Synchronization of Coupled Networks with Mixed Delays by Intermittent Control

Abstract

The synchronization of coupled networks with mixed delays is investigated by employing Lyapunov functional method and intermittent control. A sufficient condition is derived to ensure the global synchronization of coupled networks, which is controlled by the designed intermittent controller. Finally, a numerical simulation is constructed to justify the theoretical analysis.

1. Introduction

Various large-scale and complicated systems can be modelled by complex networks, such as the Internet, genetic networks, ecosystems, electrical power grids, and the social networks. A complex network is a large set of interconnected nodes, which can be described by the graph with the nodes representing individuals in the graph and the edges representing the connections among them. The most remarkable recent advances in study of complex networks are the developments of the small-world network model [1] and scale-free network model [2], which have been shown to be very closer to most real-world networks as compared with the random-graph model [3, 4]. Thereafter, small-world and scale-free networks have been extensively investigated.

The dynamical behaviors of complex networks have become a focal topic of great interest, particularly the synchronization phenomena, which is observed in natural, social, physical, and biological systems and has been widely applied in a variety of fields, such as secure communication, image processing, and harmonic oscillation generation. It is noted that the dynamical behavior of a complex network is determined not only by the dynamical rules governing the isolated nodes, referred to as self-dynamics, but also by information flow along the edges, which depends on the topology of the network. Synchronization in an array of linearly coupled dynamical systems was investigated in [5]. Later, many results on local, global, and partial synchronization in various coupled systems have also been obtained in [6–15]. As a special case of coupled systems, coupled neural networks with time delay have also been found to exhibit complex behaviors. The estimation and diagnosis for time delay systems are discussed in [16, 17], and synchronization for coupled neural networks with time delay has been investigated by many researchers, for example, [8–15].

In the case that the whole network cannot synchronize by itself, some controllers should be designed and applied to force the network to synchronize. Recently, another interesting intermittent control was introduced and studied, that is, the control time is periodic, and in any period the time is composed of work time and rest time. It is a straightforward engineering approach to process control of any typelan approach that has been used for a variety of purposes in such engineering fields as manufacturing, transportation, and communication. Intermittent control has been introduced to control nonlinear dynamical systems [18] and has been studied in [19–26]. In [18], the authors investigated numerically chaos synchronization under the condition that the interacting systems, that is, master and slave systems are coupled intermittently. In [19, 20], the stabilization problems of chaotic systems with or without delays by periodically intermittent control were discussed. Huang et al. discussed the synchronization of coupled chaotic systems with delay by using intermittent state feedback in [21]. In [25], the authors synchronize coupled networks using pinning control and intermittent control. In [26], cluster synchronization was studied for coupled networks without time delay using adaptive intermittent control.

Another type of time delays, namely, distributed delays, has begun to receive research attention. The main reason is that a neural network usually has a spatial nature due to the presence of an amount of parallel pathways of a variety of axon sizes and lengths, and it is desirable to model them by introducing continuously distributed delays over a certain duration of time, such that the distant past has less influence compared to the recent behavior of the state [27]. Therefore, both discrete and distributed time delays should be taken into account [28–33]. Although synchronization has been investigated under intermittent control, [25, 26], there is still no theoretical result of synchronization for coupled networks with mixed delay.

Motivated by the above discussion, the intermittent controller will be designed to achieve the synchronization for coupled networks with mixed delay. The rest of the paper is organized as follows. In Section 2, some preliminary definitions and lemmas are briefly outlined. Some synchronization criteria are given and intermittent controller are designed in Section 3. An illustrative simulation is given to verify the theoretical analysis in Section 4. Conclusions are finally drawn.

Notations
ℝ^𝑛 is the 𝑛-dimensional Euclidean space; ℝ^𝑚×𝑛 denotes the set of 𝑚×𝑛 real matrix. 𝐼 is the identity matrix with appropriate dimension, and the superscript “𝑇” represents the transpose. Matrix dimensions, if not explicitly stated, are assumed to be compatible for algebraic operations.

