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Mathematical Problems in Engineering
Volume 2011 (2011), Article ID 831695, 22 pages
Research Article

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

1Key Laboratory of Measurement and Control of CSE, School of Automation, Southeast University, Ministry of Education, Nanjing 210096, China
2School of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 2010016, China

Received 23 November 2010; Accepted 9 March 2011

Academic Editor: Bin Liu

Copyright © 2011 Haitao Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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 [628].

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 [1228]. 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 [1625, 28]. Yet, the system forms addressed in [1625] 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. 𝐑𝑛 denotes the 𝑛-dimensional Euclidean space, and 𝐑𝑛×𝑚 is the set of all 𝑛×𝑚 real matrices. For the symmetric matrices 𝑋,𝑌,𝑋>𝑌 (respectively, 𝑋𝑌) means that 𝑋𝑌>0(𝑋𝑌0) is a positive-definite (respectively, positive-semidefinite) matrix; 𝐴𝑇 represents the transpose of the matrix 𝐴; 𝜆max(𝐴),𝜆min(𝐴) denote the maximum eigenvalue and minimum one of matrix 𝐴, respectively; 𝐼 represents the identity matrix of an appropriate dimension; 𝑋𝑌𝑌𝑇𝑍=𝑋𝑌𝑍.

2. Problem Formulations and Preliminaries

Suppose the nodes are coupled with states 𝑥𝑖(𝑡), 𝑖{1,,𝑁}, then the delayed Cohen-Grossberg neural network models can be formulated as follows:̇𝑥𝑖(𝑡)=𝛼𝑥𝑖(𝑡)𝛽𝑥𝑖(𝑡)𝐴𝑓𝑥𝑖(𝑡)𝐵𝑓𝑥𝑖(𝑡𝜏(𝑡))𝐈(𝑡)+𝑁𝑗=1,𝑗𝑖𝑙𝑖𝑗𝐹𝑥𝑗(𝑡)𝑥𝑖(𝑡)+𝑁𝑗=1,𝑗𝑖𝑙𝑖𝑗𝐾𝑥𝑗(𝑡𝜏(𝑡))𝑥𝑖(𝑡)+𝑁𝑗=1,𝑗𝑖𝑙𝑖𝑗𝐽𝑥𝑗(𝑡𝜏(𝑡))𝑥𝑖(𝑡𝜏(𝑡)),(2.1) in which 𝑥𝑖(𝑡)=[𝑥𝑖1(𝑡),,𝑥𝑖𝑛(𝑡)]𝑇𝐑𝑛 is the state vector of the 𝑖th network at time 𝑡, 𝛼(𝑥𝑖)=diag{𝛼1(𝑥𝑖1),,𝛼𝑛(𝑥in)} represents the amplification function, 𝛽(𝑥𝑖)=[𝛽1(𝑥𝑖1),,𝛽𝑛(𝑥in)]𝑇 is the behaved function, 𝐴=[𝑎𝑖𝑗]𝑛×𝑛, 𝐵=[𝑏𝑖𝑗]𝑛×𝑛, 𝑓(𝑥𝑖)=[𝑓1(𝑥𝑖1),,𝑓𝑛(𝑥𝑖𝑛)]𝑇; I(𝑡)=[𝐈1(t),,𝐈𝑛(t)]𝑇𝐑𝑛 is the external input vector; 𝐹=[𝑓𝑖𝑗]𝑛×𝑛, 𝐾=[𝑘𝑖𝑗]𝑛×𝑛, and 𝐽=[𝑗𝑖𝑗]𝑛×𝑛 are respectively the inner coupling matrices between the connected nodes 𝑖 and 𝑗 at times 𝑡 and 𝑡𝜏(𝑡).

For system (2.1), the following assumptions are introduced throughout this paper. (A1)𝜏(𝑡) denotes an interval time-varying delay satisfying0𝜏0𝜏(𝑡)𝜏𝑚,̇𝜏(𝑡)𝜇<+,(2.2)

and we set 𝜏𝑚=𝜏𝑚𝜏0. (A2)𝐿=[𝑙𝑖𝑗]𝑁×𝑁 is the configuration matrix that is irreducible and satisfies𝑙𝑖𝑗=𝑙𝑗𝑖,𝑖𝑗,𝑙𝑖𝑖=𝑁𝑗=1,𝑗𝑖𝑙𝑖𝑗.(2.3) Here 𝑙𝑖𝑗>0 if there is a connection between node 𝑖 and the one 𝑗 and otherwise, 𝑙𝑖𝑗=0. (A3) For 𝑖{1,2,,𝑛}, each 𝛼𝑖() is Lipschitz continuous and there exists the positive scalars 𝑎𝑖,𝑎𝑖 satisfying 0<𝑎𝑖𝛼𝑖()𝑎𝑖; and there exist the positive scalars 𝜋𝑖,𝛾𝑖 such that each function 𝛽𝑖() satisfies 0<𝛾𝑖(𝛽𝑖(𝑥)𝛽𝑖(𝑦)/𝑥𝑦)𝜋𝑖, anḋ𝛽𝑖(𝑥)̇𝛽𝑖(𝑦)𝜌𝑖(𝑥𝑦)̇𝛽𝑖(𝑥)̇𝛽𝑖(𝑦)𝜌+𝑖(𝑥𝑦)0𝑥,𝑦𝐑,𝑖=1,,𝑛,(2.4) in which 𝜌𝑖,𝜌+𝑖 are given constants. Here we set Λ=diag{𝑎1,,𝑎𝑛}, Ψ=diag{𝑎1,,𝑎𝑛}, Γ=diag{𝛾1,,𝛾𝑛}, Π=diag{𝜋1,,𝜋𝑛}, andΛ1=diag𝛾1𝜋1,,𝛾𝑛𝜋𝑛,Λ2=diag𝜋1+𝛾12,,𝜋𝑛+𝛾𝑛2,Υ1=diag𝜌+1𝜌1,,𝜌+𝑛𝜌𝑛,Υ2=diag𝜌+1+𝜌12,,𝜌+𝑛+𝜌𝑛2.(2.5)(A4) For any 𝛼,𝛽𝐑, and 𝜌𝑖,𝜌+𝑖 for 𝑖{1,2,,𝑛}, the activation function 𝑓𝑖() satisfies𝑓𝑖(𝛼)𝑓𝑖(𝛽)𝜎+𝑖(𝛼𝛽)𝑓𝑖(𝛼)𝑓𝑖(𝛽)𝜎𝑖(𝛼𝛽)0.(2.6) Here we denote Σ1=diag{𝜎+1𝜎1,,𝜎+𝑛𝜎𝑛} and Σ2=diag{(𝜎+1+𝜎1)/2,,(𝜎+𝑛+𝜎𝑛)/2}.

