Synchronization for an Array of Coupled Cohen-Grossberg Neural Networks with Time-Varying Delay

Abstract

This paper makes some great attempts to investigate the global exponential synchronization for arrays of coupled delayed Cohen-Grossberg neural networks with both delayed coupling and one single delayed one. By resorting to free-weighting matrix and Kronecker product techniques, two novel synchronization criteria via linear matrix inequalities (LMIs) are presented based on convex combination, in which these conditions are heavily dependent on the bounds of both the delay and its derivative. Owing to that the addressed system can include some famous neural network models as the special cases, the proposed methods can extend and improve those earlier reported ones. The efficiency and applicability of the presented conditions can be demonstrated by two numerical examples with simulations.

1. Introduction

In recent years, synchronization of various chaotic systems has gained considerable attention since the pioneering works of Pecora and Carroll [1, 2]. It is widely known that many benefits of having synchronization or chaos synchronization can be existent in various engineering fields such as secure communication, image processing, and harmonic oscillation generation. Thus recently, the problem on synchronization in chaotic systems has been extensively studied owing to its potential applications in many engineering areas. Especially, since chaos synchronization in arrays of linearly coupled dynamical systems was firstly studied in [3], arrays of coupled systems including delayed chaotic ones have received much attention as they can exhibit some interesting phenomena [4, 5], and a great number of elegant results have been derived [6–28].

As a typical complex systems, delayed neural networks (DNNs) have been verified to exhibit some complex and unpredictable behaviors such as stable equilibria, periodic oscillations, bifurcation, and chaotic attractors. Thus recently, chaos synchronization for arrays of coupled DNNs have been discussed and some elegant results have been proposed [12–28]. The global synchronization of linearly coupled DNNs with delayed coupling was investigated in [12], in which the dynamical behavior of the uncoupled system could be chaotic or others. The authors in [13] have considered the robust synchronization of coupled DNNs under general impulsive control. In [14], this paper has proposed an adaptive procedure to the synchronization for coupled identical Yang-Yang fuzzy DNNs based on one simple adaptive controller. In [15], with all the parameters unknown, the authors focused on the robust synchronization between two coupled DNNs that were linearly and unidirectionally coupled, in which neither symmetry nor negative (positive) definiteness of the coupling matrix were required. However, those above-mentioned results were presented in terms of some complicated inequalities, which makes them uneasily checked and applied to real ceases by the most recently developed algorithms. By employing Kronecker product and LMI technique, the global synchronization and cluster one have been studied for DNNs with couplings, and some easy-to-test sufficient conditions have been obtained [16–25, 28]. Yet, the system forms addressed in [16–25] seemed simple and some improved techniques have not been utilized to reduce the conservatism, which make the above-mentioned results inapplicable to tackle DNNs of more general forms.

The Cohen-Grossberg neural network (CGNN) model, first proposed by Cohen and Grossberg in 1983 [29], has recently gained particular research attention, since it is quite general to include many famous network models as its special case and has promising application potentials for tasks of associative memory, parallel computation, and nonlinear optimization problems. Meanwhile, owing to complexity of CGNNs themselves, there were few works studying the global synchronization for the coupled delayed CGNNs, except for that some researchers have studied the slave-master synchronization for continuous CGNNs in [26, 27] and synchronization for coupled discrete delayed CGNNs in [28]. Thus, it is urgent and challenging to establish some easy-to-check and less conservative results ensuring the global synchronization of coupled continuous-time delayed CGNNs, which constitutes the main focus of this presented work.

In this paper, the global exponential synchronization of 𝑁 identical delayed CGNNs with both delayed coupling and one single delayed one is considered and two novel LMI-based conditions are derived by using Kronecker product technique, which has not been studied in the present literature. It shows that the chaos synchronization can be ensured by a suitable design of inner coupled linking matrix and the inner delayed linking ones. Moreover, some effective mathematical techniques are employed to reduce the conservatism. Finally, the efficiency of the derived criteria can be illustrated by utilizing two numerical examples.

Notations. 𝐑^𝑛 denotes the 𝑛-dimensional Euclidean space, and 𝐑^𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices. For the symmetric matrices 𝑋,𝑌,𝑋>𝑌 (respectively, 𝑋≥𝑌) means that 𝑋−𝑌>0(𝑋−𝑌≥0) is a positive-definite (respectively, positive-semidefinite) matrix; 𝐴^𝑇 represents the transpose of the matrix 𝐴; 𝜆_max(𝐴),𝜆_min(𝐴) denote the maximum eigenvalue and minimum one of matrix 𝐴, respectively; 𝐼 represents the identity matrix of an appropriate dimension; ^𝑋𝑌_𝑌^𝑇_𝑍=^𝑋𝑌_∗𝑍.

2. Problem Formulations and Preliminaries

Suppose the nodes are coupled with states 𝑥_𝑖(𝑡), 𝑖∈{1,…,𝑁}, then the delayed Cohen-Grossberg neural network models can be formulated as follows:𝑥̇_𝑖(𝑡)=−𝛼𝑥_𝑖(𝑡)𝛽𝑥_𝑖(𝑡)−𝐴𝑓𝑥_𝑖(𝑡)−𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))−𝐈(𝑡)+^𝑁_{𝑗=1,𝑗≠𝑖}𝑙_𝑖𝑗𝐹𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)+^𝑁_{𝑗=1,𝑗≠𝑖}𝑙_𝑖𝑗𝐾𝑥_𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡)+^𝑁_{𝑗=1,𝑗≠𝑖}𝑙_𝑖𝑗𝐽𝑥_𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡−𝜏(𝑡)),(2.1) in which 𝑥_𝑖(𝑡)=[𝑥_𝑖1(𝑡),…,𝑥_𝑖𝑛(𝑡)]^𝑇∈𝐑^𝑛 is the state vector of the 𝑖th network at time 𝑡, 𝛼(𝑥_𝑖)=diag{𝛼₁(𝑥_𝑖1),…,𝛼_𝑛(𝑥_in)} represents the amplification function, 𝛽(𝑥_𝑖)=[𝛽₁(𝑥_𝑖1),…,𝛽_𝑛(𝑥_in)]^𝑇 is the behaved function, 𝐴=[𝑎_𝑖𝑗]_𝑛×𝑛, 𝐵=[𝑏_𝑖𝑗]_𝑛×𝑛, 𝑓(𝑥_𝑖)=[𝑓₁(𝑥_𝑖1),…,𝑓_𝑛(𝑥_𝑖𝑛)]^𝑇; I(𝑡)=[𝐈₁(t),…,𝐈_𝑛(t)]^𝑇∈𝐑^𝑛 is the external input vector; 𝐹=[𝑓_𝑖𝑗]_𝑛×𝑛, 𝐾=[𝑘_𝑖𝑗]_𝑛×𝑛, and 𝐽=[𝑗_𝑖𝑗]_𝑛×𝑛 are respectively the inner coupling matrices between the connected nodes 𝑖 and 𝑗 at times 𝑡 and 𝑡−𝜏(𝑡).

For system (2.1), the following assumptions are introduced throughout this paper. (A1)𝜏(𝑡) denotes an interval time-varying delay satisfying0≤𝜏₀≤𝜏(𝑡)≤𝜏_𝑚,𝜏̇(𝑡)≤𝜇<+∞,(2.2)

and we set 𝜏_𝑚=𝜏_𝑚−𝜏₀. (A2)𝐿=[𝑙_𝑖𝑗]_𝑁×𝑁 is the configuration matrix that is irreducible and satisfies𝑙_𝑖𝑗=𝑙_𝑗𝑖,𝑖≠𝑗,𝑙_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝑙_𝑖𝑗.(2.3) Here 𝑙_𝑖𝑗>0 if there is a connection between node 𝑖 and the one 𝑗 and otherwise, 𝑙_𝑖𝑗=0. (A3) For 𝑖∈{1,2,…,𝑛}, each 𝛼_𝑖(⋅) is Lipschitz continuous and there exists the positive scalars 𝑎_𝑖,𝑎_𝑖 satisfying 0<𝑎_𝑖≤𝛼_𝑖(⋅)≤𝑎_𝑖; and there exist the positive scalars 𝜋_𝑖,𝛾_𝑖 such that each function 𝛽_𝑖(⋅) satisfies 0<𝛾_𝑖≤(𝛽_𝑖(𝑥)−𝛽_𝑖(𝑦)/𝑥−𝑦)≤𝜋_𝑖, and𝛽̇‌_𝑖(𝑥)−𝛽̇‌_𝑖(𝑦)−𝜌⁻_𝑖(𝑥−𝑦)𝛽̇‌_𝑖(𝑥)−𝛽̇‌_𝑖(𝑦)−𝜌⁺_𝑖(𝑥−𝑦)≤0∀𝑥,𝑦∈𝐑,𝑖=1,…,𝑛,(2.4) in which 𝜌⁻_𝑖,𝜌⁺_𝑖 are given constants. Here we set Λ=diag{𝑎₁,…,𝑎_𝑛}, Ψ=diag{𝑎₁,…,𝑎_𝑛}, Γ=diag{𝛾₁,…,𝛾_𝑛}, Π=diag{𝜋₁,…,𝜋_𝑛}, andΛ₁=diag𝛾₁𝜋₁,…,𝛾_𝑛𝜋_𝑛,Λ₂=diag𝜋₁+𝛾₁2,…,𝜋_𝑛+𝛾_𝑛2,Υ₁=diag𝜌⁺₁𝜌⁻₁,…,𝜌⁺_𝑛𝜌⁻_𝑛,Υ₂=diag⎧⎨⎩𝜌⁺₁+𝜌⁻₁2,…,𝜌⁺_𝑛+𝜌⁻_𝑛2⎫⎬⎭.(2.5)(A4) For any 𝛼,𝛽∈𝐑, and 𝜌⁻_𝑖,𝜌⁺_𝑖 for 𝑖∈{1,2,…,𝑛}, the activation function 𝑓_𝑖(⋅) satisfies𝑓_𝑖(𝛼)−𝑓_𝑖(𝛽)−𝜎⁺_𝑖(𝛼−𝛽)𝑓_𝑖(𝛼)−𝑓_𝑖(𝛽)−𝜎⁻_𝑖(𝛼−𝛽)≤0.(2.6) Here we denote Σ₁=diag{𝜎⁺₁𝜎⁻₁,…,𝜎⁺_𝑛𝜎⁻_𝑛} and Σ₂=diag{(𝜎⁺₁+𝜎⁻₁)/2,…,(𝜎⁺_𝑛+𝜎⁻_𝑛)/2}.

