Stabilization of the heat equation with disturbance at the flux boundary condition

Abstract

In this paper, we address the problem of boundary stabilization of the heat equation subjected to an unknown disturbance, which is assumed to be acting at the flux boundary condition. By means of Lyapunov techniques and the use of the sign multivalued operator, which is responsible of rejecting the effects of the disturbance, we design a multivalued feedback law to obtain the exponential stability of the closed‐loop system. The well‐posedness of the closed‐loop system, which is a differential inclusion, is shown with the maximal monotone operator theory.