Remark 2.1. In (A3), the assumption on the derivative of 𝛽(𝑥𝑖) 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(𝑥𝑖) can be expressed as 𝑎𝑥𝑖, 𝑎𝑥𝑖+𝑏sin(𝑥𝑖), 𝑎𝑥𝑖+𝑏sin2(𝑥𝑖), 𝑎𝑥𝑖+𝑐cos(𝑥𝑖), 𝑎𝑥𝑖+𝑐cos3(𝑥𝑖), 𝑎𝑥𝑖+𝑐tanh(𝑥𝑖), respectively. Moreover, the activation functions in system (2.1) can be of general description and those present ones in [2226, 29] are just special cases of the system (2.1).
Based on assumption(A2), system (2.1) can be rewritten as the following forms: ̇𝑥𝑖(𝑡)=𝛼𝑥𝑖(𝑡)𝛽𝑥𝑖(𝑡)𝐴𝑓𝑥𝑖(𝑡)𝐵𝑓𝑥𝑖(𝑡𝜏(𝑡))I(𝑡)𝑙𝑖𝑖𝐾𝑥𝑖(𝑡𝜏(𝑡))𝑥𝑖(𝑡)+𝑁𝑗=1𝑙𝑖𝑗𝐹𝑥𝑗(𝑡)+𝑁𝑗=1𝑙𝑖𝑗(𝐾+𝐽)𝑥𝑗(𝑡𝜏(𝑡)).(2.7) To address the problem, we denote the set 𝒮={x(𝑠)=[𝑥𝑇1(𝑠),,𝑥𝑇𝑁(𝑠)]𝑥𝑖(𝑠)𝒞([𝑡0𝜏𝑚,𝜏0],𝐑𝑛),𝑥𝑖(𝑠)=𝑥𝑗(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 𝜙𝑖(𝑠),𝜙𝑗(𝑠)𝐶([𝑡0𝜏𝑚,𝑡0],𝐑𝑛), 𝑖,𝑗=1,,𝑁, there exist 𝑀>0, 𝜀>0 and sufficient large 𝑇>0 such that 𝑥𝑖(𝑡)𝑥𝑗(𝑡)𝑀𝑒𝜀𝑡 for all 𝑡𝑇, 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)==𝑥𝑁(𝑡)=𝑠(𝑡), we can deduce the synchronized state equatioṅ𝑠(𝑡)=𝛼(𝑠(𝑡))[𝛽(𝑠(𝑡))𝐴𝑓(𝑠(𝑡))𝐵𝑓(𝑠(𝑡𝜏(𝑡)))𝐈(𝑡)]𝑙𝑖𝑖𝐾[𝑠(𝑡𝜏(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==𝑙𝑁𝑁=𝑙 has to be imposed on the system (2.7).

3. Delay-Dependent Synchronization Criteria

Firstly, together with the Kronecker product in [1621], we can reformulate the system (2.7) aṡx(𝑡)=𝐚(𝑥(𝑡))𝐛(𝑥(𝑡))𝐼𝑁𝐴𝐟(𝑥(𝑡))𝐼𝑁𝐵𝐟(𝑥(𝑡𝜏(𝑡)))𝐈(𝑡)l𝐼𝑁𝐾×[𝑥(𝑡𝜏(𝑡))𝑥(𝑡)]+(𝐿𝐹)𝑥(𝑡)+(𝐿(𝐾+𝐽))𝑥(𝑡𝜏(𝑡))(3.1) with 𝑥(𝑡)=[𝑥𝑇1(𝑡),,x𝑇𝑁(𝑡)]𝑇, 𝐚(x(𝑡))=diag{𝛼(𝑥1(𝑡)),,𝛼(𝑥𝑁(𝑡))}, 𝐛(x(𝑡))=[𝛽𝑇(𝑥1(𝑡)),,𝛽𝑇(𝑥𝑁(𝑡))]𝑇, 𝐟(x())=[𝑓𝑇(𝑥1()),,𝑓𝑇(𝑥𝑁())]𝑇, and 𝐈(𝑡)=[𝐼𝑇(𝑡),,𝐼𝑇(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 2𝑥𝑇𝐷𝑇𝑆𝑦𝜀1𝑥𝑇𝐷T𝑃1𝐷𝑥+𝜀𝑦𝑇𝑆𝑇𝑃𝑆𝑦.

Lemma 3.2 (17).. Let U=[𝑢𝑖𝑗]𝑁×𝑁, 𝑃𝐑𝑛×𝑛, 𝑥=[𝑥𝑇1,𝑥𝑇2,,𝑥𝑇𝑁]𝑇, and 𝑦=[𝑦𝑇1,𝑦𝑇2,,𝑦𝑇𝑁]𝑇 with 𝑥𝑖,𝑦𝑖𝐑𝑛, 𝑖=1,,𝑁. If U=U𝑇 and each row sum of 𝑈 is 0, then 𝑥𝑇(U𝑃)𝑦=1𝑖<𝑗𝑁u𝑖𝑗(𝑥𝑖𝑥j)𝑇𝑃(𝑦𝑖𝑦𝑗).