2. Model Description and Preliminaries

Consider a dynamical network consisting of 𝑁 identical and diffusively coupled nodes, with each node being an 𝑛-dimensional delayed neural network. The state equations of the network are 𝑥̇_𝑖(𝑡)=−+𝐼𝐷(𝑥𝑡_𝑖)(+𝑡)+_𝑁𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑔𝑥_𝑖(𝑡−𝜏)+𝐶^{𝑡𝑡−𝜏}ℎ𝑥_𝑖(𝑣)𝑑𝑣_𝑗=_{1,𝑗≠𝑖}𝐺_𝑖𝑗Γ𝑥_𝑗(𝑡)−𝑥_𝑖(𝑡),(2.1) where 𝑥_𝑖(𝑡)=(𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡),…,𝑥_𝑖𝑛(𝑡))^𝑇∈ℝ^𝑛 is the state vector of the 𝑖th node; 𝐷=diag(𝑑₁,𝑑₂,…,𝑑_𝑛)>0 denotes the rate with which the cell 𝑖 resets its potential to the resting state when isolated from other cells and inputs; 𝐴∈ℝ^𝑛×𝑛, 𝐵∈ℝ^𝑛×𝑛, and 𝐶∈ℝ^𝑛×𝑛 represent the connection weight matrix, the discretely delayed connection weight matrix, and the distributively delayed connection weights, respectively; 𝑓(𝑥_𝑖(⋅))=[𝑓₁(𝑥_𝑖1(⋅)),𝑓₂(𝑥_𝑖2(⋅)),…,𝑓_𝑛(𝑥_𝑖𝑛(⋅))]^𝑇∈ℝ^𝑛, 𝑔(𝑥_𝑖(⋅))=[𝑔₁(𝑥_𝑖1(⋅)),𝑔₂(𝑥_𝑖2(⋅)),…,𝑔_𝑛(𝑥_𝑖𝑛(⋅))]^𝑇∈ℝ^𝑛 and ℎ(𝑥_𝑖(⋅))=[ℎ₁(𝑥_𝑖1(⋅)),ℎ₂(𝑥_𝑖2(⋅)),…,ℎ_𝑛(𝑥_𝑖𝑛(⋅))]^𝑇∈ℝ^𝑛 are activation functions; 𝐼(𝑡) is the input vector of each node; Γ∈ℝ^𝑛×𝑛 is the inner coupling matrix; 𝐺=(𝐺_𝑖𝑗)_𝑁×𝑁 is the coupling configuration matrix representing the topological structure of the network, where 𝐺_𝑖𝑗 is defined as follows: if there exists a connection between node 𝑖 and node 𝑗, 𝐺_𝑖𝑗>0, otherwise 𝐺_𝑖𝑗=0(𝑗≠𝑖), and the diagonal elements of matrix 𝐺 are defined by 𝐺_𝑖𝑖=−^𝑁_𝑗=_{1,𝑗≠𝑖}𝐺_𝑖𝑗,(2.2) which ensures the diffusion that ∑^𝑁𝑗=1𝐺_𝑖𝑗=0. Equivalently, network (2.1) can be rewritten in a form as follows: 𝑥̇_𝑖(𝑡)=+−𝐷𝐼(𝑥𝑡_𝑖)(+𝑡)+_𝑁𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑔𝑥_𝑖(𝑡−𝜏)+𝐶^{𝑡𝑡−𝜏}ℎ𝑥_𝑖(𝑣)𝑑𝑣_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑖=1,2,…,𝑁.(2.3) Suppose that the coupled network (2.3) is connected in the sense that there are no isolated clusters, then the coupling matrix 𝐺 is irreducible.