Remark 2.1. In (A3), the assumption on the derivative of 𝛽(𝑥_𝑖) in (2.4) is reasonable and does not result in the conservatism in many cases such as that, choosing the appropriate scalars 𝑎,𝑏,𝑐, the function 𝛽(𝑥_𝑖) can be expressed as 𝑎𝑥_𝑖, 𝑎𝑥_𝑖+𝑏sin(𝑥_𝑖), 𝑎𝑥_𝑖+𝑏sin²(𝑥_𝑖), 𝑎𝑥_𝑖+𝑐cos(𝑥_𝑖), 𝑎𝑥_𝑖+𝑐cos³(𝑥_𝑖), 𝑎𝑥_𝑖+𝑐tanh(𝑥_𝑖), respectively. Moreover, the activation functions in system (2.1) can be of general description and those present ones in [22–26, 29] are just special cases of the system (2.1).
Based on assumption(A2), system (2.1) can be rewritten as the following forms: 𝑥̇_𝑖(𝑡)=−𝛼𝑥_𝑖(𝑡)𝛽𝑥_𝑖(𝑡)−𝐴𝑓𝑥_𝑖(𝑡)−𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))−I(𝑡)−𝑙_𝑖𝑖𝐾𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡)+^𝑁_𝑗=1𝑙_𝑖𝑗𝐹𝑥_𝑗(𝑡)+^𝑁_𝑗=1𝑙_𝑖𝑗(𝐾+𝐽)𝑥_𝑗(𝑡−𝜏(𝑡)).(2.7) To address the problem, we denote the set 𝒮={x(𝑠)=[𝑥^𝑇₁(𝑠),…,𝑥^𝑇_𝑁(𝑠)]∶𝑥_𝑖(𝑠)∈𝒞([𝑡₀−𝜏_𝑚,𝜏₀],𝐑^𝑛),𝑥_𝑖(𝑠)=𝑥_𝑗(𝑠),𝑖,𝑗=1,2,…,𝑁} as the synchronization manifold for system (2.7).

Definition 2.2 (see [16]). Dynamical network (2.7) is said to be asymptotically synchronized, if for any initial conditions 𝜙_𝑖(𝑠),𝜙_𝑗(𝑠)∈𝐶([𝑡₀−𝜏_𝑚,𝑡₀],𝐑^𝑛), 𝑖,𝑗=1,…,𝑁, there exist 𝑀>0, 𝜀>0 and sufficient large 𝑇>0 such that ‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖≤𝑀𝑒^−𝜀𝑡 for all 𝑡≥𝑇, where 𝜀 and 𝑀 are said to be the decay rate and the decay coefficient, respectively. Here ‖⋅‖ denotes the Euclidean norm.
Due to the communication delay, the array of coupled nodes cannot be decoupled, and the synchronized state is always not the trajectory of an isolated node but a modified one as (2.7). Furthermore, delayed coupling matrix and the degree of the node play the important roles in the synchronized state, which has been illustrated in [21]. In the paper, we give an improved discussion for such synchronization. In the case, system (2.7) reaches the synchronization, that is, 𝑥₁(𝑡)=⋯=𝑥_𝑁(𝑡)=𝑠(𝑡), we can deduce the synchronized state equation𝑠̇(𝑡)=−𝛼(𝑠(𝑡))[𝛽(𝑠(𝑡))−𝐴𝑓(𝑠(𝑡))−𝐵𝑓(𝑠(𝑡−𝜏(𝑡)))−𝐈(𝑡)]−𝑙_𝑖𝑖𝐾[𝑠(𝑡−𝜏(𝑡))−𝑠(𝑡)],(2.8) where 𝑖=1,2,…,𝑁. Obviously, the synchronization is invariant for the coupled system (2.7). Therefore, to realize complete synchronization, the assumption 𝑙₁₁=⋯=𝑙_𝑁𝑁=𝑙 has to be imposed on the system (2.7).

3. Delay-Dependent Synchronization Criteria

Firstly, together with the Kronecker product in [16–21], we can reformulate the system (2.7) asẋ‌(𝑡)=−𝐚(𝑥(𝑡))𝐛(𝑥(𝑡))−𝐼_𝑁⊗𝐴𝐟(𝑥(𝑡))−𝐼_𝑁⊗𝐵𝐟(𝑥(𝑡−𝜏(𝑡)))−𝐈(𝑡)−l𝐼_𝑁⊗𝐾×[𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡)]+(𝐿⊗𝐹)𝑥(𝑡)+(𝐿⊗(𝐾+𝐽))𝑥(𝑡−𝜏(𝑡))(3.1) with 𝑥(𝑡)=[𝑥^𝑇₁(𝑡),…,x^𝑇_𝑁(𝑡)]^𝑇, 𝐚(x(𝑡))=diag{𝛼(𝑥₁(𝑡)),…,𝛼(𝑥_𝑁(𝑡))}, 𝐛(x(𝑡))=[𝛽^𝑇(𝑥₁(𝑡)),…,𝛽^𝑇(𝑥_𝑁(𝑡))]^𝑇, 𝐟(x(⋅))=[𝑓^𝑇(𝑥₁(⋅)),…,𝑓^𝑇(𝑥_𝑁(⋅))]^𝑇, and 𝐈(𝑡)=[𝐼^𝑇(𝑡),…,𝐼^𝑇(𝑡)]^𝑇.

In order to derive our results, the following lemmas are essential for obtaining the synchronization criteria.

Lemma 3.1. Let 𝐷,𝑆, and 𝑃>0 be real matrices of appropriate dimensions and 𝜀>0. Then for any vectors 𝑥 and 𝑦 with appropriate dimensions, one gets 2𝑥^𝑇𝐷^𝑇𝑆𝑦≤𝜀⁻¹𝑥^𝑇𝐷^T𝑃⁻¹𝐷𝑥+𝜀𝑦^𝑇𝑆^𝑇𝑃𝑆𝑦.

Lemma 3.2 (17).. Let U=[𝑢_𝑖𝑗]_𝑁×𝑁, 𝑃∈𝐑^𝑛×𝑛, 𝑥=[𝑥^𝑇₁,𝑥^𝑇₂,…,𝑥^𝑇_𝑁]^𝑇, and 𝑦=[𝑦^𝑇₁,𝑦^𝑇₂,…,𝑦^𝑇_𝑁]^𝑇 with 𝑥_𝑖,𝑦_𝑖∈𝐑^𝑛, 𝑖=1,…,𝑁. If U=U^𝑇 and each row sum of 𝑈 is 0, then 𝑥^𝑇(U⊗𝑃)𝑦=−∑_{1≤𝑖<𝑗≤𝑁}u_𝑖𝑗(𝑥_𝑖−𝑥_j)^𝑇𝑃(𝑦_𝑖−𝑦_𝑗).

Then by utilizing the most improved techniques for achieving the criteria in [30], we state and investigate the global exponential synchronization for the system (3.1).