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, 𝐿𝑖(𝑖=1,2,3,4), 𝑛×𝑛 matrices 𝑃𝑙>0, 𝑄𝑙>0, 𝑅𝑙(𝑙=1,2,3) making 𝑃𝑙𝑅𝑙𝑄𝑙0, 𝑛×𝑛 diagonal matrices 𝑅>0, 𝑄>0, 𝐺>0, 𝐸>0, 𝑈>0, 𝑉>0, 𝑊>0, 𝐻>0, 𝑇𝑖>0(𝑖=1,2), 13𝑛×𝑛 matrices 𝑁𝑖(𝑖=1,2,3), and one scalar 𝛿>0 such that, for 1𝑖<𝑗𝑁, the LMIs in (3.2) hold Ω𝑖𝑗+$+$𝑇𝐻𝜏0𝑁1𝜏𝑚𝑁2𝛿𝐼00𝑆0𝑍<0,Ω𝑖𝑗+$+$𝑇𝐻𝜏0𝑁1𝜏𝑚𝑁3𝛿𝐼00𝑆0𝑍<0,(3.2) where $=[𝑁1𝑁1+𝑁2𝑁306𝑛13𝑛𝑁2+𝑁303𝑛13𝑛], 𝐻=[0𝑛13𝑛𝑄(Λ1Ψ1)0𝑛𝑛]𝑇, Ω𝑖𝑗=Ξ1100Ξ1400Ξ17𝐿𝑇10Ξ1,10𝑙𝐿𝑇1𝐾Ξ1,12Ξ1,13Ξ2200Ξ2500000000Ξ3300Ξ360000000Ξ44000𝐴𝑇𝐿3𝐴𝑇𝐿400𝐴𝑇𝑄𝑇0Ξ5500000000Ξ660000000Ξ77𝐿𝑇20Ξ7,10𝑙𝐿𝑇2𝐾Ψ1𝑅0Ξ88𝐿𝑇30𝐿𝑇3𝐵𝐿𝑇30Ξ990𝐿𝑇4𝐵𝐿𝑇40Ξ10,10Ξ10,11Ξ10,120Ξ11,11𝐵𝑇𝑄𝑇0Ξ12,120𝑇2(3.3) with Ξ11=P2+𝑙(𝐿𝑇1𝐾+𝐾𝑇𝐿1)𝑙𝑖𝑗𝑁(𝐿𝑇1𝐹+𝐹𝑇𝐿1)2Γ𝑇𝐺UΣ1𝑇1Π1𝑇2Υ1, Ξ14=𝑅2+UΣ2, Ξ17=𝑃𝐿𝑇1𝑙𝑖𝑗𝑁𝐹𝑇𝐿2+2Π𝑇𝑅Ψ12Γ𝑇𝑄Λ1, Ξ1,10=𝑙𝐿𝑇1𝐾𝑙𝑖𝑗𝑁𝐿𝑇1(𝐾+𝐽), Ξ1,12=𝑙𝐾𝑇𝑄Λ1𝑙𝑖𝑗𝑁𝐹𝑇𝑄Λ1+𝐺𝑇+𝑇1Π2, Ξ1,13=𝑇2Υ2+𝑄Λ1𝑅Ψ1, Ξ22=𝑃2+𝑃1+𝑃3𝑊Σ1, Ξ25=𝑅2+𝑅1+𝑅3+𝑊Σ2, Ξ33=𝑃3𝐻Σ1, Ξ36=𝐻Σ2𝑅3, Ξ44=𝑈+𝑄2, Ξ55=𝑄2+𝑄1+𝑄3𝑊, Ξ66=𝑄3𝐻, Ξ77=𝐿𝑇2𝐿2+𝜏0𝑆+𝜏𝑚𝑍, Ξ7,10=𝑙𝐿𝑇2𝐾𝑙𝑖𝑗𝑁𝐿𝑇2(𝐾+𝐽), Ξ88=𝐸+𝛿𝐼𝑛, Ξ99=𝐿𝑇4𝐿4+Ψ𝑇𝐸Ψ, Ξ10,10=(1𝜇)𝑃1𝑉Σ1, Ξ10,11=(1𝜇)𝑅1+𝑉Σ2, Ξ10,12=𝑙𝐾𝑇𝑄Λ1𝑙𝑖𝑗𝑁(𝐾+𝐽)𝑇𝑄Λ1, Ξ11,11=(1𝜇)𝑄1𝑉, Ξ12,12=2𝑄𝑇𝑇1.

