Feedback Stabilizability of a generalized Burgers-Huxley equation with kernel around a non-constant steady state

Abstract: In this article, we investigate a generalized Burgers-Huxley equation with a smooth kernel defined in a bounded domain $\Omega\subset\mathbb{R}^d$, $d\in{1,2,3}$, focusing on feedback stabilizability around a non-constant steady state. Initially, employing the Banach fixed point theorem, we establish the local existence and uniqueness of a strong solution, which is subsequently extended globally using an energy estimate. To analyze stabilizability, we linearize the model around a non-constant steady state and examine the stabilizability of the principal system. For the principal system, we develop a feedback control operator by solving an appropriate algebraic Riccati equation. This allows for the construction of both finite and infinite-dimensional feedback operators. By applying this feedback operator and establishing necessary regularity results, we utilize the Banach fixed point theorem to demonstrate the stabilizability of the entire system. Furthermore, we also explore the stabilizability of the model problem around the zero steady state and validate our findings through numerical simulations using the finite element method for both zero and non-constant steady states.