Theorem 3.3. Supposing that assumptions (A₁)–(A₄) hold, then the dynamical system (3.1) is globally exponentially synchronized, if there exist 𝑛×𝑛 matrices 𝑃>0, 𝑆>0, 𝑍>0, 𝐿_𝑖(𝑖=1,2,3,4), 𝑛×𝑛 matrices 𝑃_𝑙>0, 𝑄_𝑙>0, 𝑅_𝑙(𝑙=1,2,3) making ^{𝑃𝑙𝑅𝑙∗𝑄}_𝑙≥0, 𝑛×𝑛 diagonal matrices 𝑅>0, 𝑄>0, 𝐺>0, 𝐸>0, 𝑈>0, 𝑉>0, 𝑊>0, 𝐻>0, 𝑇_𝑖>0(𝑖=1,2), 13𝑛×𝑛 matrices 𝑁_𝑖(𝑖=1,2,3), and one scalar 𝛿>0 such that, for 1≤𝑖<𝑗≤𝑁, the LMIs in (3.2) hold ⎡⎢⎢⎢⎢⎣Ω_𝑖𝑗+$+$^𝑇𝐻√𝜏₀𝑁₁√𝜏_𝑚𝑁₂∗−𝛿𝐼00∗∗−𝑆0∗∗∗−𝑍⎤⎥⎥⎥⎥⎦<0,⎡⎢⎢⎢⎢⎣Ω_𝑖𝑗+$+$^𝑇𝐻√𝜏₀𝑁₁√𝜏_𝑚𝑁₃∗−𝛿𝐼00∗∗−𝑆0∗∗∗−𝑍⎤⎥⎥⎥⎥⎦<0,(3.2) where $=[𝑁₁−𝑁₁+𝑁₂−𝑁₃0_{6𝑛⋅13𝑛}−𝑁₂+𝑁₃0_{3𝑛⋅13𝑛}], 𝐻=[0_{𝑛⋅13𝑛}𝑄(Λ⁻¹−Ψ⁻¹)0_𝑛⋅𝑛]^𝑇, Ω_𝑖𝑗=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Ξ₁₁00Ξ₁₄00Ξ₁₇𝐿^𝑇₁0Ξ_1,10𝑙𝐿^𝑇₁𝐾Ξ_1,12Ξ_1,13∗Ξ₂₂00Ξ₂₅00000000∗∗Ξ₃₃00Ξ₃₆0000000∗∗∗Ξ₄₄000𝐴^𝑇𝐿₃𝐴^𝑇𝐿₄00𝐴^𝑇𝑄^𝑇0∗∗∗∗Ξ₅₅00000000∗∗∗∗∗Ξ₆₆0000000∗∗∗∗∗∗Ξ₇₇−𝐿^𝑇₂0Ξ_7,10𝑙𝐿^𝑇₂𝐾−Ψ⁻¹𝑅0∗∗∗∗∗∗∗Ξ₈₈−𝐿^𝑇₃0𝐿^𝑇₃𝐵−𝐿^𝑇₃0∗∗∗∗∗∗∗∗Ξ₉₉0𝐿^𝑇₄𝐵−𝐿^𝑇₄0∗∗∗∗∗∗∗∗∗Ξ_10,10Ξ_10,11Ξ_10,120∗∗∗∗∗∗∗∗∗∗Ξ_11,11𝐵^𝑇𝑄^𝑇0∗∗∗∗∗∗∗∗∗∗∗Ξ_12,120∗∗∗∗∗∗∗∗∗∗∗∗−𝑇₂⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(3.3) with Ξ₁₁=P₂+𝑙(𝐿^𝑇₁𝐾+𝐾^𝑇𝐿₁)−𝑙_𝑖𝑗𝑁(𝐿^𝑇₁𝐹+𝐹^𝑇𝐿₁)−2Γ^𝑇𝐺−UΣ₁−𝑇₁Π₁−𝑇₂Υ₁, Ξ₁₄=𝑅₂+UΣ₂, Ξ₁₇=𝑃−𝐿^𝑇₁−𝑙_𝑖𝑗𝑁𝐹^𝑇𝐿₂+2Π^𝑇𝑅Ψ⁻¹−2Γ^𝑇𝑄Λ⁻¹, Ξ_1,10=−𝑙𝐿^𝑇₁𝐾−𝑙_𝑖𝑗𝑁𝐿^𝑇₁(𝐾+𝐽), Ξ_1,12=𝑙𝐾^𝑇𝑄Λ⁻¹−𝑙_𝑖𝑗𝑁𝐹^𝑇𝑄Λ⁻¹+𝐺^𝑇+𝑇₁Π₂, Ξ_1,13=𝑇₂Υ₂+𝑄Λ⁻¹−𝑅Ψ⁻¹, Ξ₂₂=−𝑃₂+𝑃₁+𝑃₃−𝑊Σ₁, Ξ₂₅=−𝑅₂+𝑅₁+𝑅₃+𝑊Σ₂, Ξ₃₃=−𝑃₃−𝐻Σ₁, Ξ₃₆=𝐻Σ₂−𝑅₃, Ξ₄₄=−𝑈+𝑄₂, Ξ₅₅=−𝑄₂+𝑄₁+𝑄₃−𝑊, Ξ₆₆=−𝑄₃−𝐻, Ξ₇₇=−𝐿^𝑇₂−𝐿₂+𝜏₀𝑆+𝜏_𝑚𝑍, Ξ_7,10=−𝑙𝐿^𝑇₂𝐾−𝑙_𝑖𝑗𝑁𝐿^𝑇₂(𝐾+𝐽), Ξ₈₈=−𝐸+𝛿𝐼_𝑛, Ξ₉₉=−𝐿^𝑇₄−𝐿₄+Ψ^𝑇𝐸Ψ, Ξ_10,10=−(1−𝜇)𝑃₁−𝑉Σ₁, Ξ_10,11=−(1−𝜇)𝑅₁+𝑉Σ₂, Ξ_10,12=−𝑙𝐾^𝑇𝑄Λ⁻¹−𝑙_𝑖𝑗𝑁(𝐾+𝐽)^𝑇𝑄Λ⁻¹, Ξ_11,11=−(1−𝜇)𝑄₁−𝑉, Ξ_12,12=−2𝑄^𝑇−𝑇₁.

