Adaptive Synchronization of Complex Networks with Mixed Probabilistic Coupling Delays via Pinning Control

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 742956, 9 pages http://dx.doi.org/10.1155/2014/742956 Research Article Adaptive Synchronization of Complex Networks with Mixed Probabilistic Coupling Delays via Pinning Control Jian-An Wang School of Electronics Information Engineering, Taiyuan University of Science and Technology, Shanxi 030024, China Correspondence should be addressed to Jian-An Wang; Received 26 February 2014; Accepted 22 June 2014; Published 15 July 2014 Academic Editor: Qing-Wen Wang Copyright © 2014 Jian-An Wang. 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. The problem of synchronization for a class of complex networks with probabilistic time-varying coupling delay and distributed time-varying coupling delay (mixed probabilistic time-varying coupling delays) using pinning control is investigated in this paper. The coupling configuration matrices are not assumed to be symmetric or irreducible. By adding adaptive feedback controllers to a small fraction of network nodes, a low-dimensional pinning sufficient condition is obtained, which can guarantee that the network asymptotically synchronizes to a homogenous trajectory in mean square sense. Simultaneously, two simple pinning synchronization criteria are derived from the proposed condition. Numerical simulation is provided to verify the effectiveness of the theoretical results. 1. Introduction During the past few decades, synchronization in complex networks has gained increasing research attention [1–7]. There are many different kinds of methods in the study of network synchronization behavior such as adaptive feedback control [8–10], impulsive control [11, 12], passive method [13, 14], intermittent control [15, 16], and sampled-data control [17–19]. As we know, since the real-world complex networks usually have a large number of nodes, it is impossible to realize network synchronization by adding controllers to all nodes. To reduce the number of controlled nodes, pinning control, in which some local feedback controllers are only applied to a fraction of network nodes, has been introduced in many works [20–29]. Pinning control is an effective synchronization strategy because it is easily realized in practice. In [20], the authors found that one can pin the linearly coupled networks by introducing fewer locally negative feedback controllers. They also found out that the pinning strategy based on highest connection degree has better performance than totally randomly pinning. Chen et al. in [21] pinned a complex network to a homogenous solution by a single controller under a large enough coupling strength. By using adaptive pinning control method, the authors in [22] investigated local and global pinning synchronization of complex networks and presented some low-dimensional pinning synchronization criteria. In [23], Yu et al. showed that the nodes with low degrees should be pinned first when the coupling strength is small, which is different from the traditional results. The authors in [24] considered the pinning synchronization of a complex network with nonderivative and derivative coupling. Song and Cao in [25] presented some low-dimensional pinning schemes for global synchronization of both directed and undirected complex networks and proposed specifically pinning schemes to select pinned nodes by investigating the relationship among pinning synchronization, network topology, and the coupling strength. Furthermore, Song et al. in [26] investigated the pinning controlled synchronization of a general complex dynamical network with discrete-delay coupling and distributed-delay coupling. Some sufficient conditions for the synchronization to require the minimum number of pinning nodes were derived in [27], and the method for calculating the number of pinning nodes was given by using the decreasing law of maximum eigenvalues of the principal submatrixes. Recently, the pinning sampled-data synchronization problem was addressed in [28]. Time delay is ubiquitous in many physical systems due to the finite switching speed of amplifiers, finite signal 2 propagation time in biological networks, memory effects, and so on. In order to give a more precise description of dynamical network, time delay should be considered inevitably. Therefore, much effort has been devoted to the study of the synchronization of complex networks with coupling delays. It is worth pointing out that, among most existing results, the network synchronization problem has been predominantly studied for complex networks with deterministic delays. However, as reported in [30], the probability distribution of time delay in an interval is an important characteristic in networked control systems [30]. The probability of the delay appearing in lower interval is large and long delay happens with a low probability, which will lead to some conservatism if only the information of variation range of time delay is considered. Thus, coupling delay in complex networks may exist in a random form and take values according to probability in different interval ranges [31]. In addition, it is noted that time delays can be generally categorized as discrete ones and distributed ones. Moreover, it has been observed that they usually have a spatial nature due to the presence of a number of parallel pathways of a variety of axon sizes and lengths in a network. To the best of the authors’ knowledge, up to now, little attention has been paid to the study of pinning synchronization problem for complex networks with probabilistic time-varying coupling delay and distributed time-varying coupling delay, which motivates our investigation. In this paper, we are concerned with the synchronization problem in an array of hybrid-coupled complex networks with mixed probabilistic time-varying coupling delays by pinning control scheme. The coupling configuration matrices are not assumed to be symmetric or irreducible. Under a low-dimensional condition, the network can be asymptotically pinned to a homogenous state in mean square sense by applying adaptive feedback control actions to a small fraction of nodes. Also, two pinning synchronization criteria are obtained for simple cases. A numerical example is given to demonstrate the effectiveness of the proposed results. The rest of this paper is organized as follows. In Section 2, the model of complex dynamical network with mixed probabilistic time-varying coupling delays is presented and some preliminaries are also provided. Pinning adaptive synchronization criterion is discussed in Section 3. Numerical simulations are given in Section 4. Finally, a conclusion is presented in Section 5. Notations. 𝑅𝑛 and 𝑅𝑚×𝑛 denote the 𝑛-dimensional Euclidean space and the set of all 𝑚 × 𝑛 real matrices, respectively. The superscript “𝑇” (...truncated)