Pinning Control Strategies for Synchronization of Linearly Coupled Neural Networks With Reaction–Diffusion Terms

Two types of coupled neural networks with reaction-diffusion terms are considered in this paper. In the first one, the nodes are coupled through their states. In the second one, the nodes are coupled through the spatial diffusion terms. For the former, utilizing Lyapunov functional method and pinning control technique, we obtain some sufficient conditions to guarantee that network can realize synchronization. In addition, considering that the theoretical coupling strength required for synchronization may be much larger than the needed value, we propose an adaptive strategy to adjust the coupling strength for achieving a suitable value. For the latter, we establish a criterion for synchronization using the designed pinning controllers. It is found that the coupled reaction-diffusion neural networks with state coupling under the given linear feedback pinning controllers can realize synchronization when the coupling strength is very large, which is contrary to the coupled reaction-diffusion neural networks with spatial diffusion coupling. Moreover, a general criterion for ensuring network synchronization is derived by pinning a small fraction of nodes with adaptive feedback controllers. Finally, two examples with numerical simulations are provided to demonstrate the effectiveness of the theoretical results.

I. Introduction

Many systems in nature and society, such as food webs, communication networks, social networks, power grids, cellular networks, World Wide Web, metabolic systems, and disease transmission networks, can be modeled as complex networks. Therefore, complex networks have been considered as a fundamental tool to understand the dynamical behavior and the response of real systems, and the analysis and control of dynamical behaviors in complex networks have received much attention in recent years.