊₁𝑓(𝑥(𝑘))=−_{1≤𝑖<𝑗≤𝑁}𝑢_𝑖𝑗Λ_𝑖−Λ_𝑗𝑊₁𝑓𝑥_𝑖(𝑘)−𝑓𝑥_𝑗(𝑘)=_{1≤𝑖<𝑗≤𝑁}𝑗−𝑖𝑊₁𝑓𝑧_𝑖𝑗(𝑘),Λ𝑈⊗𝑊₂𝑓(𝑥(𝑘−ℎ(𝑘)))=−_{1≤𝑖<𝑗≤𝑁}𝑢_𝑖𝑗Λ_𝑖−Λ_𝑗𝑊₂×𝑓𝑥_𝑖(𝑡−ℎ(𝑘))−𝑓𝑥_𝑗(𝑡−ℎ(𝑘))=_{1≤𝑖<𝑗≤𝑁}𝑗−𝑖𝑊₂𝑓𝑧_𝑖𝑗((𝑘,Λ()𝑝(𝑘)𝑘−ℎ))𝑈⊗𝐷=−_{1≤𝑖<𝑗≤𝑁}𝑢_𝑖𝑗Λ_𝑖−Λ_𝑗𝐷𝑝_𝑖(𝑘)−𝑝_𝑗(𝑘)=_{1≤𝑖<𝑗≤𝑁}𝑗−𝑖𝐷𝑝_𝑖𝑗(𝑘). (44) Then, (42) can be rewritten as follows: 0_𝑛×1=𝔼_{1≤𝑖<𝑗≤𝑁}×𝑗−𝑖−𝐼_𝑛,𝐴,0_𝑛,−𝑁𝑔_𝑖𝑗Γ,0_𝑛,−𝐼_𝑛,𝑊₁,𝑊₂,𝐷_Υ𝑖𝑗×𝜁_𝑖𝑗(𝑘). (45) Therefore, if the zero equality (40) holds, then a synchronization condition for the network (14) is 𝔼_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑘)⁷_𝑙=1Ξ_𝑙+𝜖Ψ^𝑇Ψ𝜁_𝑖𝑗(𝑘)<0 (46) subject to 𝔼_{1≤𝑖<𝑗≤𝑁}𝑗−𝑖Υ_𝑖𝑗𝜁_𝑖𝑗(𝑘)=0_𝑛×1. (47) Here, if inequality (47) holds, then there exists a positive scalar 𝜀 such that ∑⁷_𝑙=1Ξ_𝑙+𝜖Ψ^𝑇Ψ<−𝜀𝐼_9𝑛. From (39) and (47), we have 𝔼{Δ𝑉(𝑘)}≤𝔼{−𝜀∑_{1≤𝑖<𝑗≤𝑁}‖𝑥_𝑖(𝑘)−𝑥_𝑗(𝑘)‖²}. Thus, by Lyapunov theorem and Definition 3, it can be guaranteed that the subnetworks in the coupled discrete-time neural networks (14) are asymptotically synchronized. Also, condition (47) is equivalent to the following inequality: _{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑘)⁷_𝑙=1Ξ_𝑙+𝜖Ψ^𝑇Ψ𝜁_𝑖𝑗(𝑘)<0 (48) subject to 𝑗−𝑖Υ_{1≤𝑖<𝑗≤𝑁𝑖𝑗}𝜁_𝑖𝑗(𝑘)=0_𝑛×1. (49) Finally, by the use of Lemma 6, condition (49) is equivalent to the following inequality: 𝑗−𝑖Υ_{1≤𝑖<𝑗≤𝑁𝑖𝑗}^⟂𝑇7_𝑙=1Ξ_𝑙+𝜖Ψ^𝑇Ψ𝑗−𝑖Υ_𝑖𝑗^⟂<0, (50) and applying Schur complement [25] leads to 𝑗−𝑖Υ_{1≤𝑖<𝑗≤𝑁}∑_𝑖𝑗^⟂𝑇7_𝑙=1Ξ_𝑙𝑗−𝑖Υ_𝑖𝑗^⟂⋆𝜖Ψ𝑗−𝑖Υ_𝑖𝑗^⟂−𝜖𝐼_𝑛<0, (51) which can be rewritten by 𝑗−𝑖Υ_{1≤𝑖<𝑗≤𝑁𝑖𝑗}^⟂0_9𝑛×𝑛0_𝑛×8𝑛𝐼_𝑛^𝑇7_𝑙∑₌₁Ξ_𝑙𝜖Ψ^𝑇⋆−𝜖𝐼_𝑛×𝑗−𝑖Υ_𝑖𝑗^⟂0_9𝑛×𝑛0_𝑛×8𝑛𝐼_𝑛<0. (52) From inequality (52), if the LMIs (22) are satisfied, then stability condition (47) holds. This completes our proof.

