Finite-Time Bounded Synchronization of the Growing Complex Network with Nondelayed and Delayed Coupling

Xu, Yuhua; Zhang, Jincheng; Zhou, Wuneng; Tong, Dongbing

doi:https://doi.org/10.1155/2017/6501583

Discrete Dynamics in Nature and Society

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Abstract Introduction Model and Preliminaries Conclusion Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2017 | Article ID 6501583 | https://doi.org/10.1155/2017/6501583

Finite-Time Bounded Synchronization of the Growing Complex Network with Nondelayed and Delayed Coupling

Yuhua Xu,^1,2,3Jincheng Zhang ,⁴Wuneng Zhou,⁵and Dongbing Tong⁶

Academic Editor: Seenith Sivasundaram

Received17 Sept 2016

Accepted13 Dec 2016

Published30 Jan 2017

Abstract

The objective of this paper is to discuss finite-time bounded synchronization for a class of the growing complex network with nondelayed and delayed coupling. In order to realize finite-time synchronization of complex networks, a new finite-time stable theory is proposed; effective criteria are developed to realize synchronization of the growing complex dynamical network in finite time. Moreover, the error of two growing networks is bounded simultaneously in the process of finite-time synchronization. Finally, some numerical examples are provided to verify the theoretical results established in this paper.

1. Introduction

In recent years, as finite-time control is an important indicator of test control of complex networks, which not only has important theoretical significance, but also has important practical value in practical engineering, finite-time control of complex networks has become one of the most interesting subjects in control theory of complex networks. To get fast convergence speed, much finite-time stabilization theory is proposed to investigate finite-time control of complex networks [1–10]. For example, in [11], Shen et al. investigated finite-time synchronization control of uncertain Markov jump neural networks by input constraints. In [12], Zhang et al. discussed finite-time stabilization of time-varying nonlinear systems by state feedback control. In [13], Wang et al. discussed finite-time global synchronization of Markovian jump complex networks by partially unknown transition rates with feedback control. In [14], Liu et al. considered finite-time synchronization of Markovian switching neutral complex networks based on pinning controller. In [15], Shi studied finite-time control of linear systems with time-varying sampling, and so on. It is noticed that most of the studies on finite-time synchronization of dynamical network have been mainly focused on static networks.

In addition, as bounded control of complex system is relevant in practical applications, for example, power networks cannot achieve complete synchronization, so it is desirable to obtain conditions within given bound such that the rotor phase differences between the generators remain [16]. In recent years, bounded control of complex dynamical network concerns the existence of bounded synchronization region globally stabilizing the complex system, which has grown very quickly, and it is one of the important issues in control theory and applications [17–21].

In [1–17], the authors studied the numbers of the nodes in the networks invariant about synchronization of complex dynamical networks. As the structures of many complex systems are typically dynamic, some new nodes can enter the network as time goes on [22]. Therefore, how to achieve synchronization of a growing dynamical network with bounded error is a very interesting and indeed important subject for research. Motivated by the existing works, the aim of this paper is to discuss finite-time synchronization of the growing complex dynamical network with nondelayed and delayed coupling. A new finite-time stabilization theory is proposed to investigate finite-time bounded synchronization of complex dynamical networks. Finite-time stabilization theory presented in this paper extends the conclusions of literatures [11–15] using finite-time stabilization theory.

The rest of this paper is organized as follows: Section 2 presents model and preliminaries. Section 3 gives the sufficient conditions of finite-time bounded synchronization. Section 4 presents an example and relates simulation results. Section 5 gives the conclusions of this paper.

2. Model and Preliminaries

Firstly, we introduce the network model and give some useful mathematical preliminaries.

Considering the growing complex dynamical network consisting of identical coupled nodes at time , with each node being an -dimensional dynamical system. The state equations of complex dynamical network can be described bywhere , standing for the activity of an individual subsystem is a vector value function, and and are two coupling strengths of the network. If there exists a link from node to at time , and , .

Then the controlled dynamical network can beSo, the error equation iswhere , .

We give the following preliminaries for obtaining the main result.

Assumption 1. We assume that is Lipschitz with respect to its argument; that is,

Lemma 2 (see [23]). The following matrix inequality holds: for any vectors and positive-definite matrix . If not specified otherwise, inequality means is a positive (or negative, or semipositive, or seminegative) definite matrix.

Lemma 3 (see [24]). Let and . Then .

