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 [615]. 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, [815].

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 [1926]. 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 [2833]. 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.

𝑛 is the 𝑛-dimensional Euclidean space; 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 ̇𝑥𝑖(𝑡)=𝐷𝑥𝑖(𝑥𝑡)+𝐴𝑓𝑖(𝑥𝑡)+𝐵𝑔𝑖(𝑡𝜏)+𝐶𝑡𝑡𝜏𝑥𝑖(𝑣)𝑑𝑣+𝐼(𝑡)+𝑁𝑗=1,𝑗𝑖𝐺𝑖𝑗Γ𝑥𝑗(𝑡)𝑥𝑖,(𝑡)(2.1) where 𝑥𝑖(𝑡)=(𝑥𝑖1(𝑡),𝑥𝑖2(𝑡),,𝑥𝑖𝑛(𝑡))𝑇𝑛 is the state vector of the 𝑖th node; 𝐷=diag(𝑑1,𝑑2,,𝑑𝑛)>0 denotes the rate with which the cell 𝑖 resets its potential to the resting state when isolated from other cells and inputs; 𝐴𝑛×𝑛, 𝐵𝑛×𝑛, and 𝐶𝑛×𝑛 represent the connection weight matrix, the discretely delayed connection weight matrix, and the distributively delayed connection weights, respectively; 𝑓(𝑥𝑖())=[𝑓1(𝑥𝑖1()),𝑓2(𝑥𝑖2()),,𝑓𝑛(𝑥𝑖𝑛())]𝑇𝑛, 𝑔(𝑥𝑖())=[𝑔1(𝑥𝑖1()),𝑔2(𝑥𝑖2()),,𝑔𝑛(𝑥𝑖𝑛())]𝑇𝑛 and (𝑥𝑖())=[1(𝑥𝑖1()),2(𝑥𝑖2()),,𝑛(𝑥𝑖𝑛())]𝑇𝑛 are activation functions; 𝐼(𝑡) is the input vector of each node; Γ𝑛×𝑛 is the inner coupling matrix; 𝐺=(𝐺𝑖𝑗)𝑁×𝑁 is the coupling configuration matrix representing the topological structure of the network, where 𝐺𝑖𝑗 is defined as follows: if there exists a connection between node 𝑖 and node 𝑗, 𝐺𝑖𝑗>0, otherwise 𝐺𝑖𝑗=0(𝑗𝑖), and the diagonal elements of matrix 𝐺 are defined by 𝐺𝑖𝑖=𝑁𝑗=1,𝑗𝑖𝐺𝑖𝑗,(2.2) which ensures the diffusion that 𝑁𝑗=1𝐺𝑖𝑗=0. Equivalently, network (2.1) can be rewritten in a form as follows: ̇𝑥𝑖(𝑡)=𝐷𝑥𝑖(𝑥𝑡)+𝐴𝑓𝑖(𝑥𝑡)+𝐵𝑔𝑖(𝑡𝜏)+𝐶𝑡𝑡𝜏𝑥𝑖(𝑣)𝑑𝑣+𝐼(𝑡)+𝑁𝑗=1𝐺𝑖𝑗Γ𝑥𝑗(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(𝑡𝜏))+𝐶𝑡𝑡𝜏(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 ̇𝑥𝑖(𝑡)=𝐷𝑥𝑖(𝑥𝑡)+𝐴𝑓𝑖(𝑥𝑡)+𝐵𝑔𝑖(𝑡𝜏)+𝐶𝑡𝑡𝜏𝑥𝑖(𝑣)𝑑𝑣+𝐼(𝑡)+𝑁𝑗=1𝐺𝑖𝑗Γ𝑥𝑗(𝑡)+𝑢𝑖,𝑖=1,2,,𝑁,(2.5) where 𝑢𝑖=𝑘𝑖𝑥(𝑡)𝑖(𝑡)𝑠(𝑡),(2.6)𝑘(𝑡) is the intermittent linear state feedback control gain defined as follows: 𝑘𝑖𝑘(𝑡)=𝑖,𝑛𝜔𝑡𝑛𝜔+𝛿,0,𝑛𝜔+𝛿<𝑡(𝑛+1)𝜔,(2.7) where 𝑘𝑖𝑅 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 𝑘𝑖 such that network (2.5) synchronize with respect to the isolated node 𝑠(𝑡). Denote 𝑒𝑖(𝑡)=𝑥𝑖(𝑡)𝑠(𝑡), then the following error dynamical system is obtained: ̇𝑒𝑖(𝑡)=𝐷𝑒𝑖𝑓𝑥(𝑡)+𝐴𝑖𝑔𝑥(𝑡)𝑓(𝑠(𝑡))+𝐵𝑖(𝑡𝜏)𝑔(𝑠(𝑡𝜏))+𝐶𝑡𝑡𝜏𝑥𝑖(𝑣)(𝑠(𝑣))𝑑𝑣+𝑁𝑗=1𝐺𝑖𝑗Γ𝑒𝑗(𝑡)𝑘𝑖𝑒𝑖(𝑡),𝑛𝜔𝑡𝑛𝜔+𝛿,̇𝑒𝑖(𝑡)=𝐷𝑒𝑖𝑓𝑥(𝑡)+𝐴𝑖𝑔𝑥(𝑡)𝑓(𝑠(𝑡))+𝐵𝑖(𝑡𝜏)𝑔(𝑠(𝑡𝜏))+𝐶𝑡𝑡𝜏𝑥𝑖(𝑣)𝑠(𝑣)𝑑𝑣+𝑁𝑗=1𝐺𝑖𝑗Γ𝑒𝑗(𝑡),𝑛𝜔+𝛿<𝑡(𝑛+1)𝜔.(2.8)(H) We assume that 𝑓, 𝑔, and are Lipschitz continuous functions; there exist positive constants 𝐿𝑓, 𝐿𝑔 and 𝐿 such that, for all 𝑥,𝑦𝑚, 𝑓(𝑥)𝑓(𝑦)𝐿𝑓𝑥𝑦,𝑔(𝑥)𝑔(𝑦)𝐿𝑔𝑥𝑦,(𝑥)(𝑦)𝐿𝑥𝑦.(2.9)

