On Several Synchronization Control Problems Of Complex Networks

A complex network is a complex aggregate of some interlinked and interactive units with some characteristics and functions. Recently, complex network has become highly active, with many new systems being studied in application areas such as math, biology and computer science. Synchronization problem is one of the most demonstrating topics in complex networks. Synchronization of complex networks is not only a common phenomenon within nonlinear systems, but also a significant part for practical applications. Therefore it is significant to study synchronization control problem of dynamical systems on complex networks. In this thesis, by using the nonlinear system theory, classical control theory and matrix analysis, three types of synchronization problems i.e. the complete synchronization of a single complex network, the outer synchronization of two complex networks and the cluster synchronization of a complex network with multiple communities are investigated. Furthermore, different sufficient stability conditions are given for the corresponding synchronous types. The main work can be summarized as follows:Firstly, considering a class of complex network with coupling delay, a sampled-data controller with stochastically-varying sampling intervals is designed combining with pinning control strategy. Based on linear matrix inequalities and Lyapunov stability theory, a sufficient stability condition is given. Numerical simulation results show that the proposed controller can achieve complete synchronization for the complex network with many Lorenz chaotic systems as its nodes. Then the influence of coupling strength and coupling delay on the synchronization effect is analyzed.Secondly, in view of two complex networks with multiple random nonlinearities and coupling time-varying delay, the outer synchronization control problem is investigated. An adaptive controller is designed to achieve the outer synchronization of complex networks. Numerical simulation results show the effectiveness of the adaptive controller. In addition, the influence of two complex networks’ topological structures on the synchronization performance is illustrated by numerical simulation with many unified chaotic systems as its nodes.Thirdly, for a class of directed complex network of non-identical dynamic systems, in which the coupling relationships between the nodes are nonlinear, a state feedback controller is designed. Furthermore, the pinning control scheme is given, which can ensure that cluster synchronization for the nonlinear coupled directed network. In this scheme, the special strategy to choose pinning nodes is given, and the relationships among the number of pinning nodes, the degree of pinning nodes and the coupling relationships are studied. Numerical simulation results show the effectiveness of the designed controller. In addition, we can get the special pinning strategy can guarantee the synchronization of a directed network by numerical simulation.Finally, the main work of this thesis is summarized and some future research directions are put forward.