New Global Synchronization Analysis for Complex Networks with Coupling Delay

Abstract

Global synchronization analysis for complex networks with coupling delay is investigated. Firstly the constant time delay is analyzed and then the case for time-varying delay is considered. Sufficient conditions for network synchronization are given based on Lyapunov functional, linear matrix inequality, and Kronecker product technique. The unknown variables in the sufficient conditions are fewer than those in the recent reference. Moreover, for the time-varying delay case, we find that the conditions are dependent on the bounds of both time delay and its derivative, and the derivative of the time-varying delay can be any value in the bounds. Finally, numerical examples are given to validate the effectiveness of the obtained results.

1. Introduction

Dynamical recurrent neural networks are used extensively in classification of patterns, associative memories, optimization [1–3], and so on. The networks are composed of a large number of highly interconnected dynamical units and exhibit very complicated dynamics. Therefore, theory analysis of complex networks has become a focal research field and attracted a great deal of attention.

Recently, it is found that synchronization is one of the most important dynamical properties of complex networks and has been extensively investigated in different ways [4–19]. In [5], Lü and Chen considered a dynamical network and gave the sufficient conditions for local synchronization. Because of the network traffic congestions as well as the finite speed of signal transmission over the links, time delays occur commonly in complex networks. Therefore, Zhou and Chen [7] and Gao et al. [8] improved the models with no delays, the case with constant coupling delays is considered, and they also analyzed the synchronization problem [9, 10]. However, in many real-world networks time delay is varying; therefore time varying coupling delays are considered in [11–13]. By using free-weighting matrices, some synchronization criteria for general complex dynamical networks with time-varying delays are proposed. However, the computation is huge, and there are a large amount of variables in the condition. Moreover, in order to derive the synchronization condition, in some papers [7–13], the time-varying delay 𝜏(𝑡) is usually confined to 0≤𝜏(𝑡)≤𝜏 (lower bound of the delay is zero), and the derivative is restricted to less than 1. Therefore, how to improve the system performance by removing the redundant variables and reducing computation still remains unsolved.

Motivated by the mentioned work, we study the synchronization problems for general complex networks with time constant coupling delays and interval time-varying delays. Using different Lyapunov functions, the synchronization conditions derived turn out to be less conservative, and the addressed systems contain some models as their special cases; the more effective mathematical techniques are employed to reduce the conservatism.

The remainder of this paper is organized as follows. In Section 2, the investigated systems are formulated and some lemmas and notations are given. In Section 3, the conditions for synchronization are derived. In Section 4, two numerical examples are presented to demonstrate the effectiveness and the advantage of the proposed method. Finally, conclusions are drawn in Section 5.

2. Problem Formulations

In this paper, 𝑅^𝑚×𝑛 denotes the set of 𝑚×𝑛 real matrixes, and 𝑋≥0 denotes that matrix 𝑋 is positive semidefinite, while 𝑋>0 denotes that matrix 𝑋 is positive definite. We use diag{⋯} to denote a block-diagonal matrix; 𝑋𝑌∗𝑍 stands for 𝑌𝑋𝑌_𝑇𝑍. The notation 𝐴⊗𝐵 denotes the Kronecker product of matrices 𝐴 and 𝐵; 𝐼_𝑛 represents the 𝑛-dimensional identity matrix. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.

We use 𝑥_𝑖(𝑡) to denote the state of coupling nodes, 𝑖∈{1,…,𝑁}, and then the dynamic neural networks (DNNs) of general form can be described by 𝑥̇_𝑖(𝑡)=−𝐶𝑥_𝑖(𝑡)+𝐷𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏)+𝐼(𝑡)+𝑐^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗𝐴𝑥_𝑗(𝑡−𝜏)−𝑥_𝑖(𝑡−𝜏)+𝑐^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗𝐴_𝜏𝑥_𝑗(𝑡−𝜏)−𝑥_𝑖(𝑡,𝑥̇_𝑖)(𝑡)=−𝐶𝑥_𝑖(𝑡)+𝐷𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))+𝐼(𝑡)+𝑐^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗𝐴𝑥_𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡−𝜏(𝑡))+𝑐^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗𝐴_𝜏𝑥_𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡), (1) in which 𝑥_𝑖(𝑡)=[𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡)]^𝑇∈𝑅^𝑛 is the state vector of the 𝑖th network at time 𝑡. The functions 𝑓(𝑥_𝑖(𝑡))=[𝑓₁(𝑥_𝑖1(𝑡)),𝑓₂(𝑥_𝑖2(𝑡)),…,𝑓_𝑛(𝑥_𝑖𝑛(𝑡))]^𝑇 are sufficiently smooth nonlinear vector fields, and 𝐼(𝑡)=[𝐼₁(𝑡),𝐼₂(𝑡),…,𝐼_𝑛(𝑡)]^𝑇∈𝑅^𝑛 is the external input vector. 𝐶=[𝑐_𝑖𝑗]_𝑛×𝑛, 𝐷=[𝑑_𝑖𝑗]_𝑛×𝑛, and 𝐵=[𝑏_𝑖𝑗]_𝑛×𝑛 are coefficient matrixes, 𝐴=[𝑎_𝑖𝑗]_𝑛×𝑛 and 𝐴_𝜏=[𝑎^′_𝑖𝑗]_𝑛×𝑛 are inner-coupling matrixes, and 𝐺=[𝐺_𝑖𝑗]_𝑁×𝑁 represents the outer-coupling connections. The constant 𝑐>0 represents the coupling strength; 𝜏 and 𝜏(𝑡) represent the time constant delay and time-varying delay, respectively.

For the networks (1), we have the following assumptions.

Assumption  (H1).  𝜏(𝑡) is the interval time-varying delay satisfying 0≤ℎ₁≤𝜏(𝑡)≤ℎ₂,𝜏̇(𝑡)ℎ≤𝜇<+∞,₂₁=ℎ₂−ℎ₁. (2)

Assumption (H2). The outer-coupling configuration matrices of the network satisfy 𝐺_𝑖𝑗𝐺≥0,𝑖≠𝑗,_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗,𝑖,𝑗=1,2,…,𝑁. (3)

Assumption  (H3).  There exist constants 𝜎⁻_𝑖, 𝜎⁺_𝑖, and the functions satisfy 𝜎⁻_𝑖≤𝑓_𝑖𝑥₁−𝑓_𝑖𝑥₂𝑥₁−𝑥₂≤𝜎⁺_𝑖,𝑖=1,2,…,𝑛. (4) We denote Λ₁=diag𝜎⁺₁𝜎⁻₁,𝜎⁺₂𝜎⁻₂,…,𝜎⁺_𝑛𝜎⁻_𝑛,Λ₂𝜎=diag⁺¹+𝜎⁻₁2,𝜎⁺₂+𝜎⁻₂2𝜎,…,^+𝑛+𝜎⁻_𝑛2. (5)

Based on Assumption (H2), system (1) can be rewritten as the following form: 𝑥̇_𝑖(𝑡)=−𝐶𝑥_𝑖(𝑡)+𝐷𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏)+𝐼(𝑡)+𝑐^𝑁_𝑗=1𝐺_𝑖𝑗𝐴𝑥_𝑗(𝑡−𝜏)+𝑐^𝑁_𝑗=1𝐺_𝑖𝑗𝐴_𝜏𝑥_𝑗(𝑡−𝜏)−𝑐𝐺_𝑖𝑖𝐴_𝜏𝑥_𝑖(𝑡−𝜏)−𝑥_𝑖(𝑡),𝑥̇_𝑖(𝑡)=−𝐶𝑥_𝑖(𝑡)+𝐷𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))+𝐼(𝑡)+𝑐^𝑁_𝑗=1𝐺_𝑖𝑗𝐴𝑥_𝑗(𝑡−𝜏(𝑡))+𝑐^𝑁_𝑗=1𝐺_𝑖𝑗𝐴_𝜏𝑥_𝑗(𝑡−𝜏(𝑡))−𝑐𝐺_𝑖𝑖𝐴_𝜏𝑥_𝑖(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡). (6)(7)

The initial conditions of (6) are given by 𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠). 𝑠∈[𝑡₀−𝜏,𝑡₀], 𝑖=1,2,…,𝑁, where 𝜙_𝑖(⋅)=[𝜙_𝑖1(⋅),𝜙_𝑖2(⋅),…,𝜙_𝑖𝑛(⋅)]^𝑇∈𝐶([𝑡₀−𝜏,𝑡₀],𝑅^𝑛). The initial conditions of (7) are given by 𝑥_𝑖(𝑠)=𝜙_𝑖(𝑠). 𝑠∈[𝑡₀−ℎ₂,𝑡₀], 𝑖=1,2,…,𝑁, where 𝜙_𝑖(⋅)=[𝜙_𝑖1(⋅),𝜙_𝑖2(⋅),…,𝜙_𝑖𝑛(⋅)]^𝑇∈𝐶([𝑡₀−ℎ₂,𝑡₀],𝑅^𝑛), and in order to simplify we set 𝐺_𝑖𝑖=−𝑙.

Remark 1. The constants 𝜎⁻_𝑖, 𝜎⁺_𝑖 in Assumption (H3) are allowed to be any value. Then, most of the previous results in similar networks are just special cases of this assumption, which means that the activation functions are more general than those of other works.
Considering the sign of Kronecker product, models (6) and (7) can be rewritten as 𝑥̇(𝑡)=−𝐼_𝑁⊗𝐶𝑥(𝑡)+𝐼_𝑁⊗𝐷𝐹(𝑥(𝑡))+𝐼_𝑁(𝑥(⊗𝐵×𝐹𝑡−𝜏))+𝐼^′(𝑡)+𝑐𝐺⊗𝐴+𝐴_𝜏𝑥()𝐼𝑡−𝜏+𝑐𝑙_𝑁⊗𝐴_𝜏[𝑥(𝑡−𝜏)−𝑥(𝑡)],𝑥̇(𝑡)=−𝐼_𝑁⊗𝐶𝑥(𝑡)+𝐼_𝑁𝐹(𝑥(𝑡+𝐼⊗𝐷))_𝑁⊗𝐵𝐹(𝑥(𝑡−𝜏(𝑡)))+𝐼^′(𝑡)+𝑐𝐺⊗𝐴+𝐴_𝜏𝑥((𝑡𝐼𝑡−𝜏))+𝑐𝑙_𝑁⊗𝐴_𝜏[𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡)], (8)(9) with 𝑥(𝑡)=[𝑥^𝑇₁(𝑡),𝑥^𝑇₂(𝑡),…,𝑥^𝑇_𝑁(𝑡)]^𝑇, 𝐹(𝑥(𝑡))=[𝑓^𝑇(𝑥₁(𝑡)),𝑓^𝑇(𝑥₂(𝑡)),…,𝑓^𝑇(𝑥_𝑁(𝑡))]^𝑇, and 𝐼^′(𝑡)=[𝐼^𝑇(𝑡),𝐼^𝑇(𝑡),…,𝐼^𝑇(𝑡)]_𝑁^𝑇.

Definition 2 (see [14]). Dynamical networks (6) are said to be global asymptotic synchronization, if for any 𝜀, any 𝜑_𝑖(𝑠),𝜑_𝑗(𝑠)∈𝐶([𝑡₀−𝜏,𝑡₀],𝑅^𝑛)(𝑖,𝑗=1,2,…,𝑁). There exists 𝑇₁>𝑡₀ such that ‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖≤𝜀, where 𝑡>𝑇₁,  𝑖,𝑗=1,2,…,𝑁, and ‖⋅‖ denotes the Euclidean norm.

Definition 3 (see [15]). Dynamical networks (7) are said to be global exponential synchronization, if for any initial conditions 𝜑_𝑖(𝑠),𝜑_𝑗(𝑠)∈𝐶([𝑡₀−ℎ₂,𝑡₀],𝑅^𝑛)(𝑖,𝑗=1,2,…,𝑁), there exist 𝑇₁>𝑡₀ and 𝜀>0 such that ‖𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)‖≤𝑀𝑒^−𝜀𝑡, in which 𝑡>𝑇₁ and ‖⋅‖ denotes the Euclidean norm.

Lemma 4. Let ⊗ denote the notation of Kronecker product; then one has the following conclusions:(1)(𝛼𝐴)⊗𝐵=𝐴⊗(𝛼𝐵)(2)(𝐴+𝐵)⊗𝐶=𝐴⊗𝐶+𝐵⊗𝐶(3)(𝐴⊗𝐵)(𝐶⊗𝐷)=(𝐴𝐶)⊗𝐵𝐷

Lemma 5 (see [16]). Let 𝑒=(1,1,…,1)^𝑇,  𝐸_𝑁=𝑒𝑒^𝑇,  𝑈=𝑁𝐼_𝑁−𝐸_𝑁, 𝐾∈𝑅^𝑛×𝑛, 𝑥=[𝑥^𝑇₁,𝑥^𝑇₂,…,𝑥^𝑇_𝑁]^𝑇, 𝑦=[𝑦^𝑇₁,𝑦^𝑇₂,…,𝑦^𝑇_𝑁]^𝑇 with 𝑥_𝑘,𝑦_𝑘∈𝑅^𝑛  (𝑘=1,2,…,𝑁), and then 𝑥^𝑇(𝑈⊗𝐾)𝑦=∑^𝑁_{1≤𝑖<𝑗≤𝑁}(𝑥_𝑖−𝑥_𝑗)^𝑇𝐾(𝑦_𝑖−𝑦_𝑗).

Lemma 6 (see [17]). For any constant matrix 𝑋∈𝑅^𝑛×𝑛, 𝑋=𝑋^𝑇≥0, a scalar functional ℎ∶=ℎ(𝑡)≥0, and a vector function 𝑥̇∶[−ℎ,0]→𝑅^𝑛, the following inequality holds −ℎ∫^𝑡_𝑡−ℎ𝑥̇(𝑠)^𝑇×𝑥̇(𝑠)𝑑𝑠≤−[𝑥(𝑡)−𝑥(𝑡−ℎ)]^𝑇×[𝑥(𝑡)−𝑥(𝑡−ℎ)].

3. Main Results

3.1. Synchronization Condition for Constant Time Delay

In this section, we state and investigate the global asymptotic synchronization for system (6).

Theorem 7. Suppose Assumptions (H1)–(H3) hold, if there exist three symmetric positive definite matrices 𝑃, 𝑄, 𝑊 and three positive diagonal matrices 𝐻, 𝐾, 𝑉, such that the LMIs hold for each 𝑖, 𝑗(𝑖<𝑗) in (1,2,…,𝑁). ConsiderΠ_𝑖𝑗=𝛼−𝑐𝑁𝐺_𝑖𝑗𝑃𝐴+𝐴_𝜏+𝑐𝑙𝑃𝐴_𝜏+𝑊∗−𝑄−𝑊−𝑉Λ₁∗∗∗∗∗∗−𝐶^𝑇𝐾^𝑇−𝑐𝑙𝐴^𝑇_𝜏𝐾^𝑇𝑃𝐷+𝐻Λ₂𝑃𝐵𝛽0𝑉Λ₂𝜏²𝑊−𝐾−𝐾^𝑇𝐾𝐷𝐾𝐵∗−𝐻0∗∗−𝑉<0, (10)where 𝛼=−𝑃𝐶−𝐶^𝑇𝑃+𝑄−𝑐𝑙𝑃𝐴_𝜏−𝑐𝑙𝐴^𝑇_𝜏𝑃−𝑊−𝐻Λ₁,  𝛽=−𝑐𝑁𝐺_𝑖𝑗(𝐴+𝐴_𝜏)^𝑇𝐾^𝑇+𝑐𝑙𝐴^𝑇_𝜏𝐾^𝑇.