Definition 2.1. For any positive integers 𝑝, 𝑞, 𝑟, 𝑠, we define the Kronecker product of two matrices 𝐴𝑝×𝑞, 𝐵𝑟×𝑠 as follows: 𝑎𝐴𝐵=11𝐵𝑎1𝑞𝐵𝑎𝑝1𝐵𝑎𝑝𝑞𝐵𝑝𝑟×𝑞𝑠.(2.10)

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

Lemma 2.3. For any vectors 𝑥, 𝑦𝑚, and positive-definite matrix 𝑄𝑚×𝑚, the following matrix inequality holds: 2𝑥𝑇𝑦𝑥𝑇𝑄𝑥+𝑦𝑇𝑄1𝑦.(2.11)

Lemma 2.4 (Jensen's inequality [34]). For any constant matrix 𝑉𝑚×𝑚, 𝑉>0, scalar 0<𝑟(𝑡)<𝑟, vector function 𝜈[0,𝑟]𝑚 such that the integrations concerned are well defined, then 𝑟(𝑡)0𝑟(𝑡)𝜈𝑇(𝑠)𝑉𝜈(𝑠)𝑑𝑠0𝑟(𝑡)𝜈(𝑠)𝑑𝑠𝑇𝑉0𝑟(𝑡).𝜈(𝑠)𝑑𝑠(2.12)

Lemma 2.5 (Halanay inequality [35]). Let 𝑉[𝜇𝜏,)[0,) be a continuous function such that 𝑑𝑉(𝑡)𝑑𝑡𝑎𝑉(𝑡)+𝑏max𝑉𝑡(2.13) is satisfied for 𝑡𝜇. If 𝑎>𝑏>0, then 𝑉(𝑡)max𝑉𝜇exp{𝑟(𝑡𝜇)},𝑡𝜇,(2.14) where max𝑉𝑡=sup𝑡𝜏𝜃𝑡𝑉(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[𝜇𝜏,)[0,) be a continuous function, such that 𝑑𝑉(𝑡)𝑑𝑡𝑎𝑉(𝑡)+𝑏max𝑉𝑡(2.16) is satisfied for 𝑡𝜇. If 𝑎>0, 𝑏>0, then 𝑉(𝑡)max𝑉𝑡max𝑉𝜇exp{(𝑎+𝑏)(𝑡𝜇)},𝑡𝜇,(2.17) where max𝑉𝑡=sup𝑡𝜏𝜃𝑡𝑉(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)𝐼𝑁(𝑄+𝑎1𝑃)+𝐺Γ𝐾𝐼𝑛0, (b)𝐼𝑁(𝑄𝑎2𝑃)+𝐺Γ0, (c)𝛽1𝐿2𝑔𝐼𝑛𝑏1𝑃0, (d)𝛾1𝐿2𝐼𝑛𝑏2𝑃0, (e)𝑎1>𝑏=𝑏1+𝜏2𝑏2(f)𝜌=𝑟(𝛿𝜏)(𝑎2+𝑏)(𝜔𝛿)>0, where 𝑄=𝑃𝐷+(𝛼/4)𝑃𝐴𝐴𝑇𝑃+𝛼1𝐿2𝑓𝐼𝑛+(𝛽/4)𝑃𝐵𝐵𝑇𝑃+(𝛾/4)𝑃𝐶𝐶𝑇𝑃+𝑎1𝑃, 𝐾=diag(𝑘1,𝑘2,,𝑘𝑁) and 𝑟 is the positive solution of 𝑟=𝑎1+𝑏𝑒𝑟𝜏.

