Abstract and Applied Analysis

Volume 2015 (2015), Article ID 381078, 12 pages

http://dx.doi.org/10.1155/2015/381078

## Semiglobal Finite-Time Synchronization of Complex Dynamical Networks via Periodically Intermittent Control

College of Science, China Three Gorges University, Yichang, Hubei 443002, China

Received 7 October 2014; Accepted 21 April 2015

Academic Editor: Agacik Zafer

Copyright © 2015 Yihan Fan 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.

#### Abstract

This paper studies the finite-time synchronization problem for a class of complex dynamical networks by means of periodically intermittent control. Based on some analysis techniques and finite-time stability theory, some novel and effective finite-time synchronization criteria are given in terms of a set of linear matrix inequalities. Particularly, the previous synchronization problem by using periodically intermittent control has been extended in this paper. Finally, numerical simulations are presented to verify the theoretical results.

#### 1. Introduction

A complex dynamical network consists of a number of nodes, which are dynamic systems, and links between the nodes. Complex networks exist in various fields of science, engineering, and society and have been deeply investigated in recent years [1]. As the major collective behavior, synchronization is one of the key issues that has been extensively addressed. Several books and reviews [2–6] have appeared which deal with this topic.

Up till now, the synchronization for nonlinear systems especially dynamical networks [7, 8] has been one of the extensive research subjects and many important and fundamental results have been reported on the synchronization and control of nonlinear systems. Meanwhile, lots of control approaches have been developed to synchronize nonlinear systems such as adaptive control [9, 10], feedback control [11, 12], observer control [13, 14], impulsive control [15–19], and intermittent control [20–26]. Among these control approaches and other control methods, the discontinuous control methods, such as impulsive control and intermittent control, have received much interest because they are practical and easily implemented in engineering fields such as transportation and communication. Though the two control methods are discontinuous control, the intermittent control is different from the impulsive control since impulsive control is activated only at some isolated instants, while intermittent control has a nonzero control width. Under some circumstances, using intermittent control is more effective and robust [27]. Hence, some synchronization criteria for nonlinear systems with or without time delays via intermittent control have been presented in recent years; see [21, 22, 28, 29].

Nevertheless, to our best knowledge, the previous results only focus on asymptotical or exponential synchronization of networks through intermittent control; there are few results concerned with finite-time synchronization via intermittent control. In view of this, the purpose of this paper is to study the synchronization of a class of systems by designing reasonable intermittent control. In addition, some previous work views the finite-time synchronization via intermittent control in [30], which will be extended in this paper. Besides, many superiority in finite-time stability has no emphasis in this paper (see [30–32]).

The main contribution of this paper lies in the following aspects. Firstly, a new central lemma is proved by using analysis method. Additionally, an intermittent controller is designed to synchronize the addressed complex networks and some new and useful finite-time synchronization criteria are obtained. Besides, the derivative of the Lyapunov function is smaller than , which enriches the previous results in [30], when controllers are added into the network. Finally, numerical examples are given to show the effectiveness of the theoretical results.

The paper is organized as follows. In Section 2, the problem statement and synchronization scheme to be studied are formulated, and some useful lemmas and preliminaries are presented. In Section 3, some finite-time synchronization criteria for the complex dynamical networks are rigorously derived. In Section 4, the effectiveness of the developed methods is shown by numerical examples. Conclusions are finally drawn in Section 5.

#### 2. Preliminaries

Consider a complex dynamical network consisting of linearly and diffusively coupled identical nodes, with each node being an -dimensional dynamical system. The state equation of the entire network is designed as follows:where is the state vector of the th dynamical node, is a smooth nonlinear vector-value function, and the constant is a coupling strength. is the inner-coupling matrix of the network. Matrix represents the coupling configuration of the network, in which is defined as follows: if there is a connection from the nodes to , then ; otherwise, , and the diagonal elements of matrices are defined as

To achieve the aim of this paper, the following assumptions and some lemmas are necessary.

*Assumption 1. *Assume that there exists a positive definite diagonal matrix and a diagonal matrix , such that satisfies the following inequality: for some , all , and .

Lemma 2 (see [33]). *Assume that a continuous, positive-definite function satisfies the following differential inequality:**where and are all constants. Then, for any given satisfies the following inequality:**with given by*

*Lemma 3. Assume that a continuous, positive-definite function satisfies the following differential inequality:where , , and are three constants. Then, for any given satisfies the following inequality: with given by for .*

*Proof. *Consider the following differential equation: By multiplying , we have Although this differential equation does not satisfy the global Lipschitz condition, the unique solution to this equation can be found as It is direct to prove that is differential for . From the comparison lemma, one obtainswith given in (9) with .

*Remark 4. *Lemma 3 is similar to Lemma 2, but our result can enrich the famous differential inequality [33] to general differential inequality, and give a direction to proof the following Lemma 5.