Then system (8) is global asymptotic synchronization.

Proof. Choose a Lyapunov-Krasovskii functional as 𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡)(𝑈⊗𝑃)𝑥(𝑡)+^𝑡_𝑡−𝜏𝑥^𝑇(𝑠)𝑥(𝑠)+)(𝑈⊗𝑄𝑑𝑠_{0−𝜏𝑡𝑡+𝜃}𝜏𝑥̇(𝑠)^𝑇(𝑈⊗𝑊)𝑥̇(𝑠)𝑑𝑠𝑑𝜃. (11)
Now by directly computing 𝑉̇(𝑥(𝑡)) along the trajectory of system (8), we have 𝑉̇(𝑥(𝑡))=2𝑥^𝑇(𝑡)(𝑈⊗𝑃)𝑥̇(𝑡)+𝑥^𝑇(𝑡)(𝑈⊗𝑄)𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)(𝑈⊗𝑄)𝑥(𝑡−𝜏)+𝜏²𝑥̇^𝑇(𝑡)×𝑥̇(𝑡)−)(𝑈⊗𝑊_{𝑡𝑡−𝜏}𝜏𝑥̇(𝑠)^𝑇(𝑈⊗𝑊)𝑥̇(𝑠)𝑑𝑠. (12)
In view of Lemma 6, we have 𝑉̇(𝑥(𝑡))≤2𝑥^𝑇(𝑡)(𝑈⊗𝑃)𝑥̇(𝑡)+𝑥^𝑇(𝑡)(𝑈⊗𝑄)𝑥(𝑡)−𝑥^𝑇(𝑡−𝜏)(𝑈⊗𝑄)𝑥(𝑡−𝜏)+𝜏²𝑥̇^𝑇(𝑡)𝑥̇(𝑡)−[𝑥(𝑡)()(𝑈⊗𝑊−𝑥𝑡−𝜏)]^𝑇(𝑈⊗𝑊)[𝑥(𝑡)−𝑥(𝑡−𝜏)]. (13)
Noting the facts that (𝑈⊗𝑃)𝐼(𝑡)=0 and 𝑈𝐺=𝑁𝐺, using Lemmas 4 and 5, we have 𝑉̇(𝑥(𝑡))≤2^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇−𝑃𝐶−𝑐𝑙𝑃𝐴_𝜏+12𝑄×𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)+𝑃𝐷𝑓𝑥_𝑖(𝑡)−𝑓𝑥_𝑗(𝑡)+𝑃𝐵𝑓𝑥_𝑖()𝑥𝑡−𝜏−𝑓_𝑗(𝑡−𝜏)−𝑐𝑁𝐺_𝑖𝑗𝑃𝐴+𝐴_𝜏×𝑥_𝑖(𝑡−𝜏)−𝑥_𝑗(𝑡−𝜏)+𝑐𝑙𝑃𝐴_𝜏×𝑥_𝑖(𝑡−𝜏)−𝑥_𝑗()−𝑡−𝜏_{𝑁−1𝑁𝑖=1𝑗=𝑖+1}𝑥_𝑖(𝑡−𝜏)−𝑥_𝑗(𝑡−𝜏)^𝑇×𝑄𝑥_𝑖(𝑡−𝜏)−𝑥_𝑗(𝑡−𝜏)+𝜏^{2𝑁−1𝑁}_{𝑖=1𝑗=𝑖+1}𝑥̇_𝑖(𝑡)−𝑥̇_𝑗(𝑡)^𝑇𝑊×𝑥̇_𝑖(𝑡)−𝑥̇_𝑗(𝑡)−^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑊𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)−2𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)^𝑇𝑊×𝑥_𝑖(𝑡−𝜏)−𝑥_𝑗()+𝑥𝑡−𝜏_𝑖(𝑡−𝜏)−𝑥_𝑗(𝑡−𝜏)^𝑇𝑊×𝑥_𝑖(𝑡−𝜏)−𝑥_𝑗(𝑡−𝜏). (14)
On the other hand, it is easy to see from the formulation of (8) that the following equation also holds for any matrices 𝐾∈𝑅^𝑛×𝑛: 2𝑥̇^𝑇(𝑡)(𝑈⊗𝐾)−𝑥̇(𝑡)−𝐼_𝑁𝑥(𝑡)+𝐼⊗𝐶_𝑁𝐹(𝑥(𝑡+𝐼⊗𝐷))_𝑁⊗𝐵𝐹(𝑥(𝑡−𝜏))+𝐼^′(𝑡)+𝑐𝐺⊗𝐴+𝐴_𝜏𝑥()𝐼𝑡−𝜏+𝑐𝑙_𝑁⊗𝐴_𝜏(𝑥(𝑡−𝜏)−𝑥(𝑡))=0. (15)
For convenience, let 𝑥_𝑖𝑗(𝑡)=𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡), 𝑥̇_𝑖𝑗(𝑡)=𝑥̇_𝑖(𝑡)−𝑥̇_𝑗(𝑡), 𝑓(𝑥_𝑖𝑗(𝑡))=𝑓(𝑥_𝑖(𝑡))−𝑓(𝑥_𝑗(𝑡)), and 𝑓(𝑥_𝑖𝑗(𝑡−𝜏))=𝑓(𝑥_𝑖(𝑡−𝜏))−𝑓(𝑥_𝑗(𝑡−𝜏)).
For any 𝑛×𝑛 diagonal matrices 𝐻>0, 𝑉>0, and Λ₁, Λ₂ from Assumption (H3), we have _{1≤𝑖<𝑗≤𝑁}−𝑥^𝑇_𝑖𝑗(𝑡)𝐻Λ₁𝑥_𝑖𝑗(𝑡)−2𝑥^𝑇_𝑖𝑗(𝑡)𝐻Λ₂𝑓𝑥_𝑖𝑗(𝑡)+𝑓^𝑇𝑥_𝑖𝑗(𝑡)𝐻𝑓𝑥_𝑖𝑗(𝑡)−𝑥^𝑇_𝑖𝑗(𝑡−𝜏)𝑉Λ₁𝑥_𝑖𝑗(𝑡−𝜏)−2𝑥^𝑇_𝑖𝑗(𝑡−𝜏)𝑉Λ₂𝑓𝑥_𝑖𝑗(𝑡−𝜏)+𝑓^𝑇𝑥_𝑖𝑗(𝑡−𝜏)𝑉𝑓𝑥_𝑖𝑗(𝑡−𝜏)≥0. (16)
Adding up (14)–(16) from both sides, we have 𝑉̇(𝑡)≤^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}2𝑥_𝑖𝑗(𝑡)^𝑇1−𝑃𝐶+2𝑄−𝑐𝑙𝑃𝐴_𝜏𝑥_𝑖𝑗(𝑡)+2𝑥_𝑖𝑗(𝑡)^𝑇𝑃𝐷𝑓𝑥_𝑖𝑗(𝑡)+2𝑥_𝑖𝑗(𝑡)^𝑇𝑃𝐵𝑓𝑥_𝑖𝑗(𝑡−𝜏)+2𝑥_𝑖𝑗(𝑡)^𝑇−𝑐𝑁𝐺_𝑖𝑗𝑃𝐴+𝐴_𝜏×𝑥_𝑖𝑗(𝑡−𝜏)+2𝑥_𝑖𝑗(𝑡)^𝑇𝑐𝑙𝑃𝐴_𝜏𝑥_𝑖𝑗(−𝑡−𝜏)_{𝑁−1𝑁𝑖=1𝑗=𝑖+1}𝑥_𝑖𝑗(𝑡−𝜏)^𝑇𝑄𝑥_𝑖𝑗()+𝑡−𝜏_{𝑁−1𝑁𝑖=1𝑗=𝑖+1}𝜏²𝑥̇_𝑖𝑗(𝑡)^𝑇𝑊𝑥̇_𝑖𝑗(𝑡)−^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝑥_𝑖𝑗(𝑡)^𝑇𝑊𝑥_𝑖𝑗(𝑡)−2𝑥_𝑖𝑗(𝑡)^𝑇𝑊𝑥_𝑖𝑗(𝑡−𝜏)+𝑥^𝑇_𝑖𝑗(𝑡−𝜏)𝑊𝑥_𝑖𝑗(−𝑡−𝜏)_{𝑁−1𝑁𝑖=1𝑗=𝑖+1}2𝑥̇^𝑇_𝑖𝑗(𝑡)𝐾𝑥̇_𝑖𝑗(𝑡)+2𝑥̇^𝑇_𝑖𝑗(𝑡)𝐾𝐶+𝑐𝑙𝐾𝐴_𝜏×𝑥_𝑖𝑗(𝑡)−2𝑥̇^𝑇_𝑖𝑗(𝑡)𝐾𝐷𝑓𝑥_𝑖𝑗(𝑡𝑥̇)−2^𝑇_𝑖𝑗(𝑡)𝐾𝐵𝑓𝑥_𝑖𝑗()𝑥̇𝑡−𝜏+2^𝑇_𝑖𝑗(𝑡)𝑐𝑁𝐺_𝑖𝑗𝐾𝐴+𝐴_𝜏𝑥_𝑖𝑗()𝑥̇𝑡−𝜏−2^𝑇_𝑖𝑗(𝑡)𝑐𝑙𝐾𝐴_𝜏𝑥_𝑖𝑗(−𝑡−𝜏)_{𝑁−1𝑁𝑖=1𝑗=𝑖+1}𝑥^𝑇_𝑖𝑗(𝑡)𝐻Λ₁𝑥_𝑖𝑗(𝑡)−2𝑥^𝑇_𝑖𝑗(𝑡)𝐻Λ₂𝑓𝑥_𝑖𝑗(𝑡)+𝑓^𝑇𝑥_𝑖𝑗(𝑡)𝐻𝑓𝑥_𝑖𝑗(𝑡)+𝑥^𝑇_𝑖𝑗(𝑡−𝜏)𝑉Λ₁𝑥_𝑖𝑗(𝑡−𝜏)−2𝑥^𝑇_𝑖𝑗(𝑡−𝜏)𝑉Λ₂𝑓𝑥_𝑖𝑗()𝑥𝑡−𝜏+𝑓^𝑇_𝑖𝑗(𝑡−𝜏)𝑉𝑓𝑥_𝑖𝑗()=𝑡−𝜏_{𝑁−1𝑁𝑖=1𝑗=𝑖+1}𝜂^𝑇_𝑖𝑗(𝑡)Π_𝑖𝑗𝜂_𝑖𝑗(𝑡), (17) where Π_𝑖𝑗 is from (10), and 𝜂^𝑇_𝑖𝑗(𝑡)=𝑥^𝑇_𝑖𝑗(𝑡)𝑥^𝑇_𝑖𝑗(𝑡−𝜏)𝑥̇^𝑇_𝑖𝑗(𝑡)𝑓𝑥^𝑇_𝑖𝑗(𝑡)𝑓𝑥^𝑇_𝑖𝑗(𝑡−𝜏). (18)
Following Theorem 7, we have 𝑉(𝑡)≤𝑉(0), and then we have 𝑥_𝑖(𝑡)−𝑥_𝑗(𝑡)→0 for all 1≤𝑖<𝑗≤𝑁. Therefore, system (8) is global asymptotic synchronization. Then end the proof.

