Exponential input-to-state stability of delay reaction-diffusion systems

This brief paper considers the exponential input-to-state stability (EISS) for delay reaction-diffusion systems (DRDSs). The distributed input and boundary input are both included in the considered model. Boundary input is an important characteristic for DRDSs which are a kind of partial differential systems. Using Lyapunov–Krasovskii functional method and Wirtinger-type inequality, a delay-dependent sufficient condition is obtained to ensure the EISS of DRDSs. Besides the effect of time delay on the EISS, this sufficient condition also shows the effect of the diffusion term on the EISS and this is a significant property of reaction–diffusion systems differing from the ordinary differential systems. At last, numerical examples are provided to show the effectiveness of the theoretical results.