Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 932058, 8 pages

http://dx.doi.org/10.1155/2013/932058

## Mean-Square Exponential Synchronization of Stochastic Complex Dynamical Networks with Switching Topology by Impulsive Control

^{1}School of Computer Engineering, Shenzhen Polytechnic, Shenzhen 518055, China^{2}College of Information and Engineering, Shenzhen University, Shenzhen 518060, China^{3}Institute of Intelligent Computing Science, Shenzhen University, Shenzhen 518060, China

Received 2 November 2012; Accepted 25 February 2013

Academic Editor: Qiru Wang

Copyright © 2013 Xuefei Wu and Chen Xu. 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 investigates the mean-square exponential synchronization issues of delayed stochastic complex dynamical networks with switching topology and impulsive control. By using the Lyapunov functional method, impulsive control theory, and linear matrix inequality (LMI) approaches, some sufficient conditions are derived to guarantee the mean-square exponential synchronization of delay complex dynamical network with switch topology, which are independent of the network size and switch topology. Numerical simulations are given to illustrate the effectiveness of the obtained results in the end.

#### 1. Introduction

Complex networks are everywhere in nature and our daily life, such as the Internet, ecosystems, social networks, World Wide Web, and neural networks. A complex network can be described by a set of nodes and edges interconnecting these nodes together. During the last two decades, complex networks have been focused on by scientists from various fields, such as mathematical, engineering, and social and economic science. There are many literatures concerning the collective behaviors of complex networks [1–11]. Among them, synchronization is the most interesting phenomenon [3–11], because the synchronization is a kind of typical collective behaviors exhibited in many natural systems.

Due to many complex systems may experience abrupt changes in their connection caused by some phenomena such as link failures, component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbance, and so forth. Thus, the synchronization of complex network with switching topology has been studied in [12–17]. In [12], Yao et al. studied the synchronization of a general complex dynamical network with switching topology and the time-varying coupling is unknown but bounded. Wang et al. [13] investigated the synchronization issues of complex dynamical networks with switching topology. Yu et al. [14] explored the synchronization of switched linearly coupled neural networks with delay. Some sufficient conditions were given to guarantee the global synchronization. In [15], Liu et al. studied the local and global exponential synchronization of complex dynamical network with switching topology and time-varying coupling delays. In [16], some sufficient conditions were given to guarantee the synchronization of leader following issues with switching connective network and coupling delay. In [17], authors investigated the consensus problem in mean square for uncertain multiagent systems with stochastic measurement noises and symmetric or asymmetric time-varying delays.

Uncertainties commonly exist in the real world, such as stochastic forces on the physical systems and noisy measurements caused by environmental uncertainties. Thus, a stochastic behavior should be produced instead of a deterministic one [18]. In fact, signals transmitted between nodes of complex networks are unavoidably subject to stochastic perturbations from environment, which may cause information contained in these signals to be lost [19]. Therefore, stochastic perturbations should be considered [18–23]. In [19–21], stochastic perturbations are all one-dimensional, which means that the signal transmitted by nodes is influenced by the same noise. In [18, 22], the authors considered stochastic synchronization of coupled neural networks, in which noise perturbations are vector forms. Vector-form perturbation means that different nodes are influenced by different noise, which is more practical in the real world. In [23], authors investigated the mean-square exponential synchronization of stochastic complex networks with Markovian switching and time-varying delays by using the pinning control method.

In many systems, the impulsive effects are common phenomena due to instantaneous perturbations at certain moments [24–28]. In the past several years, impulsive control strategies have been widely used to stabilize and synchronize coupled complex dynamical system, such as signal processing system, computer networks, automatic control systems, and telecommunications. In [24], Cai et al. investigated the robust impulsive synchronization of complex delayed dynamical networks. Yang et al. [25] studied the exponential synchronization of complex dynamical network with a coupling delay and impulsive control. Zhu et al. gave some global impulsive exponential synchronization criteria of time-delayed coupled chaotic systems [27]. In [28], some sufficient conditions were given to guarantee the consensus of nonlinear multiagent systems with switching topology.

Based on the above discussion, studying the synchronization problem of the complex dynamical network with switch topology and impulsive effects is very useful and meaningful. It should be pointed out that exponential synchronization of the coupling delay complex dynamical networks with switch topology and impulsive effects has received very little research attention.