Remark 8. In order to induce a new zero equality (40), the matrix Λ in (41) was defined. It is inspired by the concept of scaling transformation matrix. To reduce the decision variable, Finsler’s lemma (ii) Υ^⟂𝑇ΦΥ^⟂<0 without free-weighting matrices was used. At this time, a zero equality is required. If the matrix Λ is not considered, then the following description (see only (43) as an example) )𝑥()=𝑥{}(𝑈⊗𝐴𝑘−𝜏{}(𝑈⊗𝐴)₁(𝑘−𝜏),…,𝑥_𝑁(𝑘−𝜏)^𝑇=_{1≤𝑖<𝑗≤𝑁}{⋅}𝐴𝑥_𝑖(𝑘−𝜏)−𝑥_𝑗(𝑘−𝜏) (53) as shown in (53) does not hold. Thus, the derivation of zero equality in (40) is impossible. Here, to use Lemma 4, a suitable vector or matrix in the empty parentheses {} is needed. Therefore, by defining the matrix Λ, the induction of the zero equality (40) is possible.

Remark 9. In this paper, the problem of new delay-dependent synchronization for coupled stochastic discrete-time neural networks with leakage delay and parameter uncertainties is considered. By using Finsler's lemma without free-weighting matrices, the proposed robust synchronization criterion for the network is established in terms of LMIs. Here, as mentioned in the Introduction, the leakage delay is the time delay in leakage or forgetting term of the systems and a considerable factor affecting dynamics for the worse in the network. The effect of the leakage delay which cannot be negligible is shown in Figure 2. Also, the stochastic discrete-time systems with parameter uncertainties do not formulate like as the network (14) in any other literature. To do this, the vector (𝜂(𝑘)−𝑥(𝑘)) is added in the augmented vector 𝜁(𝑘). It is just like as 𝑥̇(𝑡) in continuous-time systems. This form for the systems may give more less conservative results for stability analysis. As a case of stochastic continuous-time systems with parameter uncertainties, Kwon [13] derived the delay-dependent stability criteria for uncertain stochastic dynamic systems with time-varying delays via the Lyapunov-Krasovskii's functional approach with two delay fraction numbers.

4. Numerical Examples

In this section, we provide two numerical examples to illustrate the effectiveness of the proposed synchronization criterion in this paper.

Example 10. Consider the following coupled neural networks by complex model in Figure 1: 𝑦_𝑖(𝑘+1)=(𝐴+Δ𝐴)𝑦_𝑖(𝑘−𝜏)+𝑊₁+Δ𝑊₁𝑔𝑦_𝑖(𝑘)+𝑊₂+Δ𝑊₂𝑔𝑦_𝑖((𝑘+𝑘−ℎ))_5𝑗=1𝑔_𝑖𝑗Γ𝑦_𝑗(𝑘−ℎ(𝑘))1+𝜔₁(𝑘)+𝜎_𝑖𝑘,𝑦_𝑖(𝑘),𝑦_𝑖(𝑘−ℎ(𝑘))𝜔₂(𝑘), (54) with 𝑔(𝑥)=0.5tanh(𝑥), where ,𝑊𝐴=0.2000.3₁=0.001000.005,𝑊₂=,−0.10.01−0.2−0.1Γ=0.01𝐼₂,,𝐿𝐺=−210011−311001−210011−311001−2_𝑚=0₂,𝐿_𝑝=0.5𝐼₂,𝐷=0.1𝐼₂,𝐸_𝑎=0.300−0.1,𝐸₁=,𝐸−0.400.3−0.7₂=𝐸₁,𝐻₁=0.2𝐼₂,𝐻₂=𝐻₁. (55)
For the network above, the maximum allowable delay bounds with different ℎ_𝑚 and fixed 𝜏=3 by Theorem 7 are listed in Table 1. In order to confirm the obtained results with the conditions of the time delays as listed in Table 2, the simulation results for the trajectories of state responses, 𝑥_𝑖(𝑘)  (𝑖=2,3,4,5), and synchronization errors, 𝑧_𝑖1(𝑘)=𝑥_𝑖(𝑘)−𝑥₁(𝑘), of the network (54) are shown in Figures 2, 3, 4, and 5. These figures show that the network with the errors converge to zero for given initial values of the state by 𝑥^𝑇₁(0)=[1,−3], 𝑥^𝑇₂(0)=[−1,2], 𝑥^𝑇₃(0)=[4,−5], 𝑥^𝑇₄(0)=[3,−1], and 𝑥^𝑇₅(0)=[4,2]. Specially, the simulation results in Figure 2 show state response trajectories for the values of leakage delay, 𝜏, by 3, 15, and 30 with fixed values ℎ_𝑚=5 and ℎ_𝑀=7. It is easy to illustrate that the larger value of leakage delay gives the worse dynamic behaviors of the network (54).

