Synchronization of Coupled Reaction-Diffusion Neural Networks with Time-Varying Delays via Pinning-Impulsive Controller

Abstract

In this paper, global exponential synchronization stability in an array of linearly diffusively coupled reaction-diffusion neural networks with time-varying delays is investigated by adding impulsive controller to a small fraction of nodes (pinning-impulsive controller). In order to overcome the difficulty resulting from the fact that the impulsive controller affects only the dynamical behaviors of the controlled nodes, a new analysis method is developed. By using the developed method, two known lemmas on stability of delayed functional differential equation with and without impulses, and Lyapunov stability theory, several novel and easily verified synchronization criteria guaranteeing the whole network will be pinned to a homogenous solution are derived. Moreover, the effects of the pinning-impulsive controller and the dynamics of the uncontrolled nodes and the diffusive couplings on the synchronization process are explicitly expressed in the obtained criteria. Our results also show that we can always design an appropriate pinning-impulsive controller to realize the synchronization goal as long as a conventional state feedback pinning controller or an adaptive pinning controller can achieve the synchronization goal by controlling the same nodes. Furthermore, the function extreme value theory is utilized to reduce the conservativeness of the synchronization criteria. Some existing results are improved and extended. Numerical simulations including an asymmetric coupling network and BA (Barabási--Albert) scale-free network are given to show the effectiveness of the theoretical results.