Proof. Firstly, we can represent the system (3.1) as the following form: ̇𝑥(𝑡)=𝑦(𝑡),(3.4)𝑦(𝑡)=𝑧(𝑡)𝑙𝐼𝑁𝐾[𝑥(𝑡𝜏(𝑡))𝑥(𝑡)]+(𝐿𝐹)x(𝑡)+(𝐿(𝐾+𝐽))𝑥(𝑡𝜏(𝑡)),(3.5)𝑧(𝑡)=𝐚(𝑥(𝑡))𝑤(𝑡),(3.6)𝑤(𝑡)=𝐛(𝑥(𝑡))+𝐼𝑁𝐴𝐟(𝑥(𝑡))+𝐼𝑁𝐵𝐟(𝑥(𝑡𝜏(𝑡)))+𝐈(𝑡).(3.7) Based on assumptions (A1) and (A3), and 𝑈=𝑢𝑖𝑗𝑁𝑁=𝑁111𝑁1,(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))=𝑥𝑇(𝑡)(𝑈𝑃)𝑥(𝑡)+2[Θ𝑥(𝑡)𝐛(𝑥(𝑡))]𝑇𝑈𝑅Ψ1𝑥(𝑡)+2[𝐛(𝑥(𝑡))Υ𝑥(𝑡)]𝑇𝑈𝑄Λ1𝑥(𝑡),𝑉2(𝑥(𝑡))=𝑡𝜏0𝑡𝜏(𝑡)𝑥(𝑠)𝐟(𝑥(𝑠))𝑇𝑈𝑃1𝑅1𝑄1𝑥(𝑠)𝐟(𝑥(𝑠))ds,𝑉3(𝑥(𝑡))=𝑡𝑡𝜏0𝑥(𝑠)𝐟(𝑥(𝑠))𝑇𝑈𝑃2𝑅2𝑄2𝑥(𝑠)𝐟(𝑥(𝑠))ds,𝑉4(𝑥(𝑡))=𝑡𝜏0𝑡𝜏𝑚𝑥(𝑠)𝐟(𝑥(𝑠))𝑇𝑈𝑃3𝑅3𝑄3𝑥(𝑠)𝐟(𝑥(𝑠))ds,𝑉5(𝑥(𝑡))=0𝜏0𝑡𝑡+𝜃̇𝑥𝑇(𝑠)(𝑈𝑆)̇𝑥(𝑠)𝑑𝑠𝑑𝜃+𝜏0𝜏𝑚𝑡𝑡+𝜃̇𝑥𝑇(𝑠)(𝑈𝑍)̇𝑥(𝑠)𝑑𝑠𝑑𝜃(3.10) with two diagonal matrices 𝑅>0, 𝑄>0, and setting Θ=diag{Π,Π,,Π𝑁}, Υ=diag{Γ,Γ,,Γ𝑁}. Based on (A3) and Lemma 3.1, one can easily verify the definite positiveness of 𝑉1(𝑥(𝑡)).
Now, by directly calculating ̇𝑉1(𝑥(𝑡)) along the trajectory of the system (3.1), we can deducė𝑉1(𝑥(𝑡))=2𝑥𝑇(𝑡)(𝑈𝑃)𝑦(𝑡)+2[Θ𝑥(𝑡)𝐛(𝑥(𝑡))]𝑇𝑈𝑅Ψ1𝑦(𝑡)+2Θ𝑦(𝑡)̇𝐛(𝑥(𝑡))𝑇×𝑈𝑅Ψ1𝑥(𝑡)+2[𝐛(𝑥(𝑡))Υ𝑥(𝑡)]𝑇×𝑈𝑄Λ1𝑦(𝑡)+2̇𝐛(𝑥(𝑡))Υ𝑦(𝑡)𝑇𝑈𝑄Λ1𝑥(𝑡)=2𝑥𝑇(𝑡)(𝑈𝑃)𝑦(𝑡)+2[Θ𝑥(𝑡)𝐛(𝑥(𝑡))]𝑇𝑈𝑅Ψ1𝑦(𝑡)+2𝐛𝑇(𝑥(𝑡))×𝑈𝑄Λ1𝑦(𝑡)+2̇𝐛𝑇(𝑥(𝑡))𝑈𝑄Λ1𝑅Ψ1𝑥(𝑡)+2𝑦𝑇(𝑡)×𝑈Π𝑇𝑅Ψ12Γ𝑇𝑄Λ1𝑥(𝑡).(3.11) Meanwhile by (3.5), it is easy to derive that 2𝐛𝑇(𝑥(𝑡))𝑈𝑄Λ1𝑦(𝑡)=2𝐛𝑇(𝑥(𝑡))𝑈𝑄Λ1𝑧(𝑡)+2𝐛𝑇(𝑥(𝑡))𝑈𝑄Λ1×l𝐼𝑁𝐾[𝑥(𝑡𝜏(𝑡))𝑥(𝑡)]+(𝐿𝐹)𝑥(𝑡)+(𝐿(𝐾+𝐽))𝑥(𝑡𝜏(𝑡)).(3.12) Noting that 𝑈1/20 does hold, then with Lemma 3.1 and 𝛿>0, one can estimate 2𝐛𝐓(𝑥(𝑡))(𝑈𝑄Λ1)𝑧(𝑡) as 2𝐛𝐓(𝑥(𝑡))𝑈𝑄Λ1𝑧(𝑡)=2𝐛𝐓(𝑥(𝑡))𝑈1/2𝑄Λ1𝛼1(𝑥(𝑡))T𝑈1/2𝐼𝑧(𝑡)+2𝐛𝐓(𝑥(𝑡))(𝑈𝑄)𝑤(𝑡)𝛿𝑧𝑇(𝑡)𝑈𝐼𝑛𝑧(𝑡)+𝛿1𝐛𝐓(𝑥(𝑡))×𝑈Λ𝑇Ψ𝑇𝑄𝑇𝑄Λ1Ψ1𝐛(𝑥(𝑡))+2𝐛𝐓(𝑥(𝑡))×(𝑈𝑄)𝐛(𝑥(𝑡))+𝐼𝑁𝐴𝐟(𝑥(𝑡))+𝐼𝑁𝐵𝐟(𝑥(𝑡𝜏(𝑡)))+𝐈(𝑡).