Analysis and Numerical Study of Boundary control of generalized Burgers-Huxley equation

Abstract

In this work, a boundary control problem for the following generalized Burgers-Huxley (GBH) equation: $$u_t=\nu u_{xx}-\alpha u^{\delta}u_x+\beta u(1-u^{\delta})(u^{\delta}-\gamma), $$ where $\nu,\alpha,\beta>0,$ $1\leq\delta<\infty$, $\gamma\in(0,1)$ subject to Neumann boundary conditions is analyzed. Using the Minty-Browder theorem, standard elliptic partial differential equations theory, the maximum principle and the Crandall-Liggett theorem, we first address the global existence of a unique strong solution to GBH equation. Then, for the boundary control problem, we prove that the controlled GBH equation (that is, the closed loop system) is exponentially stable in the $\mathrm{H}^1$-norm (hence pointwise) when the viscosity $\nu$ is known (non-adaptive control). Moreover, we show that a damped version of GBH equation is globally asymptotically stable (in the $\mathrm{L}^2$-norm), when $\nu$ is unknown (adaptive control). Using the Chebychev collocation method with the backward Euler method as a temporal scheme, numerical findings are reported for both the non-adaptive and adaptive situations, supporting and confirming the analytical results of both the controlled and uncontrolled systems.