Note that a solution to an isolated node satisfies 𝑑𝑠(𝑡)𝑑𝑡=−𝐷𝑠(𝑡)+𝐴𝑓(𝑠(𝑡))+𝐵𝑔(𝑠(𝑡−𝜏))+𝐶^{𝑡𝑡−𝜏}ℎ(𝑠(𝑣))𝑑𝑣+𝐼(𝑡).(2.4) To realize the synchronization of network (2.3), the intermittent strategy is selected, and the controlled network can be described by 𝑥̇_𝑖(𝑡)=+−𝐷𝐼(𝑥𝑡_𝑖)(+𝑡)+_𝑁𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑔𝑥_𝑖(𝑡−𝜏)+𝐶^{𝑡𝑡−𝜏}ℎ𝑥_𝑖(𝑣)𝑑𝑣_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗(𝑡)+𝑢_𝑖,𝑖=1,2,…,𝑁,(2.5) where 𝑢_𝑖=−𝑘_𝑖(𝑡)𝑥_𝑖(𝑡)−𝑠(𝑡),(2.6)𝑘(𝑡) is the intermittent linear state feedback control gain defined as follows: 𝑘_𝑖(𝑡)=𝑘^𝑖0,,𝑛𝑛𝜔𝜔+≤𝛿𝑡≤<𝑛𝑡𝜔≤+(𝑛𝛿,+1)𝜔,(2.7) where 𝑘_𝑖∈𝑅 is a constant control gain, 𝜔>0 is the control period, and 𝛿>0 is called the control width. In this paper, our goal is to design suitable 𝛿, 𝜔, and 𝑘_𝑖 such that network (2.5) synchronize with respect to the isolated node 𝑠(𝑡). Denote 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑠(𝑡), then the following error dynamical system is obtained: 𝑒_𝑖̇(𝑡)=−+𝐷𝐶𝑒_𝑖(𝑡_{𝑡𝑡−𝜏})+𝐴𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))+𝐵𝑔𝑥_𝑖(𝑡−𝜏)−𝑔(𝑠(𝑡−𝜏))𝑒_𝑖̇ℎ𝑥_𝑖(𝑣)−ℎ(𝑠(𝑣))𝑑𝑣+^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑒_𝑗(𝑡)−𝑘_𝑖𝑒_𝑖(𝑡),𝑛𝜔≤𝑡≤𝑛𝜔+𝛿,(𝑡)=−+𝐷𝐶𝑒_𝑖(𝑡_{𝑡𝑡−𝜏})+𝐴𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))+𝐵𝑔𝑥_𝑖(𝑡−𝜏)−𝑔(𝑠(𝑡−𝜏))ℎ𝑥_𝑖(𝑣)−𝑠(𝑣)𝑑𝑣+^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑒_𝑗(𝑡),𝑛𝜔+𝛿<𝑡≤(𝑛+1)𝜔.(2.8)(H) We assume that 𝑓, 𝑔, and ℎ are Lipschitz continuous functions; there exist positive constants 𝐿_𝑓, 𝐿_𝑔 and 𝐿_ℎ such that, for all 𝑥,𝑦∈ℝ^𝑚, ‖‖𝑓𝑔((𝑥𝑥))−−𝑓𝑔((𝑦𝑦))‖‖≤≤𝐿𝐿_𝑓𝑔‖𝑥−𝑦‖,‖ℎ(𝑥)−ℎ(𝑦)‖≤𝐿_ℎ‖𝑥−𝑦‖,‖𝑥−𝑦‖.(2.9)

Definition 2.1. For any positive integers 𝑝, 𝑞, 𝑟, 𝑠, we define the Kronecker product of two matrices 𝐴∈ℝ^𝑝×𝑞, 𝐵∈ℝ^𝑟×𝑠 as follows: 𝐴⊗𝐵=⎡⎢⎢⎢⎢⎣𝑎¹¹𝑎⋮𝐵⋱⋯𝑎⋮_1𝑞𝑝1𝐵𝐵⋯𝑎_𝑝𝑞𝐵⎤⎥⎥⎥⎥⎦∈ℝ^{𝑝𝑟×𝑞𝑠}.(2.10)

Lemma 2.2. By the definition of Kronecker product, the following properties hold:(1)(𝐴⊗𝐵)^𝑇=𝐴^𝑇⊗𝐵^𝑇; (2)(𝛼𝐴)⊗𝐵=𝐴⊗(𝛼𝐵), where 𝛼 is a real number; (3)(𝐴⊗𝐵)(𝐶⊗𝐷)=(𝐴𝐶)⊗(𝐵𝐷).