Table 1 

Maximum allowable delay bounds, ℎ_𝑀, with different ℎ_𝑚 and fixed 𝜏=3 (Example 10).

Table 2 

The conditions of simulation in Example 10.

Figure 1 

The structure of complex networks with 𝑁=5 (Example 10).

(a)

(b)

(c)

Figure 2 

State responses with C1-1 (Example 10): (a) 𝜏=3, (b) 𝜏=15, and (c) 𝜏=30.

(a)

(b)

(c)

(d)

Figure 3 

Synchronization errors trajectories with C1-1 (𝜏=3) (Example 10).

Figure 4 

State responses with C1-2 (Example 10).

(a)

(b)

(c)

(d)

Figure 5 

Synchronization errors trajectories with C1-2 (Example 10).

Example 11. Consider the following coupled neural networks by BA scale-free model [33] in Figure 6: 𝑦_𝑖(𝑘+1)=(𝐴+Δ𝐴)𝑦_𝑖(𝑘−𝜏)+𝑊₁+Δ𝑊₁𝑔𝑦_𝑖(𝑘)+𝑊₂+Δ𝑊₂𝑔𝑦_𝑖((𝑘+𝑘−ℎ))_50𝑗=1𝑔_𝑖𝑗Γ𝑦_𝑗(𝑘−ℎ(𝑘))1+𝜔₁(𝑘)+𝜎_𝑖𝑘,𝑦_𝑖(𝑘),𝑦_𝑖(𝑘−ℎ(𝑘))𝜔₂(𝑘), (56) with 𝑔(𝑥)=0.1tanh(𝑥), where ,𝑊𝐴=0.01000.02₁=0.2−0.10.3−0.2,𝑊₂=,0.30.1−0.30.2Γ=0.001𝐼₂,𝐿_𝑚=0₂,𝐿_𝑝=0.1𝐼₂,𝐷=0.1𝐼₂,𝐸_𝑎=0.7−0.200.4,𝐸₁=,𝐸0.2−0.500.3₂=𝐸₁,𝐻₁=0.2𝐼₂,𝐻₂=𝐻₁. (57)
The results of maximum allowable delay bounds with different ℎ_𝑚 and fixed 𝜏=3 by Theorem 7 are listed in Table 3. For lack of space, the outer-coupling matrix 𝐺 is omitted. It is easy that the matrix 𝐺 was expressed from Figure 6. Figures 7 and 8 show the state response trajectories, 𝑥_𝑖(𝑡)  (𝑖=1,…,50), of the network (56) with the condition of the time delays as listed in Table 4 for random initial values of the state. These figures show that the network (56) with the state responses converge to zero. This means the synchronization stability of the network (56).

Table 3 

Maximum allowable delay bounds, ℎ_𝑀, with different ℎ_𝑚 and fixed 𝜏=5 (Example 11).

Table 4 

The conditions of simulation in Example 11.

Figure 6 

The structure of BA scale-free networks with 𝑁=50 (Example 11).

(a)

(b)

Figure 7 

State responses and time-delay ℎ(𝑘) with C2-1 (Example 11).

(a)

(b)

Figure 8 

State responses and time-delay ℎ(𝑘) with C2-2 (Example 11).

5. Conclusions

In this paper, the delay-dependent robust synchronization criterion for the coupled stochastic discrete-time neural networks with interval time-varying delays in network coupling, leakage delay, and parameter uncertainties has been proposed. To do this, the suitable Lyapunov-Krasovskii’s functional was used to investigate the feasible region of stability criterion. By utilization of Finsler’s lemma with a new zero equality, a sufficient condition for guaranteeing asymptotic synchronization for the concerned networks has been derived in terms of LMIs. Two numerical examples have been given to show the effectiveness and usefulness of the presented criterion.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2012-0000479) and by a Grant of the Korea Healthcare Technology R & D Project, Ministry of Health & Welfare, Republic of Korea (A100054).