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Abstract and Applied Analysis
Volume 2015, Article ID 381078, 12 pages
http://dx.doi.org/10.1155/2015/381078
Research Article

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 [26] 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 [1519], and intermittent control [2026]. 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 [3032]).

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

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

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 .

Figure 1: The chaotic trajectories of the Lorenz system.

Consider the complex dynamical network (1) consisting of 50 identical Lorenz oscillators nodes, which is described bywhere and is a symmetrically diffusive coupling matrix with or and the coupling strength .

The corresponding controlled response system of (36) is of the form where and are the same as (63) andwhere .

The initial states of the numerical simulations in the master and slave systems are as follows: and . The initial conditions of the error system are , where . In addition, the values of the parameters for the controllers (37) are selected as and . Choosing , , and , one can easily obtain that inequality (40) holds. Besides, we can obtain the parameters of the intermittent controllers as and by using LMI toolbox in Matlab to solve (39) and (40). Therefore from Theorem 8 it can be obtained that the response system (36) will synchronize the drive system (1) under the periodically intermittent controllers (37) within the time (the average time). The synchronous error is illustrated in Figures 24.

Figure 2: Synchronization errors , , with control parameters , , and .
Figure 3: Synchronization errors , , with control parameters , , and .
Figure 4: Synchronization errors , , with control parameters , , and .

Take , , and . By calculating (41), we can obtain the convergence time . The time responses of the error variables , , , are illustrated in Figures 57, with , , and . Let , , and . By calculating (41), we can obtain the convergence time . Figures 810 describe the time responses or the error variables , , , respectively, with different parameters of .

Figure 5: Synchronization errors , , with control parameters , , and .
Figure 6: Synchronization errors , , with control parameters , , and .
Figure 7: Synchronization errors , , with control parameters , , and .
Figure 8: Synchronization errors , , with control parameters , , and .
Figure 9: Synchronization errors , , with control parameters , , and .
Figure 10: Synchronization errors , , with control parameters , , and .

Remark 16. From the above analysis and figures, we can conclude that the control rate influences the convergence time; in addition, the tunable can also influence the convergence time, which is discussed in [34]; we should omit it here. And then, it can be seen that the continuous controller can synchronize the network at . Therefore, the convergence time is shorter than the periodically intermittent control.
To further verify the effectiveness of the proposed control design, we take the control rate , namely, general continuous full control. Figures 1113 demonstrate time response of the error variables , , , respectively, with , , and . When the control variables , , namely no controlled is added to the system. The time response of the error variables , , , is displayed in Figures 1416, respectively, with no control input .

Figure 11: Synchronization errors , , with full control parameters , , and .
Figure 12: Synchronization errors , , with full control parameters , , and .
Figure 13: Synchronization errors , , with full control parameters , , and .
Figure 14: Synchronization errors , , with no control input.
Figure 15: Synchronization errors , , with no control input.
Figure 16: Synchronization errors , , with no control input.

5. Conclusion

This paper has dealt with the finite-time synchronization problem for a class of complex dynamical networks by means of periodically intermittent control. Some novel and useful synchronization criteria, given in terms of a set of linear matrix inequalities, have been obtained by some analysis techniques and finite-time stability theory. Our results reduce the previous works on the controllers that the derivative of the Lyapunov function is smaller than . Several simulations are presented to verify the effectiveness of the proposed synchronization criteria finally. The future work in this endeavor will focus on the global problem.

Conflict of Interests

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

Acknowledgments

The authors would like to thank the editor and the anonymous reviewers for their valuable comments and constructive suggestions. This project is supported by the National Science Foundation of China (11371162 and 1171129) and the Education Department of Hubei Province (T201103).

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