Lemma 2.3. For any vectors 𝑥, 𝑦∈ℝ^𝑚, and positive-definite matrix 𝑄∈ℝ^𝑚×𝑚, the following matrix inequality holds: 2𝑥^𝑇𝑦≤𝑥^𝑇𝑄𝑥+𝑦^𝑇𝑄⁻¹𝑦.(2.11)

Lemma 2.4 (Jensen's inequality [34]). For any constant matrix 𝑉∈ℝ^𝑚×𝑚, 𝑉>0, scalar 0<𝑟(𝑡)<𝑟, vector function 𝜈∶[0,𝑟]→ℝ^𝑚 such that the integrations concerned are well defined, then 𝑟(𝑡)^0𝑟(𝑡)𝜈^𝑇(𝑠)𝑉𝜈(𝑠)𝑑𝑠≥^0𝑟(𝑡)𝜈(𝑠)𝑑𝑠^𝑇𝑉^0𝑟(𝑡)𝜈(𝑠)𝑑𝑠.(2.12)

Lemma 2.5 (Halanay inequality [35]). Let 𝑉∶[𝜇−𝜏,∞)→[0,∞) be a continuous function such that 𝑑𝑉(𝑡)𝑑𝑡≤−𝑎𝑉(𝑡)+𝑏max𝑉^𝑡(2.13) is satisfied for 𝑡≥𝜇. If 𝑎>𝑏>0, then 𝑉(𝑡)≤max𝑉_𝜇exp{−𝑟(𝑡−𝜇)},𝑡≥𝜇,(2.14) where max𝑉_𝑡=sup_{𝑡−𝜏≤𝜃≤𝑡}𝑉(𝜃), and 𝑟>0 is the smallest real root of the following equation: −𝑟=−𝑎+𝑏exp{𝑟𝜏}.(2.15)

Lemma 2.6 (see [19]). Let 𝑉∶[𝜇−𝜏,∞)→[0,∞) be a continuous function, such that 𝑑𝑉(𝑡)𝑑𝑡≤𝑎𝑉(𝑡)+𝑏max𝑉^𝑡(2.16) is satisfied for 𝑡≥𝜇. If 𝑎>0, 𝑏>0, then 𝑉(𝑡)≤max𝑉_𝑡≤max𝑉_𝜇exp{(𝑎+𝑏)(𝑡−𝜇)},𝑡≥𝜇,(2.17) where max𝑉_𝑡=sup_{𝑡−𝜏≤𝜃≤𝑡}𝑉(𝜃).

3. Criteria for Synchronization

Theorem 3.1. Suppose that assumption (𝐻) holds. The controlled coupled network (2.5) globally synchronizes to (2.4) if there are positive definite matrix 𝑃, positive constants 𝛼,𝛽,𝛾,𝑎₁,𝑎₂,𝑏₁,𝑏₂ such that the following conditions hold: (a)𝐼_𝑁⊗(𝑄+𝑎₁𝑃)+𝐺⊗Γ−𝐾⊗𝐼_𝑛≤0, (b)𝐼_𝑁⊗(𝑄−𝑎₂𝑃)+𝐺⊗Γ≤0, (c)𝛽⁻¹𝐿^2𝑔𝐼_𝑛−𝑏₁𝑃≤0, (d)𝛾⁻¹𝐿^2ℎ𝐼_𝑛−𝑏₂𝑃≤0, (e)𝑎₁>𝑏=𝑏₁+𝜏²𝑏₂(f)𝜌=𝑟(𝛿−𝜏)−(𝑎₂+𝑏)(𝜔−𝛿)>0, where 𝑄=−𝑃𝐷+(𝛼/4)𝑃𝐴𝐴^𝑇𝑃+𝛼⁻¹𝐿^2𝑓𝐼_𝑛+(𝛽/4)𝑃𝐵𝐵^𝑇𝑃+(𝛾/4)𝑃𝐶𝐶^𝑇𝑃+𝑎₁𝑃, 𝐾=diag(𝑘₁,𝑘₂,…,𝑘_𝑁) and 𝑟 is the positive solution of −𝑟=−𝑎₁+𝑏𝑒^𝑟𝜏.