(3.13) Now combining with terms (3.12) and (3.13) yields ̇𝑉1(𝑥(𝑡))2𝑥𝑇(𝑡)(𝑈𝑃)𝑦(𝑡)+2[Θ𝑥(𝑡)𝐛(𝑥(𝑡))]𝑇𝑈𝑅Ψ1𝑦(𝑡)+𝛿z𝑇(𝑡)𝑈𝐼𝑛𝑧(𝑡)+𝛿1𝐛𝑇(𝑥(𝑡))𝑈Λ𝑇Ψ𝑇𝑄𝑇𝑄Λ1Ψ1𝐛(𝑥(𝑡))+2𝐛𝐓(𝑥(𝑡))(𝑈𝑄)×𝐛(𝑥(𝑡))+𝐼𝑁𝐴𝐟(𝑥(𝑡))+𝐼𝑁𝐵𝐟(𝑥(𝑡𝜏(𝑡)))+𝐈(𝑡)+2𝐛𝐓(𝑥(𝑡))×𝑈𝑄Λ1𝑙𝐼𝑁𝐾[𝑥(𝑡𝜏(𝑡))𝑥(𝑡)]+(𝐿𝐹)𝑥(𝑡)+(𝐿(𝐾+𝐽))𝑥(𝑡𝜏(𝑡))+2̇𝐛𝐓(𝑥(𝑡))𝑈𝑄Λ1𝑅Ψ1𝑥(𝑡)+2𝑦𝑇(𝑡)𝑈Π𝑇𝑅Ψ12Γ𝑇𝑄Λ1𝑥(𝑡),(3.14)̇𝑉2(𝑥(𝑡))𝑥𝑇𝑡𝜏0𝑈𝑃1𝑥𝑡𝜏0+2𝑥𝑇𝑡𝜏0𝑈𝑅1𝐟𝑥𝑡𝜏0+𝐟𝐓𝑥𝑡𝜏0×𝑈𝑄1𝐟𝑥𝑡𝜏0(1𝜇)𝑥𝑇(𝑡𝜏(𝑡))𝑈𝑃1𝑥(𝑡𝜏(𝑡))+2𝑥𝐓(𝑡𝜏(𝑡))𝑈𝑅1𝐟(𝑥(𝑡𝜏(𝑡))),+𝐟T(𝑥(𝑡𝜏(𝑡)))𝑈𝑄1𝐟(𝑥(𝑡𝜏(𝑡))),(3.15)̇𝑉3(𝑥(𝑡))=𝑥𝑇(𝑡)𝑈𝑃2𝑥(𝑡)+2𝑥𝑇(𝑡)𝑈𝑅2𝐟(𝑥(𝑡))+𝐟𝐓(𝑥(𝑡))𝑈𝑄2𝐟(𝑥(𝑡))𝑥𝑇𝑡𝜏0𝑈𝑃2𝑥𝑡𝜏0+2𝑥𝑇𝑡𝜏0𝑈𝑅2𝐟𝑥𝑡𝜏0+𝐟𝐓𝑥𝑡𝜏0𝑈𝑄2𝐟𝑥𝑡𝜏0,(3.16)̇𝑉4(𝑥(𝑡))=𝑥𝑇𝑡𝜏0𝑈𝑃3𝑥𝑡𝜏0+2𝑥𝑇𝑡𝜏0𝑈𝑅3𝐟𝑥𝑡𝜏0+𝐟𝐓𝑥𝑡𝜏0𝑈𝑄3×𝐟𝑥𝑡𝜏0𝑥𝑇𝑡𝜏𝑚𝑈𝑃3𝑥𝑡𝜏𝑚+2𝑥𝑇𝑡𝜏𝑚𝑈𝑅3𝐟𝑥𝑡𝜏𝑚+𝐟𝐓𝑥𝑡𝜏𝑚𝑈𝑄3𝐟𝑥𝑡𝜏𝑚,(3.17)̇𝑉5(𝑥(𝑡))=𝑦𝑇(𝑡)𝜏0(𝑈𝑆)+𝜏𝑚(𝑈𝑍)𝑦(𝑡)𝑡𝑡𝜏0𝑦𝑇(𝑠)(𝑈𝑆)𝑦(𝑠)𝑑𝑠𝑡𝜏0𝑡𝜏𝑚𝑦𝑇(𝑠)(𝑈𝑍)𝑦(𝑠)𝑑𝑠.(3.18) For any 𝑛×𝑛 matrices 𝐿𝑖(𝑖=1,2,3,4), it follows from (3.5) and (3.7) that 0=2𝑥𝑇(𝑡)𝑈𝐿𝑇1+𝑦𝑇(𝑡)𝑈𝐿𝑇2×𝑦(𝑡)+𝑧(𝑡)𝑙𝐼𝑁𝐾[𝑥(𝑡𝜏(𝑡))𝑥(𝑡)]+(𝐿𝐹)𝑥(𝑡)+(𝐿(𝐾+𝐽))𝑥(𝑡𝜏(𝑡)),(3.19)0=2𝑧𝑇(𝑡)𝑈𝐿𝑇3+𝑤𝑇(𝑡)𝑈𝐿𝑇4×𝑤(𝑡)𝐛(𝑥(𝑡))+𝐼𝑁𝐴𝐟(𝑥(𝑡))+𝐼𝑁𝐵𝐟(𝑥(𝑡𝜏(𝑡)))+𝐈(𝑡).(3.20) By utilizing (A3) and (3.6), for any 𝑛×𝑛 diagonal matrices 𝐺0, 𝐸0, the following inequality holds 02𝐛𝐓(𝑥(𝑡))(𝑈𝐺)𝑥(𝑡)𝑥𝑇(𝑡)𝑈Γ𝑇𝐺𝑥(𝑡)+𝑤𝑇(𝑡)𝑈Ψ𝑇𝐸Ψ𝑤(𝑡)𝑧𝑇(𝑡)(𝑈𝐸)𝑧(𝑡).(3.21) Meanwhile, based on (3.14) and (3.19), it is easy to check that 𝑈𝐿=𝑁𝐿, and 𝑈𝑄Λ1(𝐿𝐹)=(𝑁𝐿)𝑄Λ1𝐹,𝑈𝑄Λ1(𝐿(𝐾+𝐽))=(𝑁𝐿)𝑄Λ1(𝐾+𝐽),𝑈𝐿𝑇𝑖(𝐿𝐹)=(𝑁𝐿)𝐿𝑇𝑖𝐹,𝑈𝐿𝑇𝑖(𝐿(𝐾+𝐽))=(𝑁𝐿)𝐿𝑇𝑖(𝐾+𝐽),𝑖=1,2.(3.22) Here we can employ the following notations to simplify the subsequent proof 𝑥𝑖𝑗=𝑥𝑖𝑥𝑗,𝑦𝑖𝑗=𝑦𝑖𝑦𝑗,𝑧𝑖𝑗=𝑧𝑖𝑧𝑗,𝑤𝑖𝑗=𝑤𝑖𝑤𝑗,𝛽𝑥𝑖𝑗=𝛽𝑥𝑖𝛽𝑥𝑗,𝑓𝑥𝑖𝑗=𝑓𝑥𝑖𝑓𝑥𝑗.