Proof. Consider the following Lyapunov function: 1𝑉(𝑡)=2𝑁𝑖=1𝑒𝑖(𝑡)𝑇𝑃𝑒𝑖1(𝑡)=2𝑒𝑇𝐼𝑁𝑃𝑒(𝑡),(3.1) where 𝑒(𝑡)=[𝑒𝑇1(𝑡),𝑒𝑇2(𝑡),,𝑒𝑇𝑁(𝑡)]𝑇. Calculate the derivative 𝑉(𝑡) with respect to time 𝑡 along the trajectory of error system (2.8), and estimate it.
For 𝑙𝜔𝑡𝑙𝜔+𝛿, using Lemma 2.3 and assumption, we have the following: ̇𝑉(𝑡)=𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑃̇𝑒𝑖=(𝑡)𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑃𝐷𝑒𝑖𝑓𝑥(𝑡)+𝐴𝑖𝑔𝑥(𝑡)𝑓(𝑠(𝑡))+𝐵𝑖(𝑡𝜏)𝑔(𝑠(𝑡𝜏))+𝐶𝑡𝑡𝜏𝑥𝑖(𝑣)𝑑𝑣𝑡𝑡𝜏+(𝑠(𝑣))𝑑𝑣𝑁𝑗=1𝐺𝑖𝑗Γ𝑒𝑗(𝑡)𝑘𝑖𝑒𝑖(𝑡)𝑁𝑖=1𝑒𝑇𝑖(𝑡)𝑃𝐷𝑒𝑖𝛼(𝑡)+4𝑒𝑇𝑖(𝑡)𝑃𝐴𝐴𝑇𝑃𝑒𝑖(𝑡)+𝛼1𝑓𝑥𝑖(𝑡)𝑓(𝑠(𝑡))2+𝛽4𝑒𝑇𝑖(𝑡)𝑃𝐵𝐵𝑇𝑃𝑒𝑖(𝑡)+𝛽1𝑔𝑥𝑖(𝑡𝜏)𝑔(𝑠(𝑡𝜏))2+𝛾4𝑒𝑇𝑖(𝑡)𝑃𝐶𝐶𝑇𝑃𝑒𝑖(𝑡)+𝛾1𝑡𝑡𝜏𝑥𝑖(𝑣)𝑑𝑣𝑡𝑡𝜏(𝑠(𝑣))𝑑𝑣2+𝑁𝑁𝑖=1𝑗=1𝐺𝑖𝑗𝑒𝑇𝑖(𝑡)Γ𝑒𝑗(𝑡)𝑁𝑖=1𝑘𝑖𝑒𝑇𝑖(𝑡)𝑒𝑖(𝑡)𝑒𝑇(𝑡)𝐼𝑁𝛼𝑃𝐷+4𝑃𝐴𝐴𝑇𝑃+𝛼1𝐿2𝑓𝐼𝑛+𝛽4𝑃𝐵𝐵𝑇𝛾𝑃+4𝑃𝐶𝐶𝑇𝑃+𝑎1𝑃𝑒(𝑡)𝑎1𝑒𝑇𝐼(𝑡)𝑁𝑃𝑒(𝑡)+𝑒𝑇𝐼(𝑡𝜏)𝑁𝛽1𝐿2𝑔𝐼𝑛𝑏1𝑃𝑒(𝑡𝜏)+𝑏1𝑒𝑇𝐼(𝑡𝜏)𝑁+𝑃𝑒(𝑡𝜏)𝑡𝑡𝜏𝑒(𝑣)𝑑𝑣𝑇𝐼𝑁𝛾1𝐿2𝐼𝑛𝑏2𝑃𝑡𝑡𝜏𝑒(𝑣)𝑑𝑣+𝑏2𝑡𝑡𝜏𝑒(𝑣)𝑑𝑣𝑇𝐼𝑁𝑃𝑡𝑡𝜏𝑒(𝑣)𝑑𝑣+𝑒𝑇(𝑡)𝐺Γ𝐾𝐼𝑛𝑒(𝑡).(3.2) From Jensen's inequality in Lemma 2.4, we have 𝑏2𝑡𝑡𝜏𝑒(𝑣)𝑑𝑣𝑇𝐼𝑁𝑃𝑡𝑡𝜏𝑒(𝑣)𝑑𝑣𝜏𝑏2𝑡𝑡𝜏𝑒𝑇𝐼(𝑣)𝑁𝑃𝑒(𝑣)𝑑𝑣.(3.3) By condition (a), (c), (d), and (3.3), one has ̇𝑉(𝑡)𝑎1𝑉(𝑡)+𝑏max𝑉𝑡(3.4) For 𝑙𝜔+𝛿𝑡(𝑙+1)𝜔, from conditions (b), (c), and (d), one has ̇𝑉(𝑡)<𝑎2𝑉(𝑡)+𝑏𝑉𝑡.(3.5) Next, we will prove the error 𝑒(𝑡)0.
From Lemma 2.5 and (3.2), one has 𝑉(𝑡)𝑉(0)𝜏𝑒𝑟𝑡,for0𝑡𝛿,(3.6) where 𝑟 is the unique positive solution of 𝑟=𝑎1+𝑏𝑒𝑟𝜏.
