Observer-based feedback stabilization of a reaction-diffusion equation with variable coefficients and boundary input delay

In this paper, we consider the stabilization issues of a reaction-diffusion equation with variable coefficients and boundary input delay. At first, we design an observer based on the system output to estimate the state of the system. Due to the present of time delay in control, we design a dynamic feedback controller based on the state information of observer, that is called the integral-type controller. By selecting appropriate kernel functions, we prove that the closed-loop system is exponentially stable. Herein, our approach mainly is based on the idea of ‘feedback equivalence’. By some equivalence transformations, we establish connection between the closed-loop system and a stable system.