(3.23) Then together with (𝑈𝑄)𝐈(𝑡)=0 in (3.14) and (𝑈𝐿𝑇3)𝐈(𝑡)=(𝑈𝐿𝑇4)𝐈(𝑡)=0 in (3.19), it follows from Lemma 3.2 and (3.14)–(3.22) that ̇𝑉(𝑥(𝑡))1𝑖<𝑗𝑁𝐮𝑖𝑗2𝑥𝑇𝐢𝐣(𝑡)𝑃𝑦𝑖𝑗(𝑡)+2Π𝑥𝑖𝑗(𝑡)𝛽𝑥𝑖𝑗(𝑡)𝑇𝑅Ψ1𝑦𝑖𝑗(𝑡)+𝛿𝑧𝑇𝑖𝑗(𝑡)𝐼𝑛𝑧𝑖𝑗(𝑡)+𝛿1𝛽𝑇𝑥𝑖𝑗(𝑡)Λ𝑇Ψ𝑇𝑄𝑇𝑄Λ1Ψ1𝛽𝑥𝑖𝑗(𝑡)+2𝛽𝑇𝑥𝑖𝑗(𝑡)×𝑄𝛽𝑥𝑖𝑗(𝑡)+𝐴𝑓𝑥𝑖𝑗(𝑡)+𝐵𝑓𝑥𝑖𝑗(𝑡𝜏(𝑡))+2𝛽𝑇𝑥𝑖𝑗(𝑡)𝑄Λ1×𝑙𝐾𝑥𝑖𝑗(𝑡𝜏(𝑡))𝑥𝑖𝑗(𝑡)+2̇𝛽𝑇𝑥𝑖𝑗(𝑡)𝑄Λ1𝑅Ψ1𝑥𝑖𝑗(𝑡)+2𝑦𝑇𝑖𝑗(𝑡)×Π𝑇𝑅Ψ12Γ𝑇𝑄Λ1𝑥𝑖𝑗(𝑡)+𝑥𝑇𝑖𝑗𝑡𝜏0𝑃1𝑃2+𝑃3𝑥𝑖𝑗𝑡𝜏0+2𝑥𝑇𝑖𝑗𝑡𝜏0𝑅1𝑅2+𝑅3×𝑓𝑥𝑖𝑗𝑡𝜏0+𝑓𝑇𝑥𝑖𝑗𝑡𝜏0𝑄1𝑄2+𝑄3𝑓𝑥𝑖𝑗𝑡𝜏0×𝑥𝑇𝑖𝑗(𝑡𝜏(𝑡))𝑃1𝑥𝑖𝑗(𝑡𝜏(𝑡))+2𝑥𝑇𝑖𝑗(𝑡𝜏(𝑡))𝑅1𝑓𝑥𝑖𝑗(𝑡𝜏(𝑡))+𝑓𝑇𝑥𝑖𝑗(𝑡𝜏(𝑡))𝑄1𝑓𝑥𝑖𝑗(𝑡𝜏(𝑡))×𝑥𝑇𝑖𝑗(𝑡)𝑃2𝑥𝑖𝑗(𝑡)+2𝑥𝑇𝑖𝑗(𝑡)𝑅2𝑓𝑥𝑖𝑗(𝑡)+𝑓𝑇𝑥𝑖𝑗(𝑡)𝑄2𝑓𝑥𝑖𝑗(𝑡)𝑥𝑇𝑖𝑗𝑡𝜏𝑚𝑃3𝑥𝑖𝑗𝑡𝜏𝑚+2𝑥𝑇𝑖𝑗𝑡𝜏𝑚𝑅3𝑓𝑥𝑖𝑗𝑡𝜏𝑚+𝑓𝑇𝑥𝑖𝑗𝑡𝜏𝑚Q3𝑓𝑥𝑖𝑗𝑡𝜏𝑚+𝑦𝑇𝑖𝑗(𝑡)𝜏0𝑆+𝜏𝑚𝑍𝑦𝑖𝑗(𝑡)𝑡𝑡𝜏0𝑦𝑇𝑖𝑗(𝑠)𝑆𝑦𝑖𝑗(𝑠)𝑑𝑠𝑡𝜏0𝑡𝜏𝑚𝑦𝑇𝑖𝑗(𝑠)𝑍𝑦𝑖𝑗(𝑠)𝑑𝑠+2𝑥𝑇𝑖𝑗(𝑡)𝐿𝑇1+𝑦𝑇𝑖𝑗(𝑡)𝐿𝑇2×𝑦𝑖𝑗(𝑡)+𝑧𝑖𝑗(𝑡)𝑙𝐾𝑥𝑖𝑗(𝑡𝜏(𝑡))𝑥𝑖𝑗(𝑡)+2𝑧𝑇𝑖𝑗(𝑡)𝐿𝑇3+𝑤𝑇𝑖𝑗(𝑡)𝐿𝑇4𝑤𝑖𝑗(𝑡)𝛽𝑥𝑖𝑗(𝑡)+𝐴𝑓𝑥𝑖𝑗(𝑡)+𝐵𝑓𝑥𝑖𝑗(𝑡𝜏(𝑡))+2𝛽𝑇𝑥𝑖𝑗(𝑡)𝐺𝑥𝑖𝑗(𝑡)𝑥𝑇𝑖𝑗(𝑡)Γ𝑇𝐺𝑥𝑖𝑗(𝑡)+𝑤𝑇𝑖𝑗(𝑡)Ψ𝑇𝐸Ψ𝑤𝑖𝑗(𝑡)𝑧𝑇𝑖𝑗(𝑡)𝐸𝑧𝑖𝑗(𝑡)+2𝑁𝑙𝑖𝑗𝛽𝑇𝑥𝑖𝑗(𝑡)𝑄Λ1+𝑥𝑇𝑖𝑗(𝑡)𝐿𝑇1+𝑦𝑇𝑖𝑗(𝑡)𝐿𝑇2𝐹𝑥𝑖𝑗(𝑡)+𝛽𝑇𝑥𝑖𝑗(𝑡)𝑄Λ1+𝑥𝑇𝑖𝑗(𝑡)𝐿𝑇1+𝑦𝑇𝑖𝑗(𝑡)𝐿𝑇2(𝐾+𝐽)𝑥𝑖𝑗(𝑡𝜏(𝑡)).(3.24) For any 𝑛×𝑛 diagonal matrices 𝑈>0, 𝑉>0, 𝑊>0, 𝐻>0, 𝑇𝑖>0(𝑖=1,2), and Σ𝑖, Π𝑖, Υ𝑖(𝑖=1,2) in (A3)-(A4), it can be deduced that 01𝑖<𝑗𝑁𝑥𝑇𝑖𝑗(𝑡)𝑈Σ1𝑥𝑖𝑗(𝑡)2𝑥𝑇𝑖𝑗(𝑡)𝑈Σ2𝑓𝑥𝑖𝑗(𝑡)+𝑓𝑇𝑥𝑖𝑗(𝑡)𝑈𝑓𝑥𝑖𝑗(𝑡)𝑥𝑇𝑖𝑗(𝑡𝜏(𝑡))𝑉Σ1×𝑥𝑖𝑗(𝑡𝜏(𝑡))2𝑥𝑇𝑖𝑗(𝑡𝜏(𝑡))𝑉Σ2𝑓𝑥𝑖𝑗(𝑡𝜏(𝑡))+𝑓𝑇𝑥𝑖𝑗(𝑡𝜏(𝑡))𝑉𝑓𝑥𝑖𝑗(𝑡𝜏(𝑡))𝑥𝑇𝑖𝑗𝑡𝜏0𝑊Σ1𝑥𝑖𝑗𝑡𝜏02𝑥𝑇𝑖𝑗𝑡𝜏0𝑊Σ2𝑓𝑥𝑖𝑗𝑡𝜏0+𝑓𝑇𝑥𝑖𝑗𝑡𝜏0𝑊𝑓𝑥𝑖𝑗𝑡𝜏0𝑥𝑇𝑖𝑗𝑡𝜏𝑚𝐻Σ1𝑥𝑖𝑗𝑡𝜏𝑚2𝑥𝑇𝑖𝑗𝑡𝜏𝑚𝐻Σ2𝑓𝑥𝑖𝑗𝑡𝜏𝑚+𝑓𝑇𝑥𝑖𝑗𝑡𝜏𝑚𝐻𝑓𝑥𝑖𝑗𝑡𝜏𝑚𝑥𝑇𝑖𝑗(𝑡)𝑇1Π1𝑥𝑖𝑗(𝑡)2𝑥𝑇𝑖𝑗(𝑡)𝑇1Π2𝛽𝑥𝑖𝑗(𝑡)+𝛽𝑇𝑥𝑖𝑗(𝑡)𝑇1𝛽𝑥𝑖𝑗(𝑡)𝑥𝑇𝑖𝑗(𝑡)𝑇2Υ1𝑥𝑖𝑗(𝑡)2𝑥𝑇𝑖𝑗(𝑡)𝑇2Υ2̇𝛽𝑥𝑖𝑗(𝑡)+̇𝛽𝑇𝑥𝑖𝑗(𝑡)𝑇2̇𝛽𝑥𝑖𝑗(𝑡).