Proof. Firstly, we can represent the system (3.1) as the following form: 𝑥̇(𝑡)=𝑦(𝑡),(3.4)𝑦(𝑡)=𝑧(𝑡)−𝑙𝐼_𝑁⊗𝐾[𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡)]+(𝐿⊗𝐹)x(𝑡)+(𝐿⊗(𝐾+𝐽))𝑥(𝑡−𝜏(𝑡)),(3.5)𝑧(𝑡)=𝐚(𝑥(𝑡))𝑤(𝑡),(3.6)𝑤(𝑡)=−𝐛(𝑥(𝑡))+𝐼_𝑁⊗𝐴𝐟(𝑥(𝑡))+𝐼_𝑁⊗𝐵𝐟(𝑥(𝑡−𝜏(𝑡)))+𝐈(𝑡).(3.7) Based on assumptions (A1) and (A3), and 𝑈=𝑢_𝑖𝑗_𝑁⋅𝑁=⎡⎢⎢⎣𝑁−1⋯−1⋮⋱⋮−1⋯𝑁−1⎤⎥⎥⎦,(3.8) we construct the following Lyapunov-Krasovskii functional: 𝑉(𝑥(𝑡))=𝑉₁(𝑥(𝑡))+𝑉₂(𝑥(𝑡))+𝑉₃(𝑥(𝑡))+𝑉₄(𝑥(𝑡))+𝑉₅(𝑥(𝑡)),(3.9) where 𝑉₁(𝑥(𝑡))=𝑥^𝑇(𝑡)(𝑈⊗𝑃)𝑥(𝑡)+2[Θ𝑥(𝑡)−𝐛(𝑥(𝑡))]^𝑇𝑈⊗𝑅Ψ⁻¹𝑥(𝑡)+2[𝐛(𝑥(𝑡))−Υ𝑥(𝑡)]^𝑇𝑈⊗𝑄Λ⁻¹𝑥(𝑡),𝑉₂(𝑥(𝑡))=^𝑡−𝜏0_{𝑡−𝜏(𝑡)}𝑥(𝑠)𝐟(𝑥(𝑠))^𝑇𝑈⊗𝑃₁𝑅₁∗𝑄₁𝑥(𝑠)𝐟(𝑥(𝑠))ds,𝑉₃(𝑥(𝑡))=^𝑡_𝑡−𝜏0𝑥(𝑠)𝐟(𝑥(𝑠))^𝑇𝑈⊗𝑃₂𝑅₂∗𝑄₂𝑥(𝑠)𝐟(𝑥(𝑠))ds,𝑉₄(𝑥(𝑡))=^𝑡−𝜏0_{𝑡−𝜏𝑚}𝑥(𝑠)𝐟(𝑥(𝑠))^𝑇𝑈⊗𝑃₃𝑅₃∗𝑄₃𝑥(𝑠)𝐟(𝑥(𝑠))ds,𝑉₅(𝑥(𝑡))=⁰_−𝜏0^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)(𝑈⊗𝑆)𝑥̇(𝑠)𝑑𝑠𝑑𝜃+^−𝜏0_−𝜏𝑚^𝑡_𝑡+𝜃𝑥̇^𝑇(𝑠)(𝑈⊗𝑍)𝑥̇(𝑠)𝑑𝑠𝑑𝜃(3.10) with two diagonal matrices 𝑅>0, 𝑄>0, and setting Θ=diag{Π,Π,…,Π_{𝑁}}, Υ=diag{Γ,Γ,…,Γ_{𝑁}}. Based on (A3) and Lemma 3.1, one can easily verify the definite positiveness of 𝑉₁(𝑥(𝑡)).
Now, by directly calculating 𝑉₁̇‌(𝑥(𝑡)) along the trajectory of the system (3.1), we can deduce𝑉₁̇‌(𝑥(𝑡))=2𝑥^𝑇(𝑡)(𝑈⊗𝑃)𝑦(𝑡)+2[Θ𝑥(𝑡)−𝐛(𝑥(𝑡))]^𝑇𝑈⊗𝑅Ψ⁻¹𝑦(𝑡)+2Θ𝑦(𝑡)−𝐛̇‌(𝑥(𝑡))^𝑇×𝑈⊗𝑅Ψ⁻¹𝑥(𝑡)+2[𝐛(𝑥(𝑡))−Υ𝑥(𝑡)]^𝑇×𝑈⊗𝑄Λ⁻¹𝑦(𝑡)+2𝐛̇‌(𝑥(𝑡))−Υ𝑦(𝑡)^𝑇𝑈⊗𝑄Λ⁻¹𝑥(𝑡)=2𝑥^𝑇(𝑡)(𝑈⊗𝑃)𝑦(𝑡)+2[Θ𝑥(𝑡)−𝐛(𝑥(𝑡))]^𝑇𝑈⊗𝑅Ψ⁻¹𝑦(𝑡)+2𝐛^𝑇(𝑥(𝑡))×𝑈⊗𝑄Λ⁻¹𝑦(𝑡)+2𝐛̇‌^𝑇(𝑥(𝑡))𝑈⊗𝑄Λ⁻¹−𝑅Ψ⁻¹𝑥(𝑡)+2𝑦^𝑇(𝑡)×𝑈⊗Π^𝑇𝑅Ψ⁻¹−2Γ^𝑇𝑄Λ⁻¹𝑥(𝑡).(3.11) Meanwhile by (3.5), it is easy to derive that 2𝐛^𝑇(𝑥(𝑡))𝑈⊗𝑄Λ⁻¹𝑦(𝑡)=2𝐛^𝑇(𝑥(𝑡))𝑈⊗𝑄Λ⁻¹𝑧(𝑡)+2𝐛^𝑇(𝑥(𝑡))𝑈⊗𝑄Λ⁻¹×−l𝐼_𝑁⊗𝐾[𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡)]+(𝐿⊗𝐹)𝑥(𝑡)+(𝐿⊗(𝐾+𝐽))𝑥(𝑡−𝜏(𝑡)).(3.12) Noting that 𝑈^1/2≥0 does hold, then with Lemma 3.1 and 𝛿>0, one can estimate 2𝐛^𝐓(𝑥(𝑡))(𝑈⊗𝑄Λ⁻¹)𝑧(𝑡) as 2𝐛^𝐓(𝑥(𝑡))𝑈⊗𝑄Λ⁻¹𝑧(𝑡)=2𝐛^𝐓(𝑥(𝑡))𝑈^1/2⊗𝑄Λ⁻¹−𝛼⁻¹(𝑥(𝑡))^T𝑈^1/2⊗𝐼𝑧(𝑡)+2𝐛^𝐓(𝑥(𝑡))(𝑈⊗𝑄)𝑤(𝑡)≤𝛿𝑧^𝑇(𝑡)𝑈⊗𝐼_𝑛𝑧(𝑡)+𝛿⁻¹𝐛^𝐓(𝑥(𝑡))×𝑈⊗Λ^−𝑇−Ψ^−𝑇𝑄^𝑇𝑄Λ⁻¹−Ψ⁻¹𝐛(𝑥(𝑡))+2𝐛^𝐓(𝑥(𝑡))×(𝑈⊗𝑄)−𝐛(𝑥(𝑡))+𝐼_𝑁⊗𝐴𝐟(𝑥(𝑡))+𝐼_𝑁⊗𝐵𝐟(𝑥(𝑡−𝜏(𝑡)))+𝐈(𝑡).(3.13) Now combining with terms (3.12) and (3.13) yields 𝑉₁̇‌(𝑥(𝑡))≤2𝑥^𝑇(𝑡)(𝑈⊗𝑃)𝑦(𝑡)+2[Θ𝑥(𝑡)−𝐛(𝑥(𝑡))]^𝑇𝑈⊗𝑅Ψ⁻¹𝑦(𝑡)+𝛿z^𝑇(𝑡)𝑈⊗𝐼_𝑛𝑧(𝑡)+𝛿⁻¹𝐛^𝑇(𝑥(𝑡))𝑈⊗Λ^−𝑇−Ψ^−𝑇𝑄^𝑇𝑄Λ⁻¹−Ψ⁻¹𝐛(𝑥(𝑡))+2𝐛^𝐓(𝑥(𝑡))(𝑈⊗𝑄)×−𝐛(𝑥(𝑡))+𝐼_𝑁⊗𝐴𝐟(𝑥(𝑡))+𝐼_𝑁⊗𝐵𝐟(𝑥(𝑡−𝜏(𝑡)))+𝐈(𝑡)+2𝐛^𝐓(𝑥(𝑡))×𝑈⊗𝑄Λ⁻¹−𝑙𝐼_𝑁⊗𝐾[𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡)]+(𝐿⊗𝐹)𝑥(𝑡)+(𝐿⊗(𝐾+𝐽))𝑥(𝑡−𝜏(𝑡))+2𝐛̇‌^𝐓(𝑥(𝑡))𝑈⊗𝑄Λ⁻¹−𝑅Ψ⁻¹𝑥(𝑡)+2𝑦^𝑇(𝑡)𝑈⊗Π^𝑇𝑅Ψ⁻¹−2Γ^𝑇𝑄Λ⁻¹𝑥(𝑡),(3.14)𝑉₂̇‌(𝑥(𝑡))≤𝑥^𝑇𝑡−𝜏₀𝑈⊗𝑃₁𝑥𝑡−𝜏₀+2𝑥^𝑇𝑡−𝜏₀𝑈⊗𝑅₁𝐟𝑥𝑡−𝜏₀+𝐟^𝐓𝑥𝑡−𝜏₀×𝑈⊗𝑄₁𝐟𝑥𝑡−𝜏₀−(1−𝜇)⎡⎢⎣𝑥^𝑇(𝑡−𝜏(𝑡))𝑈⊗𝑃₁𝑥(𝑡−𝜏(𝑡))+2𝑥^𝐓(𝑡−𝜏(𝑡))𝑈⊗𝑅₁𝐟(𝑥(𝑡−𝜏(𝑡))),+𝐟^T(𝑥(𝑡−𝜏(𝑡)))𝑈⊗𝑄₁𝐟(𝑥(𝑡−𝜏(𝑡)))⎤⎥⎦,(3.15)𝑉₃̇‌(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑈⊗𝑃₂𝑥(𝑡)+2𝑥^𝑇(𝑡)𝑈⊗𝑅₂𝐟(𝑥(𝑡))+𝐟^𝐓(𝑥(𝑡))𝑈⊗𝑄₂𝐟(𝑥(𝑡))−𝑥^𝑇𝑡−𝜏₀𝑈⊗𝑃₂𝑥𝑡−𝜏₀+2𝑥^𝑇𝑡−𝜏₀𝑈⊗𝑅₂𝐟𝑥𝑡−𝜏₀+𝐟^𝐓𝑥𝑡−𝜏₀𝑈⊗𝑄₂𝐟𝑥𝑡−𝜏₀,(3.16)𝑉₄̇‌(𝑥(𝑡))=𝑥^𝑇𝑡−𝜏₀𝑈⊗𝑃₃𝑥𝑡−𝜏₀+2𝑥^𝑇𝑡−𝜏₀𝑈⊗𝑅₃𝐟𝑥𝑡−𝜏₀+𝐟^𝐓𝑥𝑡−𝜏₀𝑈⊗𝑄₃×𝐟𝑥𝑡−𝜏₀−𝑥^𝑇𝑡−𝜏_𝑚𝑈⊗𝑃₃𝑥𝑡−𝜏_𝑚+2𝑥^𝑇𝑡−𝜏_𝑚𝑈⊗𝑅₃𝐟𝑥𝑡−𝜏_𝑚+𝐟^𝐓𝑥𝑡−𝜏_𝑚𝑈⊗𝑄₃𝐟𝑥𝑡−𝜏_𝑚,(3.17)𝑉₅̇‌(𝑥(𝑡))=𝑦^𝑇(𝑡)𝜏₀(𝑈⊗𝑆)+𝜏_𝑚(𝑈⊗𝑍)𝑦(𝑡)−^𝑡_𝑡−𝜏0𝑦^𝑇(𝑠)(𝑈⊗𝑆)𝑦(𝑠)𝑑𝑠−^𝑡−𝜏0_{𝑡−𝜏𝑚}𝑦^𝑇(𝑠)(𝑈⊗𝑍)𝑦(𝑠)𝑑𝑠.(3.18) For any 𝑛×𝑛 matrices 𝐿_𝑖(𝑖=1,2,3,4), it follows from (3.5) and (3.7) that 0=2𝑥^𝑇(𝑡)𝑈⊗𝐿^𝑇₁+𝑦^𝑇(𝑡)𝑈⊗𝐿^𝑇₂×−𝑦(𝑡)+𝑧(𝑡)−𝑙𝐼_𝑁⊗𝐾[𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡)]+(𝐿⊗𝐹)𝑥(𝑡)+(𝐿⊗(𝐾+𝐽))𝑥(𝑡−𝜏(𝑡)),(3.19)0=2𝑧^𝑇(𝑡)𝑈⊗𝐿^𝑇₃+𝑤^𝑇(𝑡)𝑈⊗𝐿^𝑇₄×−𝑤(𝑡)−𝐛(𝑥(𝑡))+𝐼_𝑁⊗𝐴𝐟(𝑥(𝑡))+𝐼_𝑁⊗𝐵𝐟(𝑥(𝑡−𝜏(𝑡)))+𝐈(𝑡).(3.20) By utilizing (A3) and (3.6), for any 𝑛×𝑛 diagonal matrices 𝐺≥0, 𝐸≥0, the following inequality holds 0≤2𝐛^𝐓(𝑥(𝑡))(𝑈⊗𝐺)𝑥(𝑡)−𝑥^𝑇(𝑡)𝑈⊗Γ^𝑇𝐺𝑥(𝑡)+𝑤^𝑇(𝑡)𝑈⊗Ψ^𝑇𝐸Ψ𝑤(𝑡)−𝑧^𝑇(𝑡)(𝑈⊗𝐸)𝑧(𝑡).(3.21) Meanwhile, based on (3.14) and (3.19), it is easy to check that 𝑈𝐿=𝑁𝐿, and 𝑈⊗𝑄Λ⁻¹(𝐿⊗𝐹)=(𝑁𝐿)⊗𝑄Λ⁻¹𝐹,𝑈⊗𝑄Λ⁻¹(𝐿⊗(𝐾+𝐽))=(𝑁𝐿)⊗𝑄Λ⁻¹(𝐾+𝐽),𝑈⊗𝐿^𝑇_𝑖(𝐿⊗𝐹)=(𝑁𝐿)⊗𝐿^𝑇_𝑖𝐹,𝑈⊗𝐿^𝑇_𝑖(𝐿⊗(𝐾+𝐽))=(𝑁𝐿)⊗𝐿^𝑇_𝑖(𝐾+𝐽),𝑖=1,2.(3.22) Here we can employ the following notations to simplify the subsequent proof 𝑥_𝑖𝑗=𝑥_𝑖−𝑥_𝑗,𝑦_𝑖𝑗=𝑦_𝑖−𝑦_𝑗,𝑧_𝑖𝑗=𝑧_𝑖−𝑧_𝑗,𝑤_𝑖𝑗=𝑤_𝑖−𝑤_𝑗,𝛽𝑥_𝑖𝑗=𝛽𝑥_𝑖−𝛽𝑥_𝑗,𝑓𝑥_𝑖𝑗=𝑓𝑥_𝑖−𝑓𝑥_𝑗.