Corollary 8. Suppose Assumptions (H1)–(H3) hold and 𝐵=0 in the system (8), if there exist three symmetric positive definite matrices 𝑃, 𝑄, 𝑊 and three positive diagonal matrices 𝐻, 𝐾, 𝑉, such that the LMIs holds for each 𝑖, 𝑗  (𝑖<𝑗) in (1,2,…,𝑁). ConsiderΠ_𝑖𝑗=𝛼−𝑐𝑁𝐺_𝑖𝑗𝑃𝐴+𝐴_𝜏+𝑐𝑙𝑃𝐴_𝜏+𝑊∗−𝑄−𝑊−𝑉Λ₁∗∗∗∗∗∗−𝐶^𝑇𝐾^𝑇−𝑐𝑙𝐴^𝑇_𝜏𝐾^𝑇𝑃𝐷+𝐻Λ₂0𝛽0𝑉Λ₂𝜏²𝑊−𝐾−𝐾^𝑇𝐾𝐷0∗−𝐻0∗∗−𝑉<0, (19)where 𝛼=−𝑃𝐶−𝐶^𝑇𝑃+𝑄−𝑐𝑙𝑃𝐴_𝜏−𝑐𝑙𝐴^𝑇_𝜏𝑃−𝑊−𝐻Λ₁,  𝛽=−𝑐𝑁𝐺_𝑖𝑗(𝐴+𝐴_𝜏)^𝑇𝐾^𝑇+𝑐𝑙𝐴^𝑇_𝜏𝐾^𝑇.

Then system (8) is global asymptotic synchronization.

The proof is obvious from Theorem 7 and we omit the details.

3.2. Time-Varying Delay Synchronization Condition

In this section, by utilizing the improved techniques used in [18], we obtain the following global exponential synchronization criterion for system (7).