Lemma 4 (see [25]). Given the following dynamical system Assume that a continuous and positive-definite satisfies the following differential inequality: Then, the origin of the dynamical system (5) is finite-time stable. The settling time satisfies

In this paper, to illustrate finite-time bounded stability further, we present the following theorem.

Theorem 5. Suppose that there exist continuous and positive-definite , , , , and such thatThen, the trajectory of system (5) is finite-time stable, and(i)if , then , , ;(ii)if , then , , .

Proof. (i) Obviously, there are such that inequality (8) can beWhen , . According to Lemma 4, the decrease of in finite time drives the trajectories of the dynamical system into . So, the trajectories of the dynamical system are bounded in finite time asAnd the time needed to reach (10) is bounded as(ii) Similarly to (i), there are such that inequality (8) can beWhen , . Therefore, the decrease of in finite time drives the trajectories of the dynamical system into , and the trajectories of the dynamical system are bounded in finite time asSo, the time needed to reach (13) is bounded as

Remark 6. Let , based on Theorem 5, we have the following conclusion for the settling time : (i) ; that is, , so the error is bounded; (ii) , that is, , so the error is also bounded.

Remark 7. When , Theorem 5 is reduced to Lemma 4.

Remark 8. In [26], the authors present new finite-time stability by turning into ; the obtained results have less conservatism than the existing one. In this paper, as , so , that is, the convergence time of Theorem 5 can be different from Lemma 4 by turning into , and is adjustable parameter.

Remark 9. In [27], the authors present new inequality theorems by turning into , and is bounded for . In this paper, is bounded for by using ; is finite-time.

3. Finite-Time Bounded Synchronization of Complex Networks

In this section, we study finite-time bounded synchronization between two networks. We can give the following main result.

Theorem 10. If Assumption 1 holds, and there exists where , , , , , and .
Then complex dynamical networks (1)-(2) can realize finite-time bounded synchronization based on the following controller: where , .

Proof. According to Theorem 5, we only need to prove .
Considering the following nonnegative function: we haveIn (18), by using Lemma 2, obviously,where , .
So,In (20), obviously,By Lemma 3, we haveHence,So By choosing appropriate , , , , then, . Based on Theorem 5 and Remark 6, the error system is finite-time bounded stability. The proof is completed.

Remark 5. By Remark 6, if , then , ; if , then , .

Remark 6. For the finite time , we can find the appropriate value by adjusting value. (i) If , then and . (ii) If , then and .

Remark 7. In the network, when the number of nodes increases, the value of will also increase, and the suitable value can always be found.

4. Illustrative Example

In this section, we present an example to illustrate the usefulness of Theorem 10 in this paper.

Consider the following Lü system [28, 29]:Obviously, , andIt is well known that Lü attractor is bounded. Here we suppose that all nodes are running in the given bounded region [30]. Our theoretical and numerical analyses show that there exist constants , , and satisfying , , so

In numerical simulation, let the initial values of the state variable be rand . When the time , the coupling network with 6 nodes is described by the coupling matrices , ,

Suppose that there are new nodes entering the network at time , the coupling network with 7 nodes is described by the coupling matrices , ,Let , , , , , , , , (network with 6 nodes), (network with 7 nodes); let . Let , obviously, two networks achieve synchronization when no longer increases. Figure 1 shows the variance of the synchronization errors. Figure 1 shows . Numerical simulation illustrates the effectiveness of Theorem 10.

5. Conclusion

In this paper, finite-time synchronization for a class of the growing complex dynamical network with nondelayed and delayed coupling was investigated. A new finite-time bounded stable theory is proposed; synchronization criterion was derived to ensure the realization of finite-time bounded synchronization. Finally, numerical simulation was given to verify the effectiveness of the proposed schemes.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (61673221, 61673257, and 11501367), the Youth Fund Project of the Humanities and Social Science Research for the Ministry of Education of China (14YJCZH173), Top-notch Academic Programs Project of Jiangsu Higher Education Institutions (Jiangsu Province Office, no. 1, PPZY2015B104), the Major Natural Science Research Projects of Jiangsu Higher Education Institutions (12KJA630001), the Key Laboratory of Financial Engineering of Jiangsu Province (NSK2015-16), the Science and Technology Research Key Program for the Education Department of Hubei Province of China (D20156001), and Applied Economics Advantage Subject Construction Project of Jiangsu Higher Education Institutions (Jiangsu Province Office, no. 37).

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Copyright © 2017 Yuhua Xu 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.

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