(3.25) For any 13 𝑛×𝑛 matrices 𝑁𝑖(𝑖=1,2,3), it follows from Newton-Leibniz formula that 0=21𝑖<𝑗𝑁𝜁𝑇𝑖𝑗(𝑡)𝑁1𝑥𝑖𝑗(𝑡)𝑥𝑖𝑗𝑡𝜏0𝑡𝜏0𝑡𝜏(𝑡)𝑦𝑖𝑗(𝑠)+𝑁2𝑥𝑖𝑗𝑡𝜏0𝑥𝑖𝑗(𝑡𝜏(𝑡))𝑡𝜏0𝑡𝜏(𝑡)𝑦𝑖𝑗(𝑠)+𝑁3𝑥𝑖𝑗(𝑡𝜏(𝑡))𝑥𝑖𝑗𝑡𝜏𝑚𝑡𝜏(𝑡)𝑡𝜏𝑚𝑦𝑖𝑗(𝑠),(3.26) where 𝜁𝑇𝑖𝑗(𝑡)=𝑥𝑇𝑖𝑗(𝑡)𝑥𝑇𝑖𝑗𝑡𝜏0𝑥𝑇𝑖𝑗𝑡𝜏𝑚𝑓𝑇𝑥𝑖𝑗(𝑡)𝑓𝑇𝑥𝑖𝑗𝑡𝜏0𝑓𝑇𝑥𝑖𝑗𝑡𝜏𝑚𝑦𝑇𝑖𝑗(𝑡)𝑧𝑇𝑖𝑗(𝑡)×𝑤𝑇𝑖𝑗(𝑡)𝑥𝑇𝑖𝑗(𝑡𝜏(𝑡))𝑓𝑇𝑥𝑖𝑗(𝑡𝜏(𝑡))𝛽𝑇𝑥𝑖𝑗(𝑡)̇𝛽𝑇𝑥𝑖𝑗(𝑡).(3.27) Now together with the terms (3.24)–(3.26), and 𝑢𝑖𝑗=1, we can deduce that ̇𝑉(𝑥(𝑡))1𝑖<𝑗𝑁𝜁𝑇𝑖𝑗(𝑡)Ω𝑖𝑗+$+$𝑇+𝛿1𝐻𝐻𝑇+𝜏0𝑁1𝑆1𝑁𝑇1+𝜏(𝑡)𝜏0𝑁2𝑍1𝑁𝑇2+𝜏𝑚𝜏(𝑡)𝑁3𝑍1𝑁𝑇3𝜁𝑖𝑗(𝑡)=1𝑖<𝑗𝑁𝜁𝑇𝑖𝑗(𝑡)Δ𝑖𝑗(𝑡)𝜁𝑖𝑗(𝑡),(3.28) where Ω𝑖𝑗,$, and 𝐻 are presented in (3.2). Through using Schur-complement and convex combination, the LMIs in (3.2) can guarantee Δ𝑖𝑗(𝑡)<0 and thus, there must exist one scalar 𝜒>0 such that Δ𝑖𝑗(𝑡)𝜒𝐼<0. Therefore, one can get ̇𝑉(𝑥(𝑡))1𝑖<𝑗𝑁𝜁𝑇𝑖𝑗(𝑡)Δ𝑖𝑗(𝑡)𝜁𝑖𝑗(𝑡)𝜒1𝑖<𝑗𝑁𝑥𝑖𝑗(𝑡)2+1𝑖<𝑗𝑁𝑥𝑖𝑗(𝑡𝜏(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 Θ𝑖>0(𝑖=1,2,3) such that𝑉(𝑥(𝑡))1𝑖<𝑗𝑁Θ1𝑥𝑖𝑗(𝑡)2+Θ2𝑡𝑡𝜏𝑚𝑥𝑖𝑗(𝑠)2𝑑𝑠+Θ3𝑡𝑡𝜏𝑚𝑥𝑖𝑗(𝑠𝜏(𝑠))2𝑑𝑠.(3.30) Letting 𝑉(𝑥(𝑡))=𝑒2𝑘𝑡𝑉(𝑥(𝑡)), one can deduce that ̇𝑉(𝑥(𝑡))=2𝑘𝑒2𝑘𝑡𝑉(𝑥(𝑡))+𝑒2𝑘𝑡̇𝑉(𝑥(𝑡)), and 𝑉(𝑥(𝑡))𝑉(𝑥(0))=𝑡𝑜̇𝑉(𝑥(𝑠))𝑑𝑠1𝑖<𝑗𝑁𝑡𝑜𝑒2𝑘𝑠𝜒𝑥𝑖𝑗(𝑠)2+𝑥𝑖𝑗(𝑠𝜏(𝑠))2+2𝑘Θ1𝑥𝑖𝑗(𝑠)2+Θ2𝑠𝑠𝜏𝑚𝑥𝑖𝑗(𝜃)2𝑑𝜃+Θ3𝑠𝑠𝜏𝑚𝑥𝑖𝑗(𝜃𝜏(𝜃))2𝑑𝜃𝑑𝑠.(3.31) By 1𝑖<𝑗𝑁 and changing the integration sequences, we have 𝑡0𝑒2𝑘𝑠𝑠𝑠𝜏𝑚𝑥𝑖𝑗(𝜃)2𝑑𝜃𝑑𝑠𝜏𝑚𝑒2𝑘𝜏𝑚0𝜏𝑚𝑥𝑖𝑗(𝜃)2𝑒2𝑘𝜃𝑑𝜃+𝑡0𝑥𝑖𝑗(𝜃)2𝑒2𝑘𝜃𝑑𝜃,𝑡0𝑒2𝑘𝑠𝑠𝑠𝜏𝑚𝑥𝑖𝑗(𝜃𝜏(𝜃))2𝑑𝜃𝑑𝑠𝜏𝑚𝑒2𝑘𝜏𝑚×02𝜏𝑚𝑥𝑖𝑗(𝜃)2𝑒2𝑘𝜃𝑑𝜃+𝑡0𝑥𝑖𝑗(𝜃𝜏(𝜃))2𝑒2𝑘𝜃𝑑𝜃.