Theorem 9. Suppose Assumptions (H1)–(H3) hold, if there exist five symmetric positive definite matrices 𝑃, 𝑄_𝑖, 𝑅_𝑖, 𝑍₁, 𝑍₂ for each 𝑖 in (1,2,3) and four positive diagonal matrices 𝐻, 𝐾, 𝑊, 𝑉, such that the LMIs holds for each 𝑖, 𝑗(𝑖<𝑗) in (1,2,…,𝑁). Consider Φ¹_𝑖𝑗=Φ_𝑖𝑗−0_𝑛,3𝑛−𝐼_𝑛𝐼_𝑛0_𝑛,3𝑛^𝑇×𝑍₂0_𝑛,3𝑛−𝐼_𝑛𝐼_𝑛0_𝑛,3𝑛<0,Φ²_𝑖𝑗=Φ_𝑖𝑗−0_𝑛,3𝑛𝐼_𝑛0_𝑛−𝐼_𝑛0_𝑛,2𝑛^𝑇×𝑍₂0_𝑛,3𝑛𝐼_𝑛0_𝑛−𝐼_𝑛0_𝑛,2𝑛<0, (20)(21) whereΦ_𝑖𝑗=Ω₁₁Ω₁₂Ω₁₃Ω₁₄𝑍₁∗Ω₂₂Ω₂₃Ω₂₄0∗∗Ω₃₃Ω₃₄0∗∗∗Ω₄₄𝑍₂∗∗∗∗Ω₅₅𝑍∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗000000000₂000𝑊Λ₂0−𝑄₂−𝑍₂−𝑉Λ₁0𝑉Λ₂∗−𝑅₁−𝑊0∗∗−𝑅₂,Ω−𝑉₁₁=−𝑃𝐶−𝐶^𝑇𝑃+³_𝑖=1𝑄_𝑖−𝑐𝑙𝑃𝐴_𝜏−𝑐𝑙𝐴^𝑇_𝜏𝑃−𝑍₁+𝐶^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐶+𝑐𝑙𝐴_𝜏+𝑐²𝑙²𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏+𝑐𝑙𝐶^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏−𝐻Λ₁,Ω₁₂=−𝐶^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐷+𝑃𝐷−𝑐𝑙𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐷+𝐻Λ₂,Ω₁₃=−𝐶^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐵−𝑐𝑙𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂Ω𝐵+𝑃𝐵,₁₄=𝐶^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝑐𝑁𝐺_𝑖𝑗𝐴+𝐴_𝜏−𝑐𝑙𝐴_𝜏−𝑐𝑁𝐺_𝑖𝑗𝑃𝐴+𝐴_𝜏+𝑐𝑙𝑃𝐴_𝜏−𝑐²𝑙𝑁𝐺_𝑖𝑗𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂×𝐴_𝜏+𝐴−𝑐²𝑙²𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏,Ω₂₂=𝐷^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐷+³_𝑖=1𝑅_𝑖Ω−𝐻,₂₃=𝐷^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂Ω𝐵,₂₄=𝐷^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂−𝑐𝑁𝐺_𝑖𝑗𝐴+𝐴_𝜏+𝑐𝑙𝐴_𝜏+𝑐𝑙𝐷^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏,Ω₃₃=𝐵^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐵−1−𝜇𝑅₃Ω−𝐾,₃₄=−𝑐𝑁𝐺_𝑖𝑗𝐵^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴+𝐴_𝜏+𝐾Λ₂+𝑐𝑙𝐵^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏,Ω₄₄=𝑐²𝑁𝐺²_𝑖𝑗𝐴+𝐴_𝜏^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴+𝐴_𝜏−1−𝜇𝑄₃−2𝑍₂−𝐾Λ₁−𝑐²𝑙𝑁𝐺_𝑖𝑗𝐴+𝐴_𝜏^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏+𝑐²𝑙²𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏,Ω₅₅=−𝑍₁−𝑍₂−𝑄₁−𝑊Λ₁. (22)

Then system (9) is global exponential synchronization.