Proof. Consider the following Lyapunov function: 𝑉(𝑡)=12^𝑁_𝑖=1𝑒_𝑖(𝑡)^𝑇𝑃𝑒_𝑖(𝑡)=12𝑒^𝑇𝐼_𝑁⊗𝑃𝑒(𝑡),(3.1) where 𝑒(𝑡)=[𝑒^𝑇1(𝑡),𝑒^𝑇2(𝑡),…,𝑒^𝑇𝑁(𝑡)]^𝑇. Calculate the derivative 𝑉(𝑡) with respect to time 𝑡 along the trajectory of error system (2.8), and estimate it.
For 𝑙𝜔≤𝑡≤𝑙𝜔+𝛿, using Lemma 2.3 and assumption, we have the following: 𝑉̇(𝑡)=^𝑁_𝑖=1=𝑒^𝑇𝑖(𝑡)𝑃𝑒_𝑖̇(𝑡)_𝑁_𝑖=1𝑒^𝑇𝑖(𝑡)𝑃−+𝐶𝐷𝑒_𝑖(𝑡)_{𝑡𝑡−𝜏}+𝐴𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))+𝐵𝑔𝑥_𝑖(𝑡−𝜏)−𝑔(𝑠(𝑡−𝜏))ℎ𝑥_𝑖(𝑣)𝑑𝑣−^{𝑡𝑡−𝜏}ℎ(𝑠(𝑣))𝑑𝑣+^𝑁_𝑗=1≤𝐺_𝑖𝑗Γ𝑒_𝑗(𝑡)−𝑘_𝑖𝑒_𝑖(𝑡)_𝑁_𝑖=1−𝑒^𝑇𝑖(𝑡)𝑃𝐷𝑒_𝑖(𝑡)+𝛼4𝑒^𝑇𝑖(𝑡)𝑃𝐴𝐴^𝑇𝑃𝑒_𝑖(𝑡)+𝛼⁻¹‖‖𝑓𝑥_𝑖(𝑡)−𝑓(𝑠(𝑡))‖‖²+𝛽4𝑒^𝑇𝑖(𝑡)𝑃𝐵𝐵^𝑇𝑃𝑒_𝑖(𝑡)+𝛽⁻¹‖‖𝑔𝑥_𝑖(𝑡−𝜏)−𝑔(𝑠(𝑡−𝜏))‖‖²+𝛾4𝑒^𝑇𝑖(𝑡)𝑃𝐶𝐶^𝑇𝑃𝑒_𝑖(𝑡)+𝛾⁻¹‖‖‖^{𝑡𝑡−𝜏}ℎ𝑥_𝑖(𝑣)𝑑𝑣−^{𝑡𝑡−𝜏}ℎ(𝑠(𝑣))𝑑𝑣‖‖‖²+_𝑁_𝑁𝑖=1_𝑗=1𝐺_𝑖𝑗𝑒^𝑇𝑖(𝑡)Γ𝑒_𝑗(𝑡)−^𝑁_𝑖=1≤𝑒_𝑇𝑘_𝑖𝑒^𝑇𝑖(𝑡)𝑒_𝑖(𝑡)(𝑡)𝐼_𝑁⊗−𝑃𝐷+𝛼4𝑃𝐴𝐴^𝑇𝑃+𝛼⁻¹𝐿^2𝑓𝐼_𝑛+𝛽4𝑃𝐵𝐵^𝑇𝑃+𝛾4𝑃𝐶𝐶^𝑇+𝑒_𝑇𝑃+𝑎₁𝑃𝑒(𝑡)−𝑎₁𝑒^𝑇(𝑡)𝐼_𝑁⊗𝑃𝑒(𝑡)+(𝑡−𝜏)𝐼_𝑁⊗𝛽⁻¹𝐿^2𝑔𝐼_𝑛−𝑏₁𝑃𝑒(𝑡−𝜏)+𝑏₁𝑒^𝑇(𝑡−𝜏)𝐼_𝑁⊗𝑃𝑒(𝑡−𝜏)_{𝑡𝑡−𝜏}+𝑏₂𝑒(𝑣)𝑑𝑣^𝑇𝐼_𝑁⊗𝛾⁻¹𝐿^2ℎ𝐼_𝑛−𝑏₂𝑃^{𝑡𝑡−𝜏}𝑒(𝑣)𝑑𝑣^{𝑡𝑡−𝜏}𝑒(𝑣)𝑑𝑣^𝑇𝐼_𝑁⊗𝑃^{𝑡𝑡−𝜏}𝑒(𝑣)𝑑𝑣+𝑒^𝑇(𝑡)𝐺⊗Γ−𝐾⊗𝐼_𝑛𝑒(𝑡).(3.2) From Jensen's inequality in Lemma 2.4, we have 𝑏₂^{𝑡𝑡−𝜏}𝑒(𝑣)𝑑𝑣^𝑇𝐼_𝑁⊗𝑃^{𝑡𝑡−𝜏}𝑒(𝑣)𝑑𝑣≤𝜏𝑏₂^{𝑡𝑡−𝜏}𝑒^𝑇(𝑣)𝐼_𝑁⊗𝑃𝑒(𝑣)𝑑𝑣.(3.3) By condition (a), (c), (d), and (3.3), one has 𝑉̇(𝑡)≤−𝑎₁𝑉(𝑡)+𝑏max𝑉_𝑡(3.4) For 𝑙𝜔+𝛿≤𝑡≤(𝑙+1)𝜔, from conditions (b), (c), and (d), one has 𝑉̇(𝑡)<𝑎₂𝑉(𝑡)+𝑏𝑉_𝑡.(3.5) Next, we will prove the error ‖𝑒(𝑡)‖→0.
From Lemma 2.5 and (3.2), one has 𝑉(𝑡)≤‖𝑉(0)‖_𝜏𝑒^−𝑟𝑡,for0≤𝑡≤𝛿,(3.6) where 𝑟 is the unique positive solution of −𝑟=−𝑎₁+𝑏𝑒^𝑟𝜏.