In this paper, we investigate the problem of exponential synchronization of coupling delay complex dynamical network with switching topology and impulsive effects, basing on the Lyapunov theory and impulsive control theory and by assuming that there exists finite connective topology of the complex dynamical network, and the connective topology may be switched (or jumped) from one to another at different moments. For controlling, the synchronization state can be any weighting average of the network states. It means that each node of the dynamical network can contribute to synchronization of the network in its weight. Finally, some sufficient conditions are given to guarantee the synchronization of the complex dynamical network, which are independent of the network size and switch topology.

#### 2. Model and Preliminaries

The switching complex dynamical networks investigated in this paper consist of nodes, whose state is described as follows: where is the state vector of the th node of the network, is a continuous vector-valued function, is an inner coupling of the networks that satisfies , , and is a switching signal, which is a piecewise constant function. Here, and are the outer-coupling matrices of the network at time at nodes , , and , respectively, such that for , , for , and . is the inner time-varying delay satisfying and is the coupling time-varying delay satisfying . Finally, and is a bounded vector-form Weiner process, satisfying is the th node impulsive gain at . The discrete set satisfies , , as , note , and . In this paper, is assumed to be irreducible in the sense that there are no isolated nodes.

The initial conditions associated with (1) are where , with the norm and our objective is to control system (1) so that it stays in the trajectory of the system

*Remark 1. *In general, the synchronization state may be an equilibrium point, a periodic orbit, or a chaotic attractor.

The following assumptions will be used throughout this paper for establishing the synchronization conditions. (H1) and are bounded and continuously differentiable functions such that , , , and . Let and . (H2) Let . Then there exist positive definite constant matrices , , and for and such that
(H3) .

Define error state ,

*Definition 2. *The complex network (1) is said to be exponentially synchronized in mean square if the trivial solution of system (6) is such that
for some and some for any initial data .

*Definition 3. *A continuous function is said to belong to the function class , denoted by for some given matrix if there exist a positive definite diagonal matrix , a diagonal matrix , and constants such that satisfies the condition
for all .

*Remark 4. *The function class QUAD includes almost all the well-known chaotic systems with or without delays such as the Lorenz system, the Rössler system, the Chen system, the delayed Chua’s circuit, the logistic delayed differential system, the delayed Hopfield neural network, and the delayed CNNs. We shall simply write

In order to derive the main results, it is necessary to propose the following lemmas.

Lemma 5 (see [16]). *The following linear matrix inequality
**
where , is equivalent to one of the following conditions: *(i)*. *(ii)*. *

#### 3. Main Result

In this section, we investigate the exponential stability condition of the error system (6). Some new criteria are presented for the exponential synchronization of network (1) based on the Lyapunov functional method, linear matrix inequality approach, and impulsive control theory.

Theorem 6. *Let assumptions (H1) and (H2) be true and let . If there exist positive constants and such that
**
where
**
then the solutions of system (6) are globally and exponentially stable. *

*Proof. *Define the Lyapunov-Krasovskii function
and let , . For , we have
Then we have
So we have
Define
and use (17) to compute the operator
which after applying the generalized Itô’s formula, gives
for any . Hence we have
By changing variable , we have
Similarly, we have
Substituting (22) and (23) into (21), we get
By using Gronwall’s inequality, we get

On the other hand, from the construction of , we have
where .

According to (8)–(11), for any , we get

Let . Because of , we have

Using Condition (12) of Theorem 6, we get . Hence, . The proof of Theorem 6 is completed.

*Remark 7. *Theorem 6 provides a sufficient condition for exponential synchronization of coupling delay switched stochastic dynamical networks with impulsive effects. If the time is sufficiently small and the impulsive gains , then exponential synchronization of the network (1) could be achieved.

If the switching signal , then the network (1) has only one coupling matrix . Suppose is irreducible and is the left eigenvector of coupling matrix corresponding to eigenvalue . Let impulsive gains are . By the proof of Theorem 6, we can derive the exponential synchronization criteria of the network (1) with only one topology, which is given as follows.

Corollary 8. *Let assumptions (H1) and (H2) be true and let . If there exist positive constants and such that
**
where
**
Then the solutions of system (6) are globally and exponentially stable.*

Let impulsive gains , and choose the synchronization state . By the proof of Theorem 6, we can derive the exponential synchronization criteria of the network (1) with the fixed impulsive gain, which is given as follows.