*Lemma 5. Suppose that function is continuous and nonnegative when and satisfies the following conditions: where , , , , , , and is a nature number; then the following inequality holds: *

*Proof. *Denote and , where . Let . It is easy to see that In the following, we will prove thatOtherwise, there exists a such that From (16), (18), and (19), we obtain which leads to a contradiction with (18). Hence inequality (17) holds.

Now, we prove that for Otherwise, there exists a such that By (22) and (23), we have which contradicts (22). Hence (21) holds.

Consequently, on the one hand, for , On the other hand, it follows from (16) and (17) that for So Similarly, we can prove the following results for , and for Now, using mathematical induction method, suppose that the following statements are true; for any integers , we can obtain .

For ,and for , Since, for any , there exists a positive integer , such that , we can conclude the following estimation of by (30) and (31).

For , and for , From the previous definition of , we have The proof of Lemma 3 is completed.

*Remark 6. *Lemmas 3 and 5 played an important role in the finite-time synchronization analysis of dynamical networks via intermittent control in this paper, because it shows the utilization of finite-time intermittent control.

*Lemma 7 (see [30]). Let be any vectors and is a real number satisfying *

*3. Criteria for Finite-Time Synchronization*

*3. Criteria for Finite-Time Synchronization**In this section, we study finite-time synchronization of system (1) with system (36) under the following intermittent controller (37).*

*In order to drive system (1) to achieve finite-time synchronization by means of periodically intermittent control, the corresponding response system is designed as follows: where , , denote the response state vector of the node of system (36). is an intermittent controller defined as follows:where is a positive constant control gain and is a tunable constant. Denote as the maximum (minimum) eigenvalue of the matrix . is the control period, is called the control width (control duration), and is the ratio of the control width to the control period called control rate. is a finite natural number set and .*

*Let be synchronization errors between the states of drive system (1) and response system (37); then the following error system can be obtained:The main results are stated as follows.*

*Theorem 8. Let Assumption 1 hold. Suppose that positive constants , , , and a positive defined diagonal matrix satisfy where , , , and is the identity matrix. Then under the periodically intermittent controllers (37), the error system (38) is synchronized in a finite time: where and is the initial condition of .*

*Proof. *Consider the following Lyapunov function: Then the time derivative of (42) along the trajectories of the first subsystem of system (38) is calculated and estimated as follows.

When , for ,Defining , we have where is a column vector of and is defined as Using Lemma 7, we have It follows from inequality (39) thatwhich shows that .

When , for , we have where is a column vector of and is defined as It follows from inequality (40) that which shows that .

Namely, defining , , and , we getAccording to Lemma 5, we have By Lemma 3, we haveThe proof of Theorem 8 is completed.

*Remark 9. *Obviously, when , the intermittent control (37) is degenerated to a continuous control input which has been extensively proposed in previous work (see [34, 35]) and focuses on [13]. However, this trivial case is not to be discussed in this paper.

If the Lyapunov function when controllers are added into the network, then it is easy to see that Theorem 8 can be restated as the following corollary.

*Corollary 10. Let Assumption 1 hold. Suppose that positive constants , , and a positive define diagonal matrix satisfy where , , , and is the identity matrix. Then under the periodically intermittent controllers (37), the error system (38) is synchronized in a finite time: where and is the initial condition of .*

*Remark 11. *Corollary 10 in this paper is the main result of Theorem 2 in [30] and the main results in [36].

*Remark 12. *According to (41) and (56) and the convergence time , we can conclude that the convergence time satisfies . We can analyse that the term should impede the convergence time. But compared with the control gain matrix in (54), it is easy to seek an appropriate control gain matrix in (39) for which synchronization happens.

*Remark 13. *It is clear to see that inequality (39) is more easily satisfied compared with inequality (54) under the same controllers and conditions via LMI Toolbox, which reveals a very interesting phenomenon; that is, the control gain under condition (39) is more easily designed than condition (54), though it can impede the convergence time.

*Remark 14. *We can find that if condition (55) is satisfied, condition (54) easily holds when the positive definite control gain matrix is anything. Then we have the following corollary.

*Corollary 15. Let Assumption 1 hold. Suppose that positive constants , , and a positive defined diagonal matrix satisfywhere , , and is the identity matrix. Then under the periodically intermittent controllers (37), the error system (38) is synchronized in a finite time: where and is the initial condition of .*

*4. Numerical Examples*

*4. Numerical Examples*

*In this section, we give some numerical examples to show the validity and effectiveness of the derived results for finite-time synchronization via periodically intermittent control.*

*In this case, to demonstrate the results above, we consider general complex dynamical networks, in which each subsystem is a Lorenz system. The dynamics of Lorenz system is described as follows:where the parameters are selected as , , and ; then the Lorenz system has a chaotic attractor (see Figure 1). Moreover, it is known that , , and . Now we will show that there exists a positive definite diagonal matrix that satisfies Assumption 1. Let , , , and ; then Hence, Then where are the maximum eigenvalues of , respectively. Let ; therefore Assumption 1 is satisfied with and .*