Proof. We construct the Lyapunov-Krasovskii functional as follows: 𝑉(𝑥(𝑡))=𝑥^𝑇(𝑡)𝑥(𝑡)+)(𝑈⊗𝑃_{𝑡(𝑡)𝑡−𝜏}𝑥^𝑇(𝑠)𝑈⊗𝑄₃𝑥(𝑠)+𝑑𝑠_{2𝑖=1𝑡𝑡−ℎ𝑖}𝑥^𝑇(𝑠)𝑈⊗𝑄_𝑖𝑥(𝑠)+𝑑𝑠_{𝑡𝑡−𝜏(𝑡)}𝑓^𝑇(𝑥(𝑠))𝑈⊗𝑅₃𝑓(𝑥(𝑠+))𝑑𝑠_{2𝑖=1𝑡𝑡−ℎ𝑖}𝑓^𝑇(𝑥(𝑠))𝑈⊗𝑅_𝑖𝑓(𝑥(𝑠+))𝑑𝑠_{0−ℎ1𝑡𝑡+𝜃}ℎ₁𝑥̇^𝑇(𝑠)𝑈⊗𝑍₁𝑥̇(𝑠)+𝑑𝑠𝑑𝜃_{−ℎ1−ℎ2𝑡𝑡+𝜃}ℎ₁₂𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)𝑑𝑠𝑑𝜃. (23)
Calculating the time derivative of 𝑉(𝑥(𝑡)) along the trajectories of system (9), from Lemmas 4 and 5, we have 𝑉̇(𝑥(𝑡))=2𝑥^𝑇(𝑡)(𝑈⊗𝑃)𝑥̇(𝑡)+³_𝑖=1𝑥^𝑇(𝑡)𝑈⊗𝑄_𝑖𝑥(𝑡)−²_𝑖=1𝑥^𝑇𝑡−ℎ_𝑖𝑈⊗𝑄_𝑖𝑥𝑡−ℎ_𝑖−1−𝜇𝑥^𝑇(𝑡−𝜏(𝑡))𝑈⊗𝑄₃×𝑥(𝑡−𝜏(𝑡))+³_𝑖=1𝑓^𝑇(𝑥(𝑡))𝑈⊗𝑅_𝑖𝑓(𝑥(𝑡−))_2𝑖=1𝑓^𝑇𝑥𝑡−ℎ_𝑖𝑈⊗𝑅_𝑖𝑓𝑥𝑡−ℎ_𝑖−1−𝜇𝑓^𝑇(𝑥(𝑡−𝜏(𝑡)))𝑈⊗𝑅₃(𝑥((𝑡+𝑥̇×𝑓𝑡−𝜏)))_𝑇(𝑡)ℎ²₁𝑈⊗𝑍₁+ℎ²₁₂𝑈⊗𝑍₂𝑥̇(𝑡)−^𝑡_𝑡−ℎ1ℎ₁𝑥̇^𝑇(𝑠)𝑈⊗𝑍₁𝑥̇(𝑠)−𝑑𝑠_{𝑡−ℎ1𝑡−ℎ2}ℎ₁₂𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)𝑑𝑠. (24)
Using Lemma 4, we have −^𝑡_𝑡−ℎ1ℎ₁𝑥̇^𝑇(𝑠)𝑈⊗𝑍₁𝑥̇(𝑠)𝑥(𝑡)𝑑𝑠≤−−𝑥𝑡−ℎ₁^𝑇𝑈⊗𝑍₁𝑥(𝑡)−𝑥𝑡−ℎ₁. (25)
On the other hand, based on the approach in [18], and using ℎ₁₂=[ℎ₂−𝜏(𝑡)]+[𝜏(𝑡)−ℎ₁], then we have −^𝑡−ℎ1_𝑡−ℎ2ℎ₁₂𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)𝑑𝑠=−^{𝑡−𝜏(𝑡)}_𝑡−ℎ2ℎ₁₂𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)𝑑𝑠−^𝑡−ℎ1_{𝑡−𝜏(𝑡)}ℎ₁₂𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)𝑑𝑠=−^{𝑡−𝜏(𝑡)}_𝑡−ℎ2ℎ₂−𝜏(𝑡)𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)−𝑑𝑠_{𝑡−𝜏(𝑡)𝑡−ℎ2}𝜏(𝑡)−ℎ₁𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)−𝑑𝑠_{𝑡−ℎ1𝑡−𝜏(𝑡)}𝜏(𝑡)−ℎ₁𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)−𝑑𝑠_{𝑡−ℎ1(𝑡)𝑡−𝜏}ℎ₂−𝜏(𝑡)𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)𝑑𝑠. (26)
And in order to combine some of the terms in (26), letting 𝜔=(𝜏(𝑡)−ℎ₁)/ℎ₂, we have −^{𝑡−𝜏(𝑡)}_𝑡−ℎ2𝜏(𝑡)−ℎ₁𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)𝑑𝑠=−𝜔^{𝑡−𝜏(𝑡)}_𝑡−ℎ2ℎ₁₂𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)𝑑𝑠≤−𝜔^{𝑡−𝜏(𝑡)}_𝑡−ℎ2ℎ₂−𝜏(𝑡)𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)−𝑑𝑠_{𝑡−ℎ1𝑡−𝜏(𝑡)}ℎ₁₂𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)()𝑑𝑠=−1−𝜔^𝑡−ℎ1_{𝑡−𝜏(𝑡)}ℎ₁₂𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)()𝑑𝑠≤−1−𝜔^𝑡−ℎ1_{(𝑡)𝑡−𝜏}𝜏(𝑡)−ℎ₁𝑥̇^𝑇(𝑠)𝑈⊗𝑍₂𝑥̇(𝑠)𝑑𝑠. (27)
For any 𝑛×𝑛, diagonal matrices 𝐻>0, 𝐾>0, 𝑊>0, and 𝑉>0, and Λ₁,Λ₂ from Assumption (H3), the following inequality holds: _{1≤𝑖<𝑗≤𝑁}−𝑥^𝑇_𝑖𝑗(𝑡)𝐻Λ₁𝑥_𝑖𝑗(𝑡)−2𝑥^𝑇_𝑖𝑗(𝑡)𝐻Λ₂𝑓𝑥_𝑖𝑗(𝑡)+𝑓^𝑇𝑥_𝑖𝑗(𝑡)𝐻𝑓𝑥_𝑖𝑗(𝑡)−𝑥^𝑇_𝑖𝑗(𝑡−𝜏(𝑡))𝐾Λ₁𝑥_𝑖𝑗(𝑡−𝜏(𝑡))−2𝑥^𝑇_𝑖𝑗(𝑡−𝜏(𝑡))𝐾Λ₂𝑓𝑥_𝑖𝑗(𝑡−𝜏(𝑡))+𝑓^𝑇𝑥_𝑖𝑗(𝑡−𝜏(𝑡))𝐾𝑓𝑥_𝑖𝑗((𝑡−𝑥𝑡−𝜏))_𝑇𝑖𝑗𝑡−ℎ₁𝑊Λ₁𝑥_𝑖𝑗𝑡−ℎ₁−2𝑥^𝑇_𝑖𝑗𝑡−ℎ₁𝑊Λ₂𝑓𝑥_𝑖𝑗𝑡−ℎ₁+𝑓^𝑇𝑥_𝑖𝑗𝑡−ℎ₁𝑊𝑓𝑥_𝑖𝑗𝑡−ℎ₁−𝑥^𝑇_𝑖𝑗𝑡−ℎ₂𝑉Λ₁𝑥_𝑖𝑗𝑡−ℎ₂−2𝑥^𝑇_𝑖𝑗𝑡−ℎ₂𝑉Λ₂𝑓𝑥_𝑖𝑗𝑡−ℎ₂+𝑓^𝑇𝑥_𝑖𝑗𝑡−ℎ₂𝑉𝑓𝑥_𝑖𝑗𝑡−ℎ₂≥0. (28)
Adding up (24)–(28) from both sides, we have 𝑉̇(𝑥(𝑡))≤_{1≤𝑖<𝑗≤𝑁}𝜉^𝑇_𝑖𝑗(𝑡)Φ_𝑖𝑗𝜉_𝑖𝑗(𝑡)−𝜔𝑥(𝑡−𝜏(𝑡))−𝑥(𝑡−ℎ₂)^𝑇×𝑈⊗𝑍₂𝑥(𝑡−𝜏(𝑡))−𝑥𝑡−ℎ₂−(1−𝜔)𝑥(𝑡−ℎ₁)−𝑥(𝑡−𝜏(𝑡))^𝑇×𝑈⊗𝑍₂𝑥𝑡−ℎ₁−𝑥(𝑡−𝜏(𝑡))=_{1<𝑖<𝑗≤𝑁}𝜉^𝑇_𝑖𝑗(𝑡)Φ_𝑖𝑗−𝜔𝐼^𝑇₁𝑍₂𝐼₁−(1−𝜔)𝐼^𝑇₂𝑍₂𝐼₂𝜉_𝑖𝑗(𝑡)=^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝜉_𝑖𝑗(𝑡)^𝑇(1−𝜔)Φ¹_𝑖𝑗+𝜔Φ²_𝑖𝑗𝜉_𝑖𝑗(𝑡), (29) where 𝜙_𝑖𝑗𝜙¹_𝑖𝑗𝜙²_𝑖𝑗 are given in (20), (21), and (22), respectively, and 𝐼₁=[0_𝑛,3𝑛−𝐼_𝑛0_𝑛−𝐼_𝑛0_𝑛,2𝑛] and  𝐼₂=[0_𝑛,3𝑛−𝐼_𝑛𝐼_𝑛0_𝑛,3𝑛]. Consider 𝜉^𝑇_𝑖𝑗(𝑡)=𝑥^𝑇_𝑖𝑗(𝑡)𝑓^𝑇𝑥_𝑖𝑗(𝑡)𝑓^𝑇𝑥_𝑖𝑗(𝑡−𝜏(𝑡))𝑥^𝑇_𝑖𝑗((𝑡𝑥𝑡−𝜏))_𝑇𝑖𝑗𝑡−ℎ₁𝑥^𝑇_𝑖𝑗𝑡−ℎ₂𝑓^𝑇𝑥_𝑖𝑗𝑡−ℎ₁𝑓^𝑇𝑥_𝑖𝑗𝑡−ℎ₂. (30)
Following Theorem 9, we have (1−𝜔)Φ¹_𝑖𝑗+𝜔Φ²_𝑖𝑗<0, and there exists a positive constant 𝑀>0 that satisfied (1−𝜔)Φ¹_𝑖𝑗+𝜔Φ²_𝑖𝑗≤−𝑀𝐼<0, such that 𝑉̇(𝑥(𝑡))≤^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝜉_𝑖𝑗(𝑡)^𝑇(1−𝜔)Φ¹_𝑖𝑗+𝜔Φ²_𝑖𝑗𝜉_𝑖𝑗(𝑡)≤−𝑀^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}𝑥_𝑖𝑗(𝑡)²+𝑥_𝑖𝑗(𝑡−𝜏(𝑡))². (31)
Furthermore, based on the proof in [12], there exist two positive scalars 𝛽>0 and 𝑘>0, such that 𝑥_𝑖𝑗(𝑡)≤𝛽^𝑁−1𝑁_{𝑖=1𝑗=𝑖+1}sup_{−2ℎ2≤𝑠≤0}𝜑_𝑖(𝑠)−𝜑_𝑗(𝑠)⋅𝑒^−𝑘𝑡 (32) for 𝑡≥𝑇₁. By Definition 3, therefore, system (9) is global exponential synchronization. Then end the proof.

Remark 10. In [10–13], the authors studied the synchronization of an array of linearly coupled networks with constant coupling delay or time-varying coupling delay, and the derivative of the time-varying delay is confined to be less than 1. We remove this restrictiveness and the derivative of the time-varying delay can be any value.

Remark 11. During the estimation, −∫^𝑡−ℎ1_𝑡−ℎ2ℎ₁₂𝑥̇^𝑇(𝑠)(𝑈⊗𝑍₂)𝑥̇(𝑠)𝑑𝑠 is separated into two parts as follows: −∫^𝑡−ℎ1_𝑡−ℎ2ℎ₁₂𝑥̇^𝑇(𝑠)(𝑈⊗𝑍₂)𝑥̇(𝑠)𝑑𝑠=−∫^{𝑡−𝜏(𝑡)}_𝑡−ℎ2ℎ₁₂𝑥̇^𝑇(𝑠)(𝑈⊗𝑍₂)𝑥̇(𝑠)𝑑𝑠−∫^𝑡−ℎ1_{𝑡−𝜏(𝑡)}ℎ₁₂𝑥̇^𝑇(𝑠)(𝑈⊗𝑍₂)𝑥̇(𝑠)𝑑𝑠, and we estimate each part, respectively, ignoring the direct estimate for −∫^𝑡−ℎ1_𝑡−ℎ2ℎ₁₂𝑥̇^𝑇(𝑠)(𝑈⊗𝑍₂)𝑥̇(𝑠)𝑑𝑠, and then we have more accurate estimate than those from the reference [12, 13].
When ℎ₁=0  in Assumption (H1), we have the following corollary from Theorem 9.