From Lemma 2.6, one obtain the following: 𝑉(𝑡)≤‖𝑉(𝛿)‖_𝜏𝑒^{(𝑎2+𝑏)(𝑡−𝛿)}=max_{𝛿−𝜏≤𝑡≤𝛿}||𝑉(𝑡)||𝑒^{(𝑎2+𝑏)(𝑡−𝛿)}≤‖𝑉(0)‖_𝜏𝑒^{−𝑟(𝛿−𝜏)}𝑒^{(𝑎2+𝑏)(𝑡−𝛿)},(3.7) for 𝛿≤𝑡≤𝜔.
Suppose that 𝜔−𝜏>𝛿, then ‖𝑉(𝜔)‖_𝜏=≤mmaaxx_{𝜔−𝜏≤𝑡≤𝜔𝜔−𝜏≤𝑡≤𝜔}||𝑉(𝑡)||=‖𝑉(0)‖_𝜏‖𝑉(0)‖_𝜏𝑒^{−𝑟(𝛿−𝜏)}𝑒^{(𝑎2+𝑏)(𝑡−𝛿)}𝑒^{−𝑟(𝛿−𝜏)}𝑒^{(𝑎2+𝑏)(𝜔−𝛿)}=‖𝑉(0)‖_𝜏𝑒^−𝜌.(3.8) Using mathematical induction, we can prove, for any positive integer 𝑙, ‖𝑉(𝑙𝜔)‖_𝜏≤‖𝑉(0)‖_𝜏𝑒^−𝑙𝜌.(3.9) Assume (3.9) holds when 𝑘≤𝑙. Now, we prove (3.9) is true when 𝑘=𝑙+1.
First, we have ‖𝑉(𝑙𝜔)‖_𝜏≤‖𝑉(0)‖_𝜏𝑒^−𝑙𝜌.(3.10) When 𝑡∈[𝑙𝜔,𝑙𝜔+𝛿], 𝑉(𝑡)≤‖𝑉(𝑙𝜔)‖_𝜏𝑒^{−𝑟(𝑡−𝑙𝜔)}≤‖𝑉(0)‖_𝜏𝑒^−𝑙𝜌𝑒^{−𝑟(𝑡−𝑙𝜔)}.(3.11) Thus, for 𝑡∈[𝑙𝜔+𝛿,(𝑙+1)𝜔], we have 𝑉(𝑡)≤‖𝑉(𝑙𝜔+𝛿)‖_𝜏𝑒^{(𝑎2+𝑏)(𝑡−𝑙𝜔−𝛿)}=max_{𝑙𝜔+𝛿−𝜏≤𝑡≤𝑙𝜔+𝛿}||𝑉(𝑡)||𝑒^{(𝑎2+𝑏)(𝑡−𝑙𝜔−𝛿)}≤max_{𝑙𝜔+𝛿−𝜏≤𝑡≤𝑙𝜔+𝛿}‖𝑉(0)‖_𝜏𝑒^−𝑙𝜌𝑒^{−𝑟(𝑡−𝑙𝜔)}×𝑒_{(𝑎2+𝑏)(𝑡−𝑙𝜔−𝛿)}‖𝑉((𝑙+1)𝜔)‖_𝜏≤‖𝑉(0)‖_𝜏𝑒^−𝑙𝜌𝑒^{−𝑟(𝛿−𝜏)}𝑒^{(𝑎2+𝑏)(𝑡−𝑙𝜔−𝛿)},≤=mmaaxx_{(𝑙+1)𝜔−𝜏≤𝑡≤(𝑙+1)𝜔(𝑙+1)𝜔−𝜏≤𝑡≤(𝑙+1)𝜔}||𝑉(𝑡)||=‖𝑉(0)‖_𝜏‖𝑉(0)‖_𝜏𝑒^−𝑙𝜌𝑒^{−𝑟(𝛿−𝜏)}𝑒^{(𝑎2+𝑏)(𝑡−𝑙𝜔−𝛿)}𝑒^−𝑙𝜌𝑒^{−𝑟(𝛿−𝜏)}𝑒^{(𝑎2+𝑏)(𝜔−𝛿)}=‖𝑉(0)‖_𝜏𝑒^{−(𝑙+1)𝜌}.(3.12) Thus, (3.9) holds for all positive integers 𝑘.
For any 𝑡>0, there is 𝑛₀≥0, such that 𝑛₀𝜔≤𝑡≤(𝑛₀+1)𝜔; 𝑉(𝑡)≤‖‖𝑉𝑛₀𝜔‖‖_𝜏𝑒^{(𝑎2+𝑏)(𝑡−𝑛0𝜔)}≤‖𝑉(0)‖_𝜏𝑒^−𝑛0𝜌𝑒^{(𝑎2+𝑏)𝜔}≤‖𝑉(0)‖_𝜏𝑒^{(𝑎2+𝑏)𝜔}𝑒^𝜌exp−𝜌𝜔𝑡.(3.13) Let 𝑀=‖𝑉(0)‖_𝜏𝑒^{(𝑎2+𝑏)𝜔}𝑒^𝜌, one has the following inequality: 𝜆_𝑚(𝑃)‖𝑒(𝑡)‖²≤𝑉(𝑡)≤𝑀exp−𝜌𝜔𝑡,for𝑡≥0.(3.14) Obviously, ‖𝑒(𝑡)‖≤𝑀𝜆_𝑚(𝑃)exp−2𝜌𝜔𝑡,(3.15) which means the coupled networks (2.5) achieve synchronization. The proof is completed.