From Lemma 2.6, one obtain the following: 𝑉(𝑡)𝑉(𝛿)𝜏𝑒(𝑎2+𝑏)(𝑡𝛿)=max𝛿𝜏𝑡𝛿||||𝑒𝑉(𝑡)(𝑎2+𝑏)(𝑡𝛿)𝑉(0)𝜏𝑒𝑟(𝛿𝜏)𝑒(𝑎2+𝑏)(𝑡𝛿),(3.7) for 𝛿𝑡𝜔.
Suppose that 𝜔𝜏>𝛿, then 𝑉(𝜔)𝜏=max𝜔𝜏𝑡𝜔||||𝑉(𝑡)max𝜔𝜏𝑡𝜔𝑉(0)𝜏𝑒𝑟(𝛿𝜏)𝑒(𝑎2+𝑏)(𝑡𝛿)=𝑉(0)𝜏𝑒𝑟(𝛿𝜏)𝑒(𝑎2+𝑏)(𝜔𝛿)=𝑉(0)𝜏𝑒𝜌.(3.8) Using mathematical induction, we can prove, for any positive integer 𝑙, 𝑉(𝑙𝜔)𝜏𝑉(0)𝜏𝑒𝑙𝜌.(3.9) Assume (3.9) holds when 𝑘𝑙. Now, we prove (3.9) is true when 𝑘=𝑙+1.
First, we have 𝑉(𝑙𝜔)𝜏𝑉(0)𝜏𝑒𝑙𝜌.(3.10) When 𝑡[𝑙𝜔,𝑙𝜔+𝛿], 𝑉(𝑡)𝑉(𝑙𝜔)𝜏𝑒𝑟(𝑡𝑙𝜔)𝑉(0)𝜏𝑒𝑙𝜌𝑒𝑟(𝑡𝑙𝜔).(3.11) Thus, for 𝑡[𝑙𝜔+𝛿,(𝑙+1)𝜔], we have 𝑉(𝑡)𝑉(𝑙𝜔+𝛿)𝜏𝑒(𝑎2+𝑏)(𝑡𝑙𝜔𝛿)=max𝑙𝜔+𝛿𝜏𝑡𝑙𝜔+𝛿||||𝑒𝑉(𝑡)(𝑎2+𝑏)(𝑡𝑙𝜔𝛿)max𝑙𝜔+𝛿𝜏𝑡𝑙𝜔+𝛿𝑉(0)𝜏𝑒𝑙𝜌𝑒𝑟(𝑡𝑙𝜔)×𝑒(𝑎2+𝑏)(𝑡𝑙𝜔𝛿)𝑉(0)𝜏𝑒𝑙𝜌𝑒𝑟(𝛿𝜏)𝑒(𝑎2+𝑏)(𝑡𝑙𝜔𝛿),𝑉((𝑙+1)𝜔)𝜏=max(𝑙+1)𝜔𝜏𝑡(𝑙+1)𝜔||||𝑉(𝑡)max(𝑙+1)𝜔𝜏𝑡(𝑙+1)𝜔𝑉(0)𝜏𝑒𝑙𝜌𝑒𝑟(𝛿𝜏)𝑒(𝑎2+𝑏)(𝑡𝑙𝜔𝛿)=𝑉(0)𝜏𝑒𝑙𝜌𝑒𝑟(𝛿𝜏)𝑒(𝑎2+𝑏)(𝜔𝛿)=𝑉(0)𝜏𝑒(𝑙+1)𝜌.(3.12) Thus, (3.9) holds for all positive integers 𝑘.
For any 𝑡>0, there is 𝑛00, such that 𝑛0𝜔𝑡(𝑛0+1)𝜔; 𝑉𝑛𝑉(𝑡)0𝜔𝜏𝑒(𝑎2+𝑏)(𝑡𝑛0𝜔)𝑉(0)𝜏𝑒𝑛0𝜌𝑒(𝑎2+𝑏)𝜔𝑉(0)𝜏𝑒(𝑎2+𝑏)𝜔𝑒𝜌𝜌exp𝜔𝑡.(3.13) Let 𝑀=𝑉(0)𝜏𝑒(𝑎2+𝑏)𝜔𝑒𝜌, one has the following inequality: 𝜆𝑚(𝑃)𝑒(𝑡)2𝜌𝑉(𝑡)𝑀exp𝜔𝑡,for𝑡0.(3.14) Obviously, 𝑒(𝑡)𝑀𝜆𝑚𝜌(𝑃)exp𝑡,2𝜔(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=𝑘𝐼𝑁 satisfies 𝑟𝑘>Φ,(3.16) where Φ(𝑟)=𝑀+𝑟+2𝐿2𝑔+2𝜏2𝐿2𝑒𝑟𝜏,𝑟=𝜔𝛿𝛿𝜏𝑀+2𝐿2𝑔+2𝜏2𝐿2.(3.17)