(3.23) Then together with (𝑈⊗𝑄)𝐈(𝑡)=0 in (3.14) and (𝑈⊗𝐿^𝑇₃)𝐈(𝑡)=(𝑈⊗𝐿^𝑇₄)𝐈(𝑡)=0 in (3.19), it follows from Lemma 3.2 and (3.14)–(3.22) that 𝑉̇‌(𝑥(𝑡))≤−_{1≤𝑖<𝑗≤𝑁}⎧⎨⎩𝐮_𝑖𝑗2𝑥^𝑇_𝐢𝐣(𝑡)𝑃𝑦_𝑖𝑗(𝑡)+2Π𝑥_𝑖𝑗(𝑡)−𝛽𝑥_𝑖𝑗(𝑡)^𝑇𝑅Ψ⁻¹𝑦_𝑖𝑗(𝑡)+𝛿𝑧^𝑇_𝑖𝑗(𝑡)𝐼_𝑛𝑧_𝑖𝑗(𝑡)+𝛿⁻¹𝛽^𝑇𝑥_𝑖𝑗(𝑡)Λ^−𝑇−Ψ^−𝑇𝑄^𝑇𝑄Λ⁻¹−Ψ⁻¹𝛽𝑥_𝑖𝑗(𝑡)+2𝛽^𝑇𝑥_𝑖𝑗(𝑡)×𝑄−𝛽𝑥_𝑖𝑗(𝑡)+𝐴𝑓𝑥_𝑖𝑗(𝑡)+𝐵𝑓𝑥_𝑖𝑗(𝑡−𝜏(𝑡))+2𝛽^𝑇𝑥_𝑖𝑗(𝑡)𝑄Λ⁻¹×−𝑙𝐾𝑥_𝑖𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖𝑗(𝑡)+2𝛽̇‌^𝑇𝑥_𝑖𝑗(𝑡)𝑄Λ⁻¹−𝑅Ψ⁻¹𝑥_𝑖𝑗(𝑡)+2𝑦^𝑇_𝑖𝑗(𝑡)×Π^𝑇𝑅Ψ⁻¹−2Γ^𝑇𝑄Λ⁻¹𝑥_𝑖𝑗(𝑡)+⎡⎢⎣𝑥^𝑇_𝑖𝑗𝑡−𝜏₀𝑃₁−𝑃₂+𝑃₃𝑥_𝑖𝑗𝑡−𝜏₀+2𝑥^𝑇_𝑖𝑗𝑡−𝜏₀𝑅₁−𝑅₂+𝑅₃×𝑓𝑥_𝑖𝑗𝑡−𝜏₀+𝑓^𝑇𝑥_𝑖𝑗𝑡−𝜏₀𝑄₁−𝑄₂+𝑄₃𝑓𝑥_𝑖𝑗𝑡−𝜏₀⎤⎥⎦×⎡⎢⎣𝑥^𝑇_𝑖𝑗(𝑡−𝜏(𝑡))𝑃₁𝑥_𝑖𝑗(𝑡−𝜏(𝑡))+2𝑥^𝑇_𝑖𝑗(𝑡−𝜏(𝑡))𝑅₁𝑓𝑥_𝑖𝑗(𝑡−𝜏(𝑡))+𝑓^𝑇𝑥_𝑖𝑗(𝑡−𝜏(𝑡))𝑄₁𝑓𝑥_𝑖𝑗(𝑡−𝜏(𝑡))⎤⎥⎦×𝑥^𝑇_𝑖𝑗(𝑡)𝑃₂𝑥_𝑖𝑗(𝑡)+2𝑥^𝑇_𝑖𝑗(𝑡)𝑅₂𝑓𝑥_𝑖𝑗(𝑡)+𝑓^𝑇𝑥_𝑖𝑗(𝑡)𝑄₂𝑓𝑥_𝑖𝑗(𝑡)−⎡⎢⎣𝑥^𝑇_𝑖𝑗𝑡−𝜏_𝑚𝑃₃𝑥_𝑖𝑗𝑡−𝜏_𝑚+2𝑥^𝑇_𝑖𝑗𝑡−𝜏_𝑚𝑅₃𝑓𝑥_𝑖𝑗𝑡−𝜏_𝑚+𝑓^𝑇𝑥_𝑖𝑗𝑡−𝜏_𝑚Q₃𝑓𝑥_𝑖𝑗𝑡−𝜏_𝑚⎤⎥⎦+𝑦^𝑇_𝑖𝑗(𝑡)𝜏₀𝑆+𝜏_𝑚𝑍𝑦_𝑖𝑗(𝑡)−^𝑡_𝑡−𝜏0𝑦^𝑇_𝑖𝑗(𝑠)𝑆𝑦_𝑖𝑗(𝑠)𝑑𝑠−^𝑡−𝜏0_{𝑡−𝜏𝑚}𝑦^𝑇_𝑖𝑗(𝑠)𝑍𝑦_𝑖𝑗(𝑠)𝑑𝑠+2𝑥^𝑇_𝑖𝑗(𝑡)𝐿^𝑇₁+𝑦^𝑇_𝑖𝑗(𝑡)𝐿^𝑇₂×−𝑦_𝑖𝑗(𝑡)+𝑧_𝑖𝑗(𝑡)−𝑙𝐾𝑥_𝑖𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖𝑗(𝑡)+2𝑧^𝑇_𝑖𝑗(𝑡)𝐿^𝑇₃+𝑤^𝑇_𝑖𝑗(𝑡)𝐿^𝑇₄−𝑤_𝑖𝑗(𝑡)−𝛽𝑥_𝑖𝑗(𝑡)+𝐴𝑓𝑥_𝑖𝑗(𝑡)+𝐵𝑓𝑥_𝑖𝑗(𝑡−𝜏(𝑡))+2𝛽^𝑇𝑥_𝑖𝑗(𝑡)𝐺𝑥_𝑖𝑗(𝑡)−𝑥^𝑇_𝑖𝑗(𝑡)Γ^𝑇𝐺𝑥_𝑖𝑗(𝑡)+𝑤^𝑇_𝑖𝑗(𝑡)Ψ^𝑇𝐸Ψ𝑤_𝑖𝑗(𝑡)−𝑧^𝑇_𝑖𝑗(𝑡)𝐸𝑧_𝑖𝑗(𝑡)+2𝑁𝑙_𝑖𝑗𝛽^𝑇𝑥_𝑖𝑗(𝑡)𝑄Λ⁻¹+𝑥^𝑇_𝑖𝑗(𝑡)𝐿^𝑇₁+𝑦^𝑇_𝑖𝑗(𝑡)𝐿^𝑇₂𝐹𝑥_𝑖𝑗(𝑡)+𝛽^𝑇𝑥_𝑖𝑗(𝑡)𝑄Λ⁻¹+𝑥^𝑇_𝑖𝑗(𝑡)𝐿^𝑇₁+𝑦^𝑇_𝑖𝑗(𝑡)𝐿^𝑇₂(𝐾+𝐽)𝑥_𝑖𝑗(𝑡−𝜏(𝑡))⎫⎬⎭.(3.24) For any 𝑛×𝑛 diagonal matrices 𝑈>0, 𝑉>0, 𝑊>0, 𝐻>0, 𝑇_𝑖>0(𝑖=1,2), and Σ_𝑖, Π_𝑖, Υ_𝑖(𝑖=1,2) in (A3)-(A4), it can be deduced that 0≤_{1≤𝑖<𝑗≤𝑁}⎧⎨⎩−𝑥^𝑇_𝑖𝑗(𝑡)𝑈Σ₁𝑥_𝑖𝑗(𝑡)−2𝑥^𝑇_𝑖𝑗(𝑡)𝑈Σ₂𝑓𝑥_𝑖𝑗(𝑡)+𝑓^𝑇𝑥_𝑖𝑗(𝑡)𝑈𝑓𝑥_𝑖𝑗(𝑡)−⎡⎢⎣𝑥^𝑇_𝑖𝑗(𝑡−𝜏(𝑡))𝑉Σ₁×𝑥_𝑖𝑗(𝑡−𝜏(𝑡))−2𝑥^𝑇_𝑖𝑗(𝑡−𝜏(𝑡))𝑉Σ₂𝑓𝑥_𝑖𝑗(𝑡−𝜏(𝑡))+𝑓^𝑇𝑥_𝑖𝑗(𝑡−𝜏(𝑡))𝑉𝑓𝑥_𝑖𝑗(𝑡−𝜏(𝑡))⎤⎥⎦−𝑥^𝑇_𝑖𝑗𝑡−𝜏₀𝑊Σ₁𝑥_𝑖𝑗𝑡−𝜏₀−2𝑥^𝑇_𝑖𝑗𝑡−𝜏₀𝑊Σ₂𝑓𝑥_𝑖𝑗𝑡−𝜏₀+𝑓^𝑇𝑥_𝑖𝑗𝑡−𝜏₀𝑊𝑓𝑥_𝑖𝑗𝑡−𝜏₀−𝑥^𝑇_𝑖𝑗𝑡−𝜏_𝑚𝐻Σ₁𝑥_𝑖𝑗𝑡−𝜏_𝑚−2𝑥^𝑇_𝑖𝑗𝑡−𝜏_𝑚𝐻Σ₂𝑓𝑥_𝑖𝑗𝑡−𝜏_𝑚+𝑓^𝑇𝑥_𝑖𝑗𝑡−𝜏_𝑚𝐻𝑓𝑥_𝑖𝑗𝑡−𝜏_𝑚−𝑥^𝑇_𝑖𝑗(𝑡)𝑇₁Π₁𝑥_𝑖𝑗(𝑡)−2𝑥^𝑇_𝑖𝑗(𝑡)𝑇₁Π₂𝛽𝑥_𝑖𝑗(𝑡)+𝛽^𝑇𝑥_𝑖𝑗(𝑡)𝑇₁𝛽𝑥_𝑖𝑗(𝑡)−𝑥^𝑇_𝑖𝑗(𝑡)𝑇₂Υ₁𝑥_𝑖𝑗(𝑡)−2𝑥^𝑇_𝑖𝑗(𝑡)𝑇₂Υ₂𝛽̇‌𝑥_𝑖𝑗(𝑡)+𝛽̇‌^𝑇𝑥_𝑖𝑗(𝑡)𝑇₂𝛽̇‌𝑥_𝑖𝑗(𝑡)⎫⎬⎭.(3.25) For any 13 𝑛×𝑛 matrices 𝑁_𝑖(𝑖=1,2,3), it follows from Newton-Leibniz formula that 0=2_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑡)𝑁₁𝑥_𝑖𝑗(𝑡)−𝑥_𝑖𝑗𝑡−𝜏₀−^𝑡−𝜏0_{𝑡−𝜏(𝑡)}𝑦_𝑖𝑗(𝑠)+𝑁₂𝑥_𝑖𝑗𝑡−𝜏₀−𝑥_𝑖𝑗(𝑡−𝜏(𝑡))−^𝑡−𝜏0_{𝑡−𝜏(𝑡)}𝑦_𝑖𝑗(𝑠)+𝑁₃𝑥_𝑖𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖𝑗𝑡−𝜏_𝑚−^{𝑡−𝜏(𝑡)}_{𝑡−𝜏𝑚}𝑦_𝑖𝑗(𝑠),(3.26) where 𝜁^𝑇_𝑖𝑗(𝑡)=𝑥^𝑇_𝑖𝑗(𝑡)𝑥^𝑇_𝑖𝑗𝑡−𝜏₀𝑥^𝑇_𝑖𝑗𝑡−𝜏_𝑚𝑓^𝑇𝑥_𝑖𝑗(𝑡)𝑓^𝑇𝑥_𝑖𝑗𝑡−𝜏₀𝑓^𝑇𝑥_𝑖𝑗𝑡−𝜏_𝑚𝑦^𝑇_𝑖𝑗(𝑡)𝑧^𝑇_𝑖𝑗(𝑡)×𝑤^𝑇_𝑖𝑗(𝑡)𝑥^𝑇_𝑖𝑗(𝑡−𝜏(𝑡))𝑓^𝑇𝑥_𝑖𝑗(𝑡−𝜏(𝑡))𝛽^𝑇𝑥_𝑖𝑗(𝑡)𝛽̇‌^𝑇𝑥_𝑖𝑗(𝑡).(3.27) Now together with the terms (3.24)–(3.26), and 𝑢_𝑖𝑗=−1, we can deduce that 𝑉̇‌(𝑥(𝑡))≤_1≤_{𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑡)⎧⎨⎩Ω_𝑖𝑗+$+$^𝑇+𝛿⁻¹𝐻𝐻^𝑇+𝜏₀𝑁₁𝑆⁻¹𝑁^𝑇₁+𝜏(𝑡)−𝜏₀𝑁₂𝑍⁻¹𝑁^𝑇₂+𝜏_𝑚−𝜏(𝑡)𝑁₃𝑍⁻¹𝑁^𝑇₃⎫⎬⎭𝜁_𝑖𝑗(𝑡)∶=_{1≤𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑡)Δ_𝑖𝑗(𝑡)𝜁_𝑖𝑗(𝑡),(3.28) where Ω_𝑖𝑗,$, and 𝐻 are presented in (3.2). Through using Schur-complement and convex combination, the LMIs in (3.2) can guarantee Δ_𝑖𝑗(𝑡)<0 and thus, there must exist one scalar 𝜒>0 such that Δ_𝑖𝑗(𝑡)≤−𝜒𝐼<0. Therefore, one can get 𝑉̇‌(𝑥(𝑡))≤_1≤_{𝑖<𝑗≤𝑁}𝜁^𝑇_𝑖𝑗(𝑡)Δ_𝑖𝑗(𝑡)𝜁_𝑖𝑗(𝑡)≤−𝜒⎡⎢⎣_1≤_{𝑖<𝑗≤𝑁}‖‖𝑥_𝑖𝑗(𝑡)‖‖²+_1≤_{𝑖<𝑗≤𝑁}‖‖𝑥_𝑖𝑗(𝑡−𝜏(𝑡))‖‖²⎤⎥⎦,(3.29) which indicates that the system (3.1) can reach the global asymptotical synchronization.
Based on (A1)–(A4), (3.9), and direct computing, there must exist three scalars Θ_𝑖>0(𝑖=1,2,3) such that𝑉(𝑥(𝑡))≤_{1≤𝑖<𝑗≤𝑁}Θ₁‖‖𝑥_𝑖𝑗(𝑡)‖‖²+Θ₂^𝑡_{𝑡−𝜏𝑚}‖‖𝑥_𝑖𝑗(𝑠)‖‖²𝑑𝑠+Θ₃^𝑡_{𝑡−𝜏𝑚}‖‖𝑥_𝑖𝑗(𝑠−𝜏(𝑠))‖‖²𝑑𝑠.(3.30) Letting 𝑉(𝑥(𝑡))=𝑒^2𝑘𝑡𝑉(𝑥(𝑡)), one can deduce that ̇‌𝑉(𝑥(𝑡))=2𝑘𝑒^2𝑘𝑡𝑉(𝑥(𝑡))+𝑒^2𝑘𝑡𝑉(̇‌𝑥(𝑡)), and 𝑉(𝑥(𝑡))−𝑉(𝑥(0))=^𝑡_𝑜̇‌𝑉(𝑥(𝑠))𝑑𝑠≤_{1≤𝑖<𝑗≤𝑁}^𝑡_𝑜𝑒^2𝑘𝑠−𝜒‖‖𝑥_𝑖𝑗(𝑠)‖‖²+‖‖𝑥_𝑖𝑗(𝑠−𝜏(𝑠))‖‖²+2𝑘Θ₁‖‖𝑥_𝑖𝑗(𝑠)‖‖²+Θ₂^𝑠_{𝑠−𝜏𝑚}‖‖𝑥_𝑖𝑗(𝜃)‖‖²𝑑𝜃+Θ₃^𝑠_{𝑠−𝜏𝑚}‖‖𝑥_𝑖𝑗(𝜃−𝜏(𝜃))‖‖²𝑑𝜃𝑑𝑠.(3.31) By 1≤𝑖<𝑗≤𝑁 and changing the integration sequences, we have ^𝑡₀𝑒^2𝑘𝑠^𝑠_{𝑠−𝜏𝑚}‖‖𝑥_𝑖𝑗(𝜃)‖‖²𝑑𝜃𝑑𝑠≤𝜏_𝑚𝑒^{2𝑘𝜏𝑚}⁰_−𝜏𝑚‖‖𝑥_𝑖𝑗(𝜃)‖‖²𝑒^2𝑘𝜃𝑑𝜃+^𝑡₀‖‖𝑥_𝑖𝑗(𝜃)‖‖²𝑒^2𝑘𝜃𝑑𝜃,^𝑡₀𝑒^2𝑘𝑠^𝑠_{𝑠−𝜏𝑚}‖‖𝑥_𝑖𝑗(𝜃−𝜏(𝜃))‖‖²𝑑𝜃𝑑𝑠≤𝜏_𝑚𝑒^{2𝑘𝜏𝑚}×⁰_−2𝜏𝑚‖‖𝑥_𝑖𝑗(𝜃)‖‖²𝑒^2𝑘𝜃𝑑𝜃+^𝑡₀‖‖𝑥_𝑖𝑗(𝜃−𝜏(𝜃))‖‖²𝑒^2𝑘𝜃𝑑𝜃.(3.32) Substituting the terms (3.32) into the relevant ones in (3.31), it is easy to have 𝑉(𝑥(𝑡))≤𝑉(𝑥(0))+_{1≤𝑖<𝑗≤𝑁}2𝑘Θ₁+2𝑘Θ₂𝜏_m𝑒^2𝑘𝜏m−𝜒^𝑡₀‖‖𝑥_𝑖𝑗(𝜃)‖‖²𝑒^2𝑘𝜃𝑑𝜃+2𝑘Θ₃𝜏_𝑚𝑒^{2𝑘𝜏𝑚}−𝜒^𝑡₀‖‖𝑥_𝑖𝑗(𝜃−𝜏(𝜃))‖‖²𝑒^2𝑘𝜃𝑑𝜃+ℎ(𝑘)(3.33) in which ℎ(𝑘)=2𝑘Θ₂𝜏_𝑚𝑒^{2𝑘𝜏𝑚}∫⁰_−𝜏𝑚‖𝑥_𝑖𝑗(𝜃)‖²𝑑𝜃+2𝑘Θ₃𝜏_𝑚𝑒^{2𝑘𝜏𝑚}∫⁰_−2𝜏𝑚‖𝑥_𝑖𝑗(𝜃)‖²𝑑𝜃. Choose an appropriate scalar 𝑘₀>0 such that 2𝑘₀Θ₁+2𝑘₀Θ₂𝜏_𝑚𝑒^{2𝑘0𝜏𝑚}−𝜒≤0,2𝑘₀Θ₃𝜏_𝑚𝑒^{2𝑘0𝜏𝑚}−𝜒≤0, one has 𝑉(𝑥(𝑡))≤ℎ(𝑘₀)+𝑉(𝑥(0)). By directly computing, there must exist a positive scalar 𝜛>0 such that ℎ𝑘₀+𝑉(𝑥(0))≤𝜛_{1≤𝑖<𝑗≤𝑁}sup_{−2𝜏𝑚≤𝑠≤0}‖‖𝜙_𝑖(𝑠)−𝜙_𝑗(𝑠)‖‖².(3.34) Meanwhile, 𝑉(𝑥(𝑡))≥𝜆_min(𝑃)∑_{1≤𝑖<𝑗≤𝑁}𝑒^2𝑘0𝑡‖𝑥_𝑖𝑗(𝑡)‖². Therefore, it can be deduced that ‖‖𝑥_𝑖𝑗(𝑡)‖‖≤𝜆⁻¹_min(𝑃)𝜛⋅_1≤_{𝑖<𝑗≤𝑁}sup_{−2𝜏𝑚≤𝑠≤0}‖‖𝜙_𝑖(𝑠)−𝜙_𝑗(𝑠)‖‖𝑒^−𝑘0𝑡,∀𝑡≥0.(3.35) By Definition 2.2, the system (3.1) is globally exponentially synchronized, and the proof is completed.

