Outer synchronization analysis of two time-varying networks with double delays based on pinning control

Abstract

It is well known that time delay is universal in complex networks. However, in most existing researches outer synchronization is realized between two networks with time delay by adding controllers to all nodes which may bring great economic costs and increase the difficulties in control in practice. In this paper, in order to deal with the problem of outer synchronization between two time-varying coupling networks with node delay and coupling delay, an adaptive pinning control scheme is proposed. First, a more realistic drive-response complex network model is constructed by introducing double delays and asymmetric coupling configuration matrices. Then, we design an adaptive pinning controller which is easy to implement, and choose an effective pinning strategy to control a crucial part of the nodes in the response network. Based on LaSalle' invariance principle and the linear matrix inequality, we may rigorously prove that the outer synchronization between the proposed drive-response networks can be achieved, and meanwhile some sufficient conditions are derived by adopting an appropriate Lyapunov-Krasovskii energy function. Finally, numerical simulation experiments are employed to verify the correctness and the effectiveness of the proposed method. Results indicate that the drive-response networks with double delays can indeed achieve outer synchronization by pinning control. Moreover, the synchronization is independent of coupling delay. And the remarkable influences of coupling delays on the synchronization speed are also revealed.