Corollary 3.2. For given control period 𝜔 and control duration 𝛿, coupled networks (2.5) achieve synchronization, if the control gain 𝐾=𝑘𝐼_𝑁 satisfies 𝑘>Φ𝑟^∗,(3.16) where Φ𝑟(_∗𝑟)=𝑀+𝑟+2𝐿^2𝑔+2𝜏²𝐿^2ℎ𝑒^𝑟𝜏,=𝜔−𝛿𝛿−𝜏𝑀+2𝐿^2𝑔+2𝜏²𝐿^2ℎ.(3.17)

Proof. In Theorem 3.1, let 𝑃=𝐼, 𝛽=𝛾=1, 𝑏₁=2𝐿^2𝑔, 𝑏₂=2𝐿^2ℎ, obviously, (c) and (d) in Theorem 3.1 hold.
Furthermore, let 𝑀=𝜆_𝑀(𝑄)+𝜆_𝑀(𝐺)×𝜆_𝑀(Γ), where 𝜆_𝑀(⋅) is the maximum of eigenvalue, 𝑎₁=𝑘−𝑀>0 and 𝑎₂=𝑀, (a) and (b) in Theorem 3.1 hold. From the above parameters and (f) in Theorem 3.1 hold if 𝑟>𝑟^∗ and 𝑟 is the positive solution of −𝑟=−𝑎₁+(2𝐿^2𝑔+2𝜏²𝐿^2ℎ)𝑒^𝑟𝜏, that is, 𝑘=𝑟+𝑀+𝐿_𝑓+(2𝐿^2𝑔+2𝜏²𝐿^2ℎ)𝑒^𝑟𝜏=Φ(𝑟). Obviously, Φ(𝑟) is increasing function.
Therefore, (a)–(f) hold if 𝑘>maxΦ𝑟^∗,𝑀=Φ𝑟^∗.(3.18)

Remark 3.3. Corollary 3.2 shows us how to determine the control gain in a simple way provided that the control period 𝜔 and control duration 𝛿 are given.

4. Numerical Example

Consider the following coupled networks: 𝑥̇_𝑖(𝑡)=−+𝐷𝐶𝑥∫_𝑖(_{𝑡𝑡−𝜏}𝑡)+𝐴𝑓𝑥_𝑖(𝑡)+𝐵𝑔𝑥_𝑖(𝑡−𝜏)𝑥ℎ𝑥_𝑖(𝑣)𝑑𝑣+∑^𝑁𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗(𝑡),_𝑖(𝑡)=𝜙(𝑡),−𝜏≤𝑡≤0,(4.1) where 𝑥_𝑖(𝑡)=[𝑥_𝑖1(𝑡),𝑥_𝑖2(𝑡)]^𝑇, 𝑖=1,2,3 are the state variable of the 𝑖th neural network. Choose 𝜏=1, 𝑓(𝑥_𝑖(𝑡))=𝑔(𝑥_𝑖(𝑡))=ℎ(𝑥_𝑖(𝑡))=(3/5)[tanh(𝑥_𝑖1),tanh(𝑥_𝑖2)]^𝑇, and 𝐷𝐵==⎡⎢⎢⎣⎡⎢⎢⎣10−−1001..52⎤⎥⎥⎦,−−02.1𝐴⎤⎥⎥⎦,=⎡⎢⎢⎣−2𝐶5.0.=0⎡⎢⎢⎣−4−−0.501.1..26⎤⎥⎥⎦,−−03..12⎤⎥⎥⎦,𝐺=⎡⎢⎢⎢⎢⎣−114−222−213⎤⎥⎥⎥⎥⎦,(4.2) and input vectors 𝐼=⁰⁰, and 𝑘_𝑖(𝑡) is the intermittent linear state feedback control gain defined as the following: 𝑘_𝑖(𝑡)=𝑘^𝑖0,,𝑘𝑘𝜔𝜔+≤𝛿𝑡≤<𝑘𝑡𝜔≤+(𝑘𝛿,+1)𝜔,(4.3) where the control gain 𝑘₁=𝑘₂=𝑘₃=0.1, the control period 𝜔=3, and the control width 𝛿=1.3. The above suitable 𝛿, 𝜔 and 𝐾 such that (4.1) synchronize. The synchronize errors are given in Figures 1 and 2.

Figure 1 

Error state 𝑒𝑖1(t).

Figure 2 

Error state 𝑒𝑖2(t).

5. Conclusion

In this paper, synchronization of coupled networks with mixed time delay has been investigated via intermittent control. Some criteria for ensuring coupled networks synchronization have been derived, and some analytical techniques have been proposed to obtain appropriate control period 𝜔, control width 𝛿, and control gain for achieving network synchronization. Finally, the simulation confirmed the effectiveness of the proposed intermittent controller.

Acknowledgments

This work was jointly supported by the National Natural Science Foundation of China under Grant 61004043 and the Specialized Research Fund for the Doctoral Program of Higher Education under Grant 2009092120066.