Remark 3.4. Theorem 3.3 presents a novel delay-dependent criterion guaranteeing arrays of coupled Cohen-Grossberg neural networks (2.7) to be globally synchronized. In [16–22], the authors considered global synchronization of an array of coupled neural networks of simple forms and in the paper, we derive a more general delayed neural networks and extended the case to the time variable one, which generalizes the earlier ones. Moreover, the conditions are expressed in terms of LMIs, therefore, by using LMI in Matlab Toolbox, it is straightforward and convenient to check the feasibility of the proposed results without tuning any parameters.
If there does not exist one single delayed coupling in system (2.1), that is, 𝐾=0, which means that the restriction 𝑙₁₁=𝑙₂₂=⋯=𝑙_𝑁𝑁=𝑙 in 𝐿=[𝑙_𝑖𝑗]_𝑁×𝑁 is removed. Then together with the proof of Theorem 3.3, we can derive the following theorem.

Theorem 3.5. Supposing that assumptions (A₁)–(A₄) hold, then the dynamical system (3.1) is globally exponentially synchronized, if there exist n×n matrices P>0, 𝑆>0, 𝑍>0, 𝐿_𝑖(𝑖=1,2,3,4), 𝑛×𝑛 matrices 𝑃_𝑙>0, 𝑄_𝑙>0, 𝑅_𝑙(l=1,2,3) making ^{𝑃𝑙𝑅𝑙∗𝑄}_𝑙≥0, n×n diagonal matrices 𝑅>0, 𝑄>0, 𝐺>0, 𝐸>0, 𝑈>0, 𝑉>0, 𝑊>0, 𝐻>0, 𝑇_𝑖>0(𝑖=1,2), 13𝑛×𝑛 matrices 𝑁_𝑖(𝑖=1,2,3), and one scalar 𝛿>0 such that, for 1≤𝑖<𝑗≤𝑁, the LMIs in (3.36) hold ⎡⎢⎢⎢⎢⎣Ω_𝑖𝑗+$+$^𝑇𝐻√𝜏₀𝑁₁√𝜏_𝑚𝑁₂∗−𝛿𝐼00∗∗−𝑆0∗∗∗−𝑍⎤⎥⎥⎥⎥⎦<0,⎡⎢⎢⎢⎢⎣Ω_𝑖𝑗+$+$^𝑇𝐻√𝜏₀𝑁₁√𝜏_𝑚𝑁₃∗−𝛿𝐼00∗∗−𝑆0∗∗∗−𝑍⎤⎥⎥⎥⎥⎦<0,(3.36) where $=[𝑁₁−𝑁₁+𝑁₂−𝑁₃0_{6𝑛⋅13𝑛}−𝑁₂+𝑁₃0_{3𝑛⋅13𝑛}], 𝐻=[0_{𝑛⋅11𝑛}𝑄(Λ⁻¹−Ψ⁻¹)0_𝑛⋅𝑛]^𝑇, Ω_𝑖𝑗=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣Ξ₁₁00Ξ₁₄00Ξ₁₇𝐿^𝑇₁0Ξ_1,100Ξ_1,12Ξ_1,13∗Ξ₂₂00Ξ₂₅00000000∗∗Ξ₃₃00Ξ₃₆0000000∗∗∗Ξ₄₄000𝐴^𝑇𝐿₃𝐴^𝑇𝐿₄00𝐴^𝑇𝑄^𝑇0∗∗∗∗Ξ₅₅00000000∗∗∗∗∗Ξ₆₆0000000∗∗∗∗∗∗Ξ₇₇−𝐿^𝑇₂0Ξ_7,100−Ψ⁻¹𝑅0∗∗∗∗∗∗∗Ξ₈₈−𝐿^𝑇₃0𝐿^𝑇₃𝐵−𝐿^𝑇₃0∗∗∗∗∗∗∗∗Ξ₉₉0𝐿^𝑇₄𝐵−𝐿^𝑇₄0∗∗∗∗∗∗∗∗∗Ξ_10,10Ξ_10,11Ξ_10,120∗∗∗∗∗∗∗∗∗∗Ξ_11,11𝐵^𝑇𝑄^𝑇0∗∗∗∗∗∗∗∗∗∗∗Ξ_12,120∗∗∗∗∗∗∗∗∗∗∗∗−𝑇₂⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦(3.37) with Ξ₁₁=𝑃₂−𝑙_𝑖𝑗𝑁(𝐿^𝑇₁𝐹+𝐹^𝑇𝐿₁)−Γ^𝑇𝐺−𝐺^𝑇Γ−𝑈Σ₁−𝑇₁Π₁−𝑇₂Υ₁, Ξ₁₄=𝑅₂+𝑈Σ₂, Ξ₁₇=𝑃−𝐿^𝑇₁−𝑙_𝑖𝑗𝑁𝐹^𝑇𝐿₂+2Π^𝑇𝑅Ψ⁻¹−2Γ^𝑇𝑄Λ⁻¹, Ξ_1,10=−𝑙_𝑖𝑗𝑁𝐿^𝑇₁𝐽, Ξ_1,12=−𝑙_𝑖𝑗𝑁𝐹^𝑇𝑄Λ⁻¹+𝐺^𝑇+𝑇₁Π₂, Ξ_1,13=𝑇₂Υ₂+𝑄Λ⁻¹−𝑅Ψ⁻¹, Ξ₂₂=−𝑃₂+𝑃₁+𝑃₃−𝑊Σ₁, Ξ₂₅=−𝑅₂+𝑅₁+𝑅₃+𝑊Σ₂, Ξ₃₃=−𝑃₃−𝐻Σ₁, Ξ₃₆=𝐻Σ₂−𝑅₃, Ξ₄₄=−𝑈+𝑄₂, Ξ₅₅=−𝑄₂+𝑄₁+𝑄₃−𝑊, Ξ₆₆=−𝑄₃−𝐻, Ξ₇₇=−𝐿^𝑇₂−𝐿₂+𝜏₀𝑆+𝜏_𝑚𝑍, Ξ_7,10=−𝑙_𝑖𝑗𝑁𝐿^𝑇₂𝐽, Ξ₈₈=−𝐸+𝛿𝐼_𝑛, Ξ₉₉=−𝐿^𝑇₄−𝐿₄+Ψ^𝑇𝐸Ψ, Ξ_10,10=−(1−𝜇)𝑃₁−𝑉Σ₁, Ξ_10,11=−(1−𝜇)𝑅₁+𝑉Σ₂, Ξ_10,12=−𝑙_𝑖𝑗𝑁𝐽^𝑇𝑄Λ⁻¹, Ξ_11,11=−(1−𝜇)𝑄₁−𝑉, Ξ_12,12=−2𝑄^𝑇−𝑇₁.

