Boundary control of generalized Korteweg-de Vries-Burgers-Huxley equation: Well-Posedness, Stabilization and Numerical Studies

Abstract

A boundary control problem for the following generalized Korteweg-de Vries-Burgers-Huxley equation: $$u_t=\nu u_{xx}-\mu u_{xxx}-\alpha u^{\delta }{u}_x+\beta u(1-u^{\delta })(u^{\delta }-\gamma ),x\in [0,1],t>0,$$ where $\nu ,\mu ,\alpha ,\beta >0,$ $\delta \in [1,\infty )$, $\gamma \in (0,1)$ subject to Neumann boundary conditions is considered in this work. We first establish the well-posedness of the Neumann boundary value problem by an application of monotonicity arguments, the Hartman-Stampacchia theorem, the Minty-Browder theorem, and the Crandall-Liggett theorem. The additional difficulties caused by the third order linear term is successfully handled by proving a proper version of the Minty-Browder theorem. By using suitable feedback boundary controls, we demonstrate ${\mathrm{L}}^2$- and ${\mathrm{H}}^1$-stability properties of the closed-loop system for sufficiently large $\nu >0$. The analytical conclusions from this work are supported and validated by numerical investigations.