Exponential Stability Of Coupled Reaction-diffusion Systems

During the past few decades, coupled systems have been a hot research topic, and many applications are developed in mechanical, electronic and biology fields. On the one hand, considering finite transmission speed of interaction between subsystems, time delays are inevitably encountered. On the other hand, the state of the subsystems is also affected by the space. So diffusion effects are completely unavoidable. Therefore, to describe more accurately, the effect of delays and reaction-diffusion on the dynamic behaviors of coupled systems cannot be neglected in modeling. Exponential stability is one of the most important dynamical properties of coupled systems. The Razumikhin methods is one of the most effective method to study the stability of coupled systems.Compared with the Lyapunov functional method, the imposed conditions of Razumikhin method have fewer restrictions. A coupled system can be described by a digraph. Therefore, utilizing graph theory and Razumikhin method, we investigate the exponential stability of coupled reaction-diffusion systems in this paper.In section 2, the exponential stability of coupled reaction-diffusion systems with mixed delays is discussed. In this system, time delays are time-varying. Not only discrete time-varying delay, but also distributed delay is taken into account in this system. Based on graph theory and Razumikin method, Razumikhin-type theorem will be provided firstly. Then the Lyapunov-type theorem can be obtained easily.Furthermore, based on the Lyapunov-type theorem, a coefficient-type theorem will be presented to judge more convenient. In the end of this section, a numerical example is given to illustrate the effectiveness and correctness of the developed theory.In section 3, the exponential stability of coupled reaction-diffusion multi-group model with delays and multi-dispersal is discussed. In this multi-group model, dispersal schema among multiple groups and each single group is affected by time delays and reaction-diffusion. Based on graph theory and Razumikin method, Razumikhin-type theorem will be provided firstly. Then the Lyapunov-type theorem can be obtained easily. To judge more convenient, based on the Lyapunov-type theorem, a coefficient-type theorem will be presented. Finally, a numerical example is given to illustrate the effectiveness of the developed theory.