Fixed-time boundary stabilisation for delay reaction–diffusion systems

Abstract

This paper considers the problem of the boundary fixed-time stabilisation for delay reaction–diffusion systems (DRDSs). A new boundary controller is designed to achieve fixed time stability of DRDSs. Firstly, a boundary fixed-time controller is presented for DRDSs. Then based on the designed boundary controller, using Lyapunov functional method, fixed-time stability lemma and Poincaré's inequality, we obtain the criteria to guarantee the fixed-time stability of DRDSs. In addition, boundary fixed-time controller is proposed for uncertain DRDSs to reach robust fixed-time stability. Both the cases, the systems with or without uncertainties, are severally followed by a fixed time, which is an upper-bound of the settling time and irrelevant to the initial conditions. Finally, two simulation examples are given to illustrate the fixed-time stability of the controlled systems, which show the validity of our theoretical results.