#### 4. Numerical Simulation

In this section, we give two numerical simulations to illustrate the feasibility and effectiveness of the theoretical results presented in the previous sections.

Consider a three-order Chua’s circuit described as follows: where and function was chosen below: where , , and .

Consider a network model consists of 5 nodes and 2-connected topology. Each node in the network is a three-order Chua’s circuit described by where , , and .

The coupling matrices are as follows:

If we choose and , then function satisfies the condition of function class , where . Let , , , , , . , , , , , , , , , and the synchronization state ; then all the conditions in Theorem 6 are satisfied (by using the Matlab LMI toolbox). The switch time is . The simulation results are given in Figures 1–3. It can be seen clearly from Figures 1, 2, and 3 that all states of the asymmetric coupled network (21) tend to the synchronization state .

#### 5. Conclusion

In this paper, the exponential synchronization of the coupling delay stochastic complex networks with switch topology and impulsive effects has been investigated. Based on the Lyapunov stability theory, LMI, and impulsive control theory, some simple, yet generic, criteria for exponential synchronization have been derived. It has shown that criteria can provide an effective control scheme to synchronize for a given coupled delay, the network size, and switch topology. Furthermore, the effectiveness of the presented method has been verified by numerical simulations.

#### Acknowledgments

This work was supported by the National Science Foundation of China under Grant no. 61070087, the Guangdong Education University Industry Cooperation Projects (2009B090300355), the Shenzhen Basic Research Project (JC200903120040A, JC201006010743A), and the Shenzhen Polytechnic Youth Innovation Project (2210k3010020). The authors are very grateful to the reviewers and editors for their valuable comments and suggestions that improved the presentation of this paper.