Corollary 12. Suppose Assumptions (H1)–(H3) hold, if there exist four symmetric positive definite matrices 𝑃, 𝑄_𝑖, 𝑅_𝑖, 𝑍₂ for each 𝑖 in (1,2) and four positive diagonal matrices 𝐻, 𝐾, 𝑉, such that the LMIs holds for each 𝑖, 𝑗(𝑖<𝑗) in (1,2,…,𝑁). Consider Φ¹_𝑖𝑗=Φ_𝑖𝑗−0_𝑛,3𝑛−𝐼_𝑛0_𝑛,2𝑛^𝑇×𝑍₂0_𝑛,3𝑛−𝐼_𝑛0_𝑛,2𝑛<0Φ²_𝑖𝑗=Φ_𝑖𝑗−0_𝑛,3𝑛𝐼_𝑛−𝐼_𝑛0_𝑛^𝑇×𝑍₂0_𝑛,3𝑛𝐼_𝑛−𝐼_𝑛0_𝑛<0Φ_𝑖𝑗=Ω₁₁Ω₁₂Ω₁₃Ω₁₄∗00Ω₂₂Ω₂₃Ω₂₄00∗∗Ω₃₃Ω₃₄00∗∗∗Ω₄₄𝑍₂0∗∗∗∗Ω₅₅𝑉Λ₂∗∗∗∗∗−𝑅₂−𝑉Ω₁₁=−𝑃𝐶−𝐶^𝑇𝑃+³_𝑖=1𝑄_𝑖−𝑐𝑙𝑃𝐴_𝜏−𝑐𝑙𝐴^𝑇_𝜏𝑃−𝑍₁+ℎ²₁₂𝐶^𝑇𝑍₂𝐶+𝑐𝑙𝐴_𝜏+𝑐²𝑙²ℎ²₁₂𝐴^𝑇_𝜏𝑍₂𝐴_𝜏+𝑐𝑙ℎ²₁₂𝐶^𝑇𝑍₂𝐴_𝜏−𝐻Λ₁Ω₁₂=−ℎ²₁₂𝐶^𝑇𝑍₂𝐷+𝑃𝐷−𝑐𝑙ℎ²₁₂𝐴^𝑇_𝜏𝑍₂𝐷+𝐻Λ₂Ω₁₃=−ℎ²₁₂𝐶^𝑇𝑍₂𝐵−𝑐𝑙ℎ²₁₂𝐴^𝑇_𝜏𝑍₂𝐵+𝑃𝐵Ω₁₄=ℎ²₁₂𝐶^𝑇𝑍₂𝑐𝑁𝐺_𝑖𝑗𝐴+𝐴_𝜏−𝑐𝑙𝐴_𝜏−𝑐𝑁𝐺_𝑖𝑗𝑃𝐴+𝐴_𝜏+𝑐𝑙𝑃𝐴_𝜏−𝑐²𝑙𝑁ℎ²₁₂𝐺_𝑖𝑗𝐴^𝑇_𝜏𝑍₂𝐴_𝜏+𝐴−𝑐²𝑙²ℎ²₁₂𝐴^𝑇_𝜏𝑍₂𝐴Ω₂₂=ℎ²₁₂𝐷^𝑇𝑍₂𝐷+³_𝑖=1𝑅_𝑖−𝐻Ω₂₃=ℎ²₁₂𝐷^𝑇𝑍₂𝐵Ω₂₄=ℎ²₁₂𝐷^𝑇𝑍₂−𝑐𝑁𝐺_𝑖𝑗𝐴+𝐴_𝜏+𝑐𝑙𝐴_𝜏+𝑐𝑙ℎ²₁₂𝐷^𝑇𝑍₂𝐴_𝜏Ω₃₃=ℎ²₁₂𝐵^𝑇𝑍₂𝐵−1−𝜇𝑅₃−𝐾Ω₃₄=−𝑐𝑁ℎ²₁₂𝐺_𝑖𝑗𝐵^𝑇𝑍₂𝐴+𝐴_𝜏+𝐾Λ₂+𝑐𝑙ℎ²₁₂𝐵^𝑇𝑍₂𝐴_𝜏Ω₄₄=𝑐²𝑁ℎ²₁₂𝐺²_𝑖𝑗𝐴+𝐴_𝜏^𝑇𝑍₂𝐴+𝐴_𝜏−1−𝜇𝑄₃−2𝑍₂−𝐾Λ₁−𝑐²𝑙𝑁ℎ²₁₂𝐺_𝑖𝑗𝐴+𝐴_𝜏^𝑇𝑍₂𝐴_𝜏+𝑐²𝑙²ℎ²₁₂𝐴^𝑇_𝜏𝑍₂𝐴_𝜏Ω₅₅=−𝑍₁−𝑍₂−𝑄₁−𝑊Λ₁. (33)

Then system (9) is global exponential synchronization. The proof is obvious from Theorem 9 and we omit the details. When 𝜏(𝑡) in Assumption (H1) is not differentiable or 𝜇 is unknown, we have the following corollary from Theorem 9.

Corollary 13. Suppose Assumptions (H1)–(H3) hold, if there exist four symmetric positive definite matrices 𝑃, 𝑄_𝑖, 𝑅_𝑖, 𝑍_𝑖 for each 𝑖 in (1,2) and four positive diagonal matrices 𝐻, 𝐾, 𝑊, 𝑉, such that the LMIs holds for each 𝑖, 𝑗(𝑖<𝑗) in (1,2,…,𝑁). Consider Φ¹_𝑖𝑗=Φ_𝑖𝑗−0_𝑛,3𝑛−𝐼_𝑛𝐼_𝑛0_𝑛,3𝑛^𝑇×𝑍₂0_𝑛,3𝑛−𝐼_𝑛𝐼_𝑛0_𝑛,3𝑛Φ<0,_2𝑖𝑗=Φ_𝑖𝑗−0_𝑛,3𝑛𝐼_𝑛0_𝑛−𝐼_𝑛0_𝑛,2𝑛^𝑇×𝑍₂0_𝑛,3𝑛𝐼_𝑛0_𝑛−𝐼_𝑛0_𝑛,2𝑛<0, (34) whereΦ_𝑖𝑗=Ω₁₁Ω₁₂Ω₁₃Ω₁₄𝑍₁∗Ω₂₂Ω₂₃Ω₂₄0∗∗Ω₃₃Ω₃₄0∗∗∗Ω₄₄𝑍₂∗∗∗∗Ω₅₅𝑍∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗000000000₂000𝑊Λ₂0−𝑄₂−𝑍₂−𝑉Λ₁0𝑉Λ₂∗−𝑅₁−𝑊0∗∗−𝑅₂.−𝑉 (35)With Ω^⌢₁₁=−𝑃𝐶−𝐶^𝑇𝑃+²_𝑖=1𝑄_𝑖−𝑐𝑙𝑃𝐴_𝜏−𝑐𝑙𝐴^𝑇_𝜏𝑃−𝑍₁+𝐶^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐶+𝑐𝑙𝐴_𝜏+𝑐²𝑙²𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏+𝑐𝑙𝐶^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏−𝐻Λ_1⌢Ω₁₂=−𝐶^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐷+𝑃𝐷−𝑐𝑙𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐷+𝐻Λ_2⌢Ω₁₃=−𝐶^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐵−𝑐𝑙𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐵+𝑃𝐵_⌢Ω₁₄=𝐶^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝑐𝑁𝐺_𝑖𝑗𝐴+𝐴_𝜏−𝑐𝑙𝐴_𝜏−𝑐𝑁𝐺_𝑖𝑗𝑃𝐴+𝐴_𝜏+𝑐𝑙𝑃𝐴_𝜏−𝑐²𝑙𝑁𝐺_𝑖𝑗𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏+𝐴−𝑐²𝑙²𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏⌢Ω₂₂=𝐷^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐷+²_𝑖=1𝑅_𝑖−𝐻_⌢Ω₂₃=𝐷^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐵_⌢Ω₂₄=𝐷^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂−𝑐𝑁𝐺_𝑖𝑗𝐴+𝐴_𝜏+𝑐𝑙𝐴_𝜏+𝑐𝑙𝐷^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏⌢Ω₃₃=𝐵^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐵−𝐾_⌢Ω₃₄=−𝑐𝑁𝐺_𝑖𝑗𝐵^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴+𝐴_𝜏+𝐾Λ₂+𝑐𝑙𝐵^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏⌢Ω₄₄=𝑐²𝑁𝐺²_𝑖𝑗𝐴+𝐴_𝜏^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴+𝐴_𝜏−2𝑍₂−𝐾Λ₁−𝑐²𝑙𝑁𝐺_𝑖𝑗𝐴+𝐴_𝜏^𝑇ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏+𝑐²𝑙²𝐴^𝑇_𝜏ℎ²₁𝑍₁+ℎ²₁₂𝑍₂𝐴_𝜏⌢Ω₅₅=−𝑍₁−𝑍₂−𝑄₁−𝑊Λ₁ (36) then system (9) is global exponential synchronization.

Remark 14. Theorem 9 can be applied to both slow and fast time-varying delays only if 𝜇 is known. But when 𝜏(𝑡) is not differentiable or 𝜇 is unknown, Theorem 9 fails to work; however, Corollary 13 can check the synchronization of system (9) instead.

4. Numerical Examples

In this section, two examples are provided to illustrate the effectiveness of the conclusion.

Example 1. Consider a lower-dimensional network model with 5 nodes, where each node is a simple three-dimensional stable linear system. To simplify, we assume that 𝐶=diag(1,2,3),  𝐷=𝐵=𝐴_𝜏=0, 𝐼(𝑡)=0, and the inner-coupling matrix is 𝐴=diag(1,1,1), and the outer-coupling matrix is defined as 𝐺=.−210011−311001−210011−311001−2 (37)
We compare the admissible delay upper bounds for 𝜏 and the number of variables with those from other references; it is clear from Table 1 that the delay is a little larger and the number of variables is less than those from [10, 12].

Table 1 

Admissible delay upper bound 𝜏 for different 𝑐.

Example 2. We consider the following DNNs: 𝑥̇_𝑖(𝑡)=−𝐶𝑥_𝑖(𝑡)+𝐷𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))+𝐼(𝑡). (38)
Let us see the synchronized states of a chaotic system: Chua’s circuit [19].
The dynamics of Chua’s circuit is 𝑑𝑥₁𝑑𝑡=10−𝑥₁+𝑥₂−𝑓𝑥₁𝑑𝑥₂𝑑𝑡=𝑥₁−𝑥₂+𝑥₃𝑑𝑥₃𝑑𝑡=−18𝑥₂, (39) where 𝑓𝑥₁=𝑏𝑥₁+0.5(𝑑−𝑏)𝑥₁+1−𝑥₁,4−1𝑑=−33,𝑏=−4. (40)
Here we choose Chua’s circuit (39) as the uncoupled system. Figure 1 illustrates the chaotic trajectories of system (39). The coupling time-varying delay 𝜏(𝑡)=0.03(2+sin(40𝑡)+cos²(80𝑡)) and the coupling strength 𝑐=1, then we consider dynamic networks consisting of three linearly coupled identical DNNs with couplings as 𝑥̇_𝑖(𝑡)=−𝐶𝑥_𝑖(𝑡)+𝐷𝑓𝑥_𝑖(𝑡)+𝐵𝑓𝑥_𝑖(𝑡−𝜏(𝑡))+𝐼(𝑡)+𝑐^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗𝐴𝑥_𝑗(𝑡−𝜏(𝑡))−𝑥_𝑖(𝑡−𝜏(𝑡))+𝑐^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗𝐴_𝜏𝑥_𝑗(𝑡−𝜏)−𝑥_𝑖(𝑡) (41) for 𝑖=1,2,3. Let ,,𝐴𝐶=10−100−11−10180,𝐷=−1000000000𝐺=−4132−4231−4,𝐴=100010001_𝜏=,0.50.1000.500.300.5 (42)𝐼(𝑡)=𝐵=0, 𝑁=3, and 𝑙=4.
By calculating we have ℎ₁=0.06, ℎ₂=0.12, and 𝜇=3.6, and the activation functions 𝑓(𝑥_𝑖) satisfy Assumption (H3). If the system reaches synchronization, we have the following synchronized state equation: 𝑠̇(𝑡)=−𝐶𝑠(𝑡)+𝐷𝑓(𝑠(𝑡))+𝐵𝑓(𝑠(𝑡−𝜏(𝑡)))+𝑙𝐴_𝜏(𝑠(𝑡−𝜏(𝑡))−𝑠(𝑡)). (43) By calculating other variables in Theorem 9 using Matlab toolbox, we realize the system synchronization, and the total error is given as error(𝑡)=∑³_𝑖=1[𝑥_1𝑖−𝑥_2𝑖]²+[𝑥_2𝑖−𝑥_3𝑖]². Figures 2 and 3 show the synchronized state of system (39) and the synchronous error, and Figures 4, 5, and 6 show the curves of state variables 𝑥_𝑖1(𝑡), 𝑥_𝑖2(𝑡), and 𝑥_𝑖3(𝑡) with the initial value randomly chosen from [0,1].

Figure 1 

The chaotic trajectories of system (39).

Figure 2 

The synchronized state of system (39).

Figure 3 

The synchronous error of system (39).

Figure 4 

The synchronized trajectory of 𝑥𝑖1(𝑡), 𝑖=1,2,3.

Figure 5 

The synchronized trajectory of 𝑥𝑖2(𝑡), 𝑖=1,2,3.

Figure 6 

The synchronized trajectory of 𝑥𝑖3(𝑡), 𝑖=1,2,3.

5. Conclusion

In this paper, global synchronization problem is investigated for general complex networks with coupling delays. Two novel theorems are obtained about synchronization of the coupled systems by employing Lyapunov-Krasovskii functional and convex combination techniques. We find that the derivative of the time-varying delay can be any given value, although most of the former results are based on assumption that the derivative of the time-varying delay should be less than one. The condition obtained in this paper is expressed in the form of linear matrix inequalities, which have less variables and are easy to be computed and checked by resorting to Matlab LMI toolbox. The proposed network model may shed some new lights on the synchronization with one delay coupling. Furthermore, there are abundant dynamical behaviors in arrays of coupled systems with different coupling schemes and they deserve to be further studied in the future. Finally, two examples are given to illustrate the usefulness of the derived methods by the simulation results.

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant nos. 61070149, 61175115, 61370113, and 61272320), the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (Grant no. CIT&TCD201304035), and Jing-Hua Talents Project of Beijing University of Technology (Grant no. 2014-JH-L06).