Study On The Synchronization Analysis And Control Of The Complex Dynamical Networks With Switching Topology

Complex networks consist of a large number of nodes with interconnection each other. For the wide applications in the natural science and engineering, complex networks have become a hot topic in recent years. The study of complex networks can acquaint and explain the dynamical phenomena in the complex nonlinear systems. By using Lyapunov theory, martingale theory, the theory of stochastic differential equations and integral equations, and state feedback control, adaptive control and pinning control, this paper investigates the synchronization control problem of complex networks. The major content of this paper are summarized as follows.(1) The local synchronization problem for a class of complex dynamical networks with switching coupling matrices and mixed time delays are investigated by using Lyapunov stability theory and main stable function method. Some sufficient conditions are derived to guarantee the local synchronization for the complex network. Moreover, these conditions are given with the form of linear matrix inequalities.(2) The synchronization control problem for a class of complex dynamical networks with mixed time-delay and switching topology is studied in terms of the Lyapunov stability theory, Kronecker product analysis technique and adaptive control method. Where, the topology of networks varies by a Markov process. Some synchronization criteria and controller design methods are provided. Furthermore, the controller depends on the lag state and Markov process.(3) The mean square exponential synchronization in Lagrange sense for uncertain complex network is investigated based on the Lyapunov stability theory and the Kronecker product analysis technique. In particular, the uncertain phenomena include varying topology structure, destroyed nodes, and the noise disturbance from working circumstance. Some conditions which guarantees the complex networks to be mean square exponential synchronization in Lagrange sense are proposed.(4) The exponential synchronization problem for a class of discrete-time uncertain complex networks with stochastic effects and time delay is investigated by using the Lyapunov stability theory and discrete Halanay inequality. Through constructing an appropriate Lyapunov function and applying inequality technique, some synchronization criteria and two control methods are obtained to ensure the considered complex network being exponential synchronization.(5) The synchronization control problem of the competitive complex networks with time delay and stochastic effects is investigated by using the stochastic technique and Lyapunov stability theory. The competitive complex networks mean that a part of nodes vary faster than other nodes. Some synchronization criteria are derived by the full controller and pinning controller, respectively.(6) The robust synchronization problem for a stochastic delayed complex network with switching topology and unmodeled dynamics is investigated by using the adaptive control scheme. The uncertain terms in the complex network only need to satisfy the bounded condition. Based on the stability theory of stochastic delayed differential equation, a novel adaptive controller which ensures that all the nodes in the complex network robustly synchronize with the target node is derived.Finally, some problems and expectations on complex networks in the future work are proposed.