Lag synchronization of complex dynamical networks with hybrid coupling via adaptive pinning control

Xiaojun Zhang, Huilan Yang, Shouming Zhong
2016 Journal of Nonlinear Science and its Applications  
In this paper, the problem of the lag synchronization between two general complex dynamical networks with mixed coupling by pinning control is studied. Based on the Lyaponov functional theory and mathematical analysis method, less conservative conditions of lag synchronization are obtained by adding the controllers to part of nodes. Moreover, the coupling configuration matrices are not required to be symmetric or irreducible. It is shown that the lag synchronization of the drive and response
more » ... ive and response systems can be realized via the linear feedback pinning control and adaptive feedback pinning control. These results remove some restrictions on the node dynamics and the number of the pinned nodes. Numerical examples are presented to illustrate the effectiveness of the theoretical results. . 9 (2016), 4678-4694 4679 synchronization of the complex network has become an important topic due to its realistic significance and study value. In recent years, synchronization and its control of complex dynamical networks have been widely studied. Many synchronization methods have been proposed including linear state feedback control [25], pinning control [8, 16, 33, 37, 38], state observer based control [11, 32] , impulsive control [20, 39] , and adaptive control [9] and so on. However, most of them focus on the inner synchronization, in which all nodes in a network achieve a coherent behavior. Different from the inner synchronization [12] , there is another kind of synchronization namely outer synchronization [28, 36] , which has quickly caught much attention since Li first proposed in 2007 [18] . In general, there are several kinds of synchronization, such as, complete synchronization [34], phase synchronization [2], lag synchronization [21], generalized synchronization [24] , and projective synchronization [13] . Among them, lag synchronization, which requires the states of response system to synchronize with the past states of the drive system, has been widely observed in many practical systems like electronic circuits, lasers and neural systems [35] . It has been proved to be a reasonable scheme from the viewpoint of engineering applications and the characteristics of channel in secure communication, parallel image processing, and pattern storage [17] . Therefore, lag synchronization has become a hot topic in many fields [10, 22, 29] . For example, in [6], the author investigated the issue of the lag synchronization between two coupled networks by adding the controllers to part of nodes. Zhao et al. [40] considered the lag synchronization problem of two different complex networks based on the approach of state observer. However, although the approach realized the lag synchronization for complex dynamical networks, there are still some problems which need to be studied. These include: (1) the coupling configuration matrices are always assumed to be irreducible and their off-diagonal entries are nonnegative, and the inner connecting matrices are diagonal positive define; (2) it is very expensive and even impractical to apply the controllers to all or many nodes, especially for the engineering applications. For this reason, as described in [38] , to achieve low cost and easy implementation, it is significant to investigate how the drive and response networks are synchronized by pinning only a small portion of nodes in a network; (3) in a real network, since the speed of signal travel between nodes is limited and the network nodes may be required to have non-local interconnections like telecommunications [15, 41] , the discrete delay coupling and distributed time coupling are inevitable in the network. Thus, the synchronization of complex networks with delayed coupling including discrete and distributed delay coupling should be considered. Sufficient conditions for adaptive lag synchronization of complex dynamical network with discrete delayed coupling have been provided in [10] . To the best of our knowledge, up to now, there has been no literature concerning the problems of lag synchronization for complex dynamical networks with mixed coupling. Inspired by the above mentioned discussions, in this paper, a lag synchronization method between two general complex dynamical networks with hybrid coupling by pinning control a small portion of nodes of the network has been proposed. The main contributions of this paper are listed as follows: first, the hybrid coupling, which is made up of non-delay coupling, discrete delay coupling and distributed delay coupling is considered; second, by applying the Lyaponov functional theory and mathematical analysis method, sufficient verifiable conditions are constructed for the lag synchronization of the drive and response networks. These results are less conservative and easy to verify through the numerical simulation. Moreover, the coupling matrices are not necessary to be symmetric and irreducible, and without assuming diagonal or positive define of the inner linking matrices; third, in numerical simulation section, we verify that pinning only one node can realize lag synchronization of the networks adequately and the node can be chosen according to the high-degree of vertex or the maximum norm of synchronization error. The rest of this paper is organized as follows: in Section 2, the complex dynamical network is introduced and some related definitions and lemmas are given; then in Section 3, the linear feedback pinning control and the adaptive feedback pinning control are designed and the corresponding lag synchronization theorems are derived respectively; in Section 4, two illustrative examples are provided to examine the effectiveness of the theoretical results; finally Section 5 concludes this paper. From now on, throughout this paper, I n denotes an n-dimensional identity matrix, n indicates the n-dimensional Euclidean space and n×n is the set of all n × n real matrices. For symmetric matrices X
doi:10.22436/jnsa.009.06.107 fatcat:jzs5fdrwxnb3ncxdb4wefkkxr4