Proof. Letting 𝐾=0 in system (2.1) and employing similar methods of proving Theorem 3.3, we can easily derive the theorem and the detailed proof is omitted here.

Remark 3.6. Theorems 3.3-3.5 require the upper bound 𝜇 of time-delay 𝜏(𝑡) to be known. If 𝜇 is unknown, by setting 𝑃₁=𝑅₁=𝑄₁ in (3.9), we can derive the delay-dependent and delay-derivative-independent synchronization criteria for the coupled systems (2.7) based on Theorems 3.3-3.5.

Remark 3.7. Together with the similar proof, we can deal with the global synchronization for arrays of coupled CGNNs with hybrid couplings described in [18]. Moreover, we still can investigate the global synchronization for arrays of coupled uncertain CGNNs with more general forms𝑥̇_𝑖(𝑡)=−𝛼𝑥_𝑖(𝑡)𝛽𝑥_𝑖(𝑡)−𝐴(𝑡)𝑓𝑥_𝑖(𝑡)−𝐵(𝑡)𝑓𝑥_𝑖(𝑡−𝜏(𝑡))−𝐈(𝑡)+^𝑁_{𝑗=1,𝑗≠𝑖}𝑙¹_𝑖𝑗𝐹𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡)+^𝑁_{𝑗=1,𝑗≠𝑖}𝑙²_𝑖𝑗𝐾𝑥_𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡)+^𝑁_{𝑗=1,𝑗≠𝑖}𝑙³_𝑖𝑗𝐽𝑥_𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡−𝜏(𝑡)),(3.38) in which 𝐴(𝑡)=𝐴+Δ𝐴(𝑡), 𝐵(𝑡)=𝐵+Δ𝐵(𝑡), and we set 𝐿^ℎ=[𝑙^ℎ_𝑖𝑗]_𝑛×𝑛 for ℎ=,1,2,3. Here Δ𝐴(𝑡) and Δ𝐵(𝑡) are unknown matrices representing variable parametric uncertainties satisfying [Δ𝐴(𝑡)Δ𝐵(𝑡)]=𝐹₀Δ(𝑡)𝐸₁𝐸₂,Δ(𝑡)=Λ(𝑡)𝐼−𝐽₀Λ(𝑡)⁻¹,𝐼−𝐽^𝑇₀𝐽₀>0,(3.39) in which 𝐹₀,𝐽₀,𝐸_𝑖(𝑖=1,2) are known appropriately as dimensional matrices and Λ(𝑡) is an unknown variable matrix function satisfying Λ^𝑇(𝑡)Λ(𝑡)≤𝐼. By utilizing the similar methods in Theorems 3.3-3.5, one can easily derive the more general results based on [30, Lemma 5].

Remark 3.8. It is worth pointing out that it is possible to extend our main results to more complex Cohen-Grossberg neural networks, such as CGNNs with distributed delay, stochastic perturbations, and Markovian jumping parameters. The corresponding results will appear in our future works.

4. Numerical Examples

In the section, two examples are provided to illustrate the effectiveness of the proposed results.

Example 4.1. We consider the delayed Cohen-Grossberg neural network models described by 𝑥̇(𝑡)=−𝛼(𝑥(𝑡))[𝛽(𝑥(𝑡))−𝐴𝑓(𝑥(𝑡))−𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))−𝐼(𝑡)](4.1) with 𝛼(𝑥)=diag{0.8+0.2|cos(2𝑥₁)|,1.0−0.2|cos(2𝑥₂)|,0.8+0.2|sin(2𝑥₃)|}, 𝐴=⎡⎢⎢⎣−101021010−100⎤⎥⎥⎦,𝐵=⎡⎢⎢⎣32−22−32−223⎤⎥⎥⎦,𝛽(𝑥)=⎡⎢⎢⎢⎣0.9𝑥₁+0.05sin2𝑥₁0.9𝑥₂+0.05cos2𝑥₂0.9𝑥₃+0.05sin2𝑥₃⎤⎥⎥⎥⎦,𝑓(𝑥)=⎡⎢⎢⎢⎣0.3||𝑥₁+1||−||𝑥₁−1||0.3||𝑥₂+1||−||𝑥₂−1||0.3||𝑥₃+1||−||𝑥₃−1||⎤⎥⎥⎥⎦,𝐼(𝑡)=0,𝜏(𝑡)=0.4+0.2sin(20𝑡)+0.05cos²(40𝑡).(4.2) One can get 𝜏₀=0.2, 𝜏_𝑚=0.65, 𝜇=8, and the functions 𝛽_𝑖(⋅),𝑓_𝑖(⋅) satisfy the assumptions (A3) and (A4), respectively. Through setting the inner linking matrix 𝐿=^−2111−21₁₁₋₂, we consider a dynamical networks comprised of three linearly coupled identical neural network models with time-delay couplings as follows: 𝑥̇_𝑖(𝑡)=−𝛼𝑥_𝑖(𝑡)𝛽𝑥_𝑖(𝑡)−𝐴𝑓𝑥_𝑖(𝑡)−𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))+³_𝑗=1𝑙_𝑖𝑗𝐹𝑥_𝑗(𝑡)+³_𝑗=1𝑙_𝑖𝑗(𝐾+𝐽)𝑥_𝑗(𝑡−𝜏(𝑡))−𝑙𝐾𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡),𝑖=1,2,3,(4.3) and choose the coupling matrices as 𝐹=⎡⎢⎢⎣100001000010⎤⎥⎥⎦,𝐾=⎡⎢⎢⎣0.050000.050000.05⎤⎥⎥⎦,𝐽=⎡⎢⎢⎣0.10000.10000.1⎤⎥⎥⎦.(4.4) Figure 1 shows that the system has a chaotic attractor. Together with Theorem 3.3 and LMI in Matlab Toolbox, it is easy to check that there exists the feasible solution to the LMIs in (3.2), which can guarantee the array of the system (4.3) to achieve the exponential synchronization. The total error is defined by error(𝑡)=³_𝑖=1𝑥_1𝑖(𝑡)−𝑥_2𝑖(𝑡)²+𝑥_2𝑖(𝑡)−𝑥_3𝑖(𝑡)²(4.5) and the synchronization error can be seen in Figure 1. During the process of simulation, the initial conditions of nodes are selected as 𝑥₁=[−0.5,−0.3,0.3]^𝑇, 𝑥₂=[0.7,−0.5,−0.6]^𝑇, and 𝑥₃=[1,0.5,0.3]^𝑇.

(a)

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Figure 1 

The synchronized trajectory and total synchronous error of system (4.3).

Example 4.2. Consider one 2-dimensional delayed Cohen-Grossberg neural networks as follows: 𝑥̇(𝑡)=−𝛼(𝑥(𝑡))[𝛽(𝑥(𝑡))−𝐴𝑓(𝑥(𝑡))−𝐵𝑓(𝑥(𝑡−𝜏(𝑡)))−𝐼(𝑡)],(4.6) where 𝛼(𝑥)=diag{0.9+0.1sin(𝑥₁),0.9+0.1cos(𝑥₁)},𝛽(𝑥)=^{1.1𝑥1+0.1sin(𝑥1)1.1𝑥}₂^+0.1sin(𝑥₂⁾, 𝐴=^2−0.2_−0.33, 𝐵=^−1.5−0.1_−0.2−2.5, 𝐈(𝑡)=^0.1_0.1, 𝑓(𝑥)=^{0.7tanh(𝑥1)+0.15(|𝑥1+1|−|𝑥1−1|)0.7tanh(𝑥}₂^)+0.15(|𝑥₂^+1|−|𝑥₂^−1|), and 𝜏(𝑡)=0.5+0.3cos(6𝑡)+0.05sin²(40𝑡). Choosing the following inner linking matrix and the coupling matrices, respectively, 𝐿=⎡⎢⎢⎣−3121−2121−3⎤⎥⎥⎦,𝐹=100010,𝐾=0,𝐽=0.1000.1,(4.7) we still consider a dynamical networks comprised of three coupled identical CGNNs with delayed couplings as 𝑥̇_𝑖(𝑡)=−𝛼𝑥_𝑖(𝑡)𝛽𝑥_𝑖(𝑡)−𝐴𝑓𝑥_𝑖(𝑡)−𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))−𝐼(𝑡)+³_𝑗=1𝑙_𝑖𝑗𝐹𝑥_𝑗(𝑡)+³_𝑗=1𝑙_𝑖𝑗𝐾𝑥_𝑗(𝑡−𝜏(𝑡)).(4.8) Then based on Theorem 3.5 and Matlab LMI Toolbox, one can get part feasible solution to (3.36) as follows: 𝑃=2.06350.02340.02342.2078,𝑆=0.0274−0.0036−0.00360.0032,𝑍=0.07570.00160.00160.0870,𝑅=0.0415000.0415,𝑄=0.0966000.0966,𝐺=1.2804001.2804,𝐸=0.0658000.0658,𝑈=9.4219009.4219,𝑉=0.6964000.6964,𝑊=1.1489001.1489,𝐻=0.6611000.6611,𝑇₁=1.9626001.9626,𝑇₂=1.6523001.6523,𝑃₁𝑅₁𝑅^𝑇₁𝑄₁=⎡⎢⎢⎢⎣0.02010.0013−0.03160.00050.00130.02860.0003−0.0281−0.03160.00030.0621−0.00470.0005−0.0281−0.00470.0289⎤⎥⎥⎥⎦,𝑃₂𝑅₂𝑅^𝑇₂𝑄₂=⎡⎢⎢⎢⎣4.27300.8822−0.4717−0.00020.882210.22510.0277−0.3621−0.47170.02772.63800.0486−0.0002−0.36210.04860.7271⎤⎥⎥⎥⎦,𝑃₃𝑅₃𝑅^𝑇₃𝑄₃=⎡⎢⎢⎢⎣2.01310.3881−0.24550.00470.38814.65320.0180−0.2932−0.24550.01801.56190.01790.0047−0.29320.01790.9050⎤⎥⎥⎥⎦(4.9) which means that the global exponential synchronization is achieved for system (4.8). The total error of the array of the system (4.8) is defined by error(𝑡)=²_𝑖=1𝑥_1𝑖(𝑡)−𝑥_2𝑖(𝑡)²+𝑥_2𝑖(𝑡)−𝑥_3𝑖(𝑡)²,(4.10) and the synchronization state and total synchronous error can be depicted in Figure 2 with the initial conditions 𝑥₁=[−0.5,−0.3]^𝑇, 𝑥₂=[0.3,0.7]^𝑇, and 𝑥₃=[−0.5,−0.6]^𝑇. Moreover, if we choose 𝛼(𝑥)=diag{0.8+0.2(𝑒^𝑥1/1+𝑒^𝑥1),0.8+0.2(𝑒^𝑥2/1+𝑒^𝑥2)}, and 𝜏(𝑡)=0.5+0.3sin(20𝑡)+0.05cos²(30𝑡), then the synchronization state and total synchronous error can be described in Figure 3.

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(b)

Figure 2 

The synchronized trajectory and total synchronous error of system (4.8).

(a)

(b)

Figure 3 

The synchronized trajectory and total synchronous error of system (4.8).

5. Conclusions

This paper has investigated the global exponential synchronization for the coupled Cohen-Grossberg neural networks with both delayed couplings and time-varying delay. Two novel conditions have been derived by employing Lyapunov-Krasovskii functional and convex combination. It is worth pointing out that, the addressed systems can include many neural network models as its special cases and some good mathematical techniques have been employed, which improve and extend those present results. The derived synchronization criteria are presented in terms of LMIs, which can be checked easily by resorting to Matlab LMI Toolbox. Finally, two numerical examples are utilized to illustrate the effectiveness of the derived methods based on the simulation results.

Acknowledgment

This work is supported by the national Natural Science Foundation of China nos. 60875035, 60804017, 60904023, 61004032, the Special Foundation of China Postdoctoral Science Funded Project no. 201003546, Jiangsu Planned Projects for Postdoctoral Research Funds no. 0901005B, and China Postdoctoral Science Foundation Funded Project no. 200904501033.