Almost sure synchronization for nonlinear complex stochastic networks with Levy noise

In this paper, almost sure synchronization is developed for a class of nonlinear complex stochastic networks with an adaptive feedback control. This class of networks are characterized under a general framework, including (1) a continuous time irreducible Markov chain which is introduced to describe the dynamical switching of the underlying network topological structure and (2) Levy process which is used to reflect the stochastic noise resulted from the external random perturbation. Different from the currently available literature which mainly focuses on the synchronization in mean, we study the conditions that ensure the almost sure synchronization of nonlinear complex networks. By introducing the convergence theorem of nonnegative semi-martingales as well as by making use of the general Ito integration for Levy process, we show that network synchronization can be achieved with a full probability via our proposed adaptive feedback control. Simulations are presented to demonstrate the effectiveness of this new approach.