Proof. In Theorem 3.1, let 𝑃=𝐼, 𝛽=𝛾=1, 𝑏1=2𝐿2𝑔, 𝑏2=2𝐿2, obviously, (c) and (d) in Theorem 3.1 hold.
Furthermore, let 𝑀=𝜆𝑀(𝑄)+𝜆𝑀(𝐺)×𝜆𝑀(Γ), where 𝜆𝑀() is the maximum of eigenvalue, 𝑎1=𝑘𝑀>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+(2𝐿2𝑔+2𝜏2𝐿2)𝑒𝑟𝜏, that is, 𝑘=𝑟+𝑀+𝐿𝑓+(2𝐿2𝑔+2𝜏2𝐿2)𝑒𝑟𝜏=Φ(𝑟). Obviously, Φ(𝑟) is increasing function.
Therefore, (a)–(f) hold if Φ𝑟𝑘>max𝑟,𝑀=Φ.(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: ̇𝑥𝑖(𝑡)=𝐷𝑥𝑖𝑥(𝑡)+𝐴𝑓𝑖𝑥(𝑡)+𝐵𝑔𝑖(𝑡𝜏)+𝐶𝑡𝑡𝜏𝑥𝑖(𝑣)𝑑𝑣+𝑁𝑗=1𝐺𝑖𝑗Γ𝑥𝑗𝑥(𝑡),𝑖(𝑡)=𝜙(𝑡),𝜏𝑡0,(4.1) where 𝑥𝑖(𝑡)=[𝑥𝑖1(𝑡),𝑥𝑖2(𝑡)]𝑇, 𝑖=1,2,3 are the state variable of the 𝑖th neural network. Choose 𝜏=1, 𝑓(𝑥𝑖(𝑡))=𝑔(𝑥𝑖(𝑡))=(𝑥𝑖(𝑡))=(3/5)[tanh(𝑥𝑖1),tanh(𝑥𝑖2)]𝑇, and ,,𝐷=1001,𝐴=𝐵=,𝐶=,𝐺=422121123(4.2) and input vectors 𝐼=00, and 𝑘𝑖(𝑡) is the intermittent linear state feedback control gain defined as the following: 𝑘𝑖𝑘(𝑡)=𝑖,𝑘𝜔𝑡𝑘𝜔+𝛿,0,𝑘𝜔+𝛿<𝑡(𝑘+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.

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.


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.