(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))+1𝑖<𝑗𝑁2𝑘Θ1+2𝑘Θ2𝜏m𝑒2𝑘𝜏m𝜒𝑡0𝑥𝑖𝑗(𝜃)2𝑒2𝑘𝜃𝑑𝜃+2𝑘Θ3𝜏𝑚𝑒2𝑘𝜏𝑚𝜒𝑡0𝑥𝑖𝑗(𝜃𝜏(𝜃))2𝑒2𝑘𝜃𝑑𝜃+(𝑘)(3.33) in which (𝑘)=2𝑘Θ2𝜏𝑚𝑒2𝑘𝜏𝑚0𝜏𝑚𝑥𝑖𝑗(𝜃)2𝑑𝜃+2𝑘Θ3𝜏𝑚𝑒2𝑘𝜏𝑚02𝜏𝑚𝑥𝑖𝑗(𝜃)2𝑑𝜃. Choose an appropriate scalar 𝑘0>0 such that 2𝑘0Θ1+2𝑘0Θ2𝜏𝑚𝑒2𝑘0𝜏𝑚𝜒0,2𝑘0Θ3𝜏𝑚𝑒2𝑘0𝜏𝑚𝜒0, one has 𝑉(𝑥(𝑡))(𝑘0)+𝑉(𝑥(0)). By directly computing, there must exist a positive scalar 𝜛>0 such that 𝑘0+𝑉(𝑥(0))𝜛1𝑖<𝑗𝑁sup2𝜏𝑚𝑠0𝜙𝑖(𝑠)𝜙𝑗(𝑠)2.(3.34) Meanwhile, 𝑉(𝑥(𝑡))𝜆min(𝑃)1𝑖<𝑗𝑁𝑒2𝑘0𝑡𝑥𝑖𝑗(𝑡)2. Therefore, it can be deduced that 𝑥𝑖𝑗(𝑡)𝜆1min(𝑃)𝜛1𝑖<𝑗𝑁sup2𝜏𝑚𝑠0𝜙𝑖(𝑠)𝜙𝑗(𝑠)𝑒𝑘0𝑡,𝑡0.(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 [1622], 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==𝑙𝑁𝑁=𝑙 in 𝐿=[𝑙𝑖𝑗]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, 𝐿𝑖(𝑖=1,2,3,4), 𝑛×𝑛 matrices 𝑃𝑙>0, 𝑄𝑙>0, 𝑅𝑙(l=1,2,3) making 𝑃𝑙𝑅𝑙𝑄𝑙0, n×n diagonal matrices 𝑅>0, 𝑄>0, 𝐺>0, 𝐸>0, 𝑈>0, 𝑉>0, 𝑊>0, 𝐻>0, 𝑇𝑖>0(𝑖=1,2), 13𝑛×𝑛 matrices 𝑁𝑖(𝑖=1,2,3), and one scalar 𝛿>0 such that, for 1𝑖<𝑗𝑁, the LMIs in (3.36) hold Ω𝑖𝑗+$+$𝑇𝐻𝜏0𝑁1𝜏𝑚𝑁2𝛿𝐼00𝑆0𝑍<0,Ω𝑖𝑗+$+$𝑇𝐻𝜏0𝑁1𝜏𝑚𝑁3𝛿𝐼00𝑆0𝑍<0,(3.36) where $=[𝑁1𝑁1+𝑁2𝑁306𝑛13𝑛𝑁2+𝑁303𝑛13𝑛], 𝐻=[0𝑛11𝑛𝑄(Λ1Ψ1)0𝑛𝑛]𝑇, Ω𝑖𝑗=Ξ1100Ξ1400Ξ17𝐿𝑇10Ξ1,100Ξ1,12Ξ1,13Ξ2200Ξ2500000000Ξ3300Ξ360000000Ξ44000𝐴𝑇𝐿3𝐴𝑇𝐿400𝐴𝑇𝑄𝑇0Ξ5500000000Ξ660000000Ξ77𝐿𝑇20Ξ7,100Ψ1𝑅0Ξ88𝐿𝑇30𝐿𝑇3𝐵𝐿𝑇30Ξ990𝐿𝑇4𝐵𝐿𝑇40Ξ10,10Ξ10,11Ξ10,120Ξ11,11𝐵𝑇𝑄𝑇0Ξ12,120𝑇2(3.37) with Ξ11=𝑃2𝑙𝑖𝑗𝑁(𝐿𝑇1𝐹+𝐹𝑇𝐿1)Γ𝑇𝐺𝐺𝑇Γ𝑈Σ1𝑇1Π1𝑇2Υ1, Ξ14=𝑅2+𝑈Σ2, Ξ17=𝑃𝐿𝑇1𝑙𝑖𝑗𝑁𝐹𝑇𝐿2+2Π𝑇𝑅Ψ12Γ𝑇𝑄Λ1, Ξ1,10=𝑙𝑖𝑗𝑁𝐿𝑇1𝐽, Ξ1,12=𝑙𝑖𝑗𝑁𝐹𝑇𝑄Λ1+𝐺𝑇+𝑇1Π2, Ξ1,13=𝑇2Υ2+𝑄Λ1𝑅Ψ1, Ξ22=𝑃2+𝑃1+𝑃3𝑊Σ1, Ξ25=𝑅2+𝑅1+𝑅3+𝑊Σ2, Ξ33=𝑃3𝐻Σ1, Ξ36=𝐻Σ2𝑅3, Ξ44=𝑈+𝑄2, Ξ55=𝑄2+𝑄1+𝑄3𝑊, Ξ66=𝑄3𝐻, Ξ77=𝐿𝑇2𝐿2+𝜏0𝑆+𝜏𝑚𝑍,