#### References

- S. H. Strogatz, “Exploring complex networks,”
*Nature*, vol. 410, no. 6825, pp. 268–276, 2001. View at Publisher · View at Google Scholar · View at Scopus - S. Boccaletti, V. Latora, Y. Moreno, M. Chavez, and D. U. Hwang, “Complex networks: structure and dynamics,”
*Physics Reports*, vol. 424, no. 4-5, pp. 175–308, 2006. View at Publisher · View at Google Scholar · View at MathSciNet - A. Arenas, A. Díaz-Guilera, J. Kurths, Y. Moreno, and C. Zhou, “Synchronization in complex networks,”
*Physics Reports*, vol. 469, no. 3, pp. 93–153, 2008. View at Publisher · View at Google Scholar · View at MathSciNet - J. H. Lü and G. R. Chen, “A time-varying complex dynamical network model and its controlled synchronization criteria,”
*IEEE Transactions on Automatic Control*, vol. 50, no. 6, pp. 841–846, 2005. View at Publisher · View at Google Scholar · View at MathSciNet - J. H. Lü, X. H. Yu, and G. R. Chen, “Chaos synchronization of general complex dynamical networks,”
*Physica A*, vol. 334, no. 1-2, pp. 281–302, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - C. G. Li and G. R. Chen, “Synchronization in general complex dynamical networks with coupling delays,”
*Physica A*, vol. 343, no. 1–4, pp. 263–278, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - W. Wu and T. Chen, “Global synchronization criteria of linearly coupled neural network systems with time-varying coupling,”
*IEEE Transactions on Neural Networks*, vol. 19, no. 2, pp. 319–332, 2008. View at Publisher · View at Google Scholar · View at Scopus - J. Zhao, J. Lu, and X. Wu, “Pinning control of general complex dynamical networks with optimization,”
*Science China: Information Sciences*, vol. 53, no. 4, pp. 813–822, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - C. W. Wu, “Synchronization in arrays of coupled nonlinear systems with delay and nonreciprocal time-varying coupling,”
*IEEE Transactions on Circuits and Systems II*, vol. 52, no. 5, pp. 282–286, 2005. View at Publisher · View at Google Scholar · View at Scopus - W. Lu and T. Chen, “Synchronization of coupled connected neural networks with delays,”
*IEEE Transactions on Circuits and Systems I*, vol. 51, no. 12, pp. 2491–2503, 2004. View at Publisher · View at Google Scholar · View at MathSciNet - X. Liu and T. Chen, “Exponential synchronization of nonlinear coupled dynamical networks with a delayed coupling,”
*Physica A*, vol. 381, no. 1-2, pp. 82–92, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. Yao, D. J. Hill, Z. H. Guan, and H. O. Wang, “Synchronization of complex dynamical networks with switching topology via adaptive control,” in
*Proceedings of the 45th IEEE Conference on Decision and Control 2006 (CDC '06)*, pp. 2819–2824, December 2006. View at Scopus - L. Wang and Q. G. Wang, “Synchronization in complex networks with switching topology,”
*Physics Letters A*, vol. 375, no. 34, pp. 3070–3074, 2011. View at Publisher · View at Google Scholar · View at Scopus - W. Yu and J. Cao, “Synchornization conrol of switched linearly coupled neural networks with delay,”
*Physics Letters A*, vol. 375, no. 4–6, pp. 3070–3074, 2010. View at Google Scholar - T. Liu, J. Zhao, and D. J. Hill, “Exponential synchronization of complex delayed dynamical networks with switching topology,”
*IEEE Transactions on Circuits and Systems I*, vol. 57, no. 11, pp. 2967–2980, 2010. View at Publisher · View at Google Scholar · View at MathSciNet - Q. Jia, W. K. S. Tang, and W. A. Halang, “Leader following of nonlinear agents with switching connective network and coupling delay,”
*IEEE Transactions on Circuits and Systems I*, vol. 58, no. 10, pp. 2508–2519, 2011. View at Publisher · View at Google Scholar · View at MathSciNet - Y. Sun, “Mean square consensus for uncertain multiagent systems with noises and delays,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 621060, 18 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. S. Yang and J. D. Cao, “Adaptive pinning synchronization of complex networks with stochastic perturbations,”
*Discrete Dynamics in Nature and Society*, vol. 2010, Article ID 416182, 21 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Cao, Z. Wang, and Y. Sun, “Synchronization in an array of linearly stochastically coupled networks with time delays,”
*Physica A*, vol. 385, no. 2, pp. 718–728, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - J. Liang, Z. Wang, Y. Liu, and X. Liu, “Global synchronization control of general delayed discrete-time networks with stochastic coupling and disturbances,”
*IEEE Transactions on Systems, Man, and Cybernetics B*, vol. 38, no. 4, pp. 1073–1083, 2008. View at Publisher · View at Google Scholar · View at Scopus - A. Pototsky and N. Janson, “Synchronization of a large number of continuous one-dimensional stochastic elements with time-delayed mean-field coupling,”
*Physica D*, vol. 238, no. 2, pp. 175–183, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - X. Yang and J. Cao, “Stochastic synchronization of coupled neural networks with intermittent control,”
*Physics Letters A*, vol. 373, no. 36, pp. 3259–3272, 2009. View at Publisher · View at Google Scholar · View at Scopus - J. Wang, C. Xu, J. Feng, M. K. Kwong, and F. Austin, “Mean-square exponential synchronization of Markovian switching stochastic complex networks with time-varying delays by pinning control,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 298095, 18 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Cai, J. Zhou, L. Xiang, and Z. Liu, “Robust impulsive synchronization of complex delayed dynamical networks,”
*Physics Letters A*, vol. 372, no. 30, pp. 4990–4995, 2008. View at Publisher · View at Google Scholar · View at Scopus - Y. Yang and J. Cao, “Exponential synchronization of the complex dynamical networks with a coupling delay and impulsive effects,”
*Nonlinear Analysis: Real World Applications*, vol. 11, no. 3, pp. 1650–1659, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Zhou, L. Xiang, and Z. Liu, “Synchronization in complex delayed dynamical networks with impulsive effects,”
*Physica A*, vol. 384, no. 2, pp. 684–692, 2007. View at Publisher · View at Google Scholar · View at Scopus - W. Zhu, D. Xu, and Y. Huang, “Global impulsive exponential synchronization of time-delayed coupled chaotic systems,”
*Chaos, Solitons and Fractals*, vol. 35, no. 5, pp. 904–912, 2008. View at Publisher · View at Google Scholar · View at Scopus - H. Jiang, Q. Bi, and S. Zheng, “Impulsive consensus in directed networks of identical nonlinear oscillators with switching topologies,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 17, no. 1, pp. 378–387, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet