Abstract

This thesis analyzes boundary control problems for a class of generalized nonlinear partial differential equations. We consider the generalized Burgers-Huxley (GBH) equation: $$ u_t = \nu u_{xx} - \alpha u^{\delta} u_x + \beta u(1-u^{\delta})(u^{\delta}-\gamma), $$ the generalized Korteweg-de Vries-Burgers-Huxley (GKdVBH) equation: $$ u_t = \nu u_{xx} - \mu u_{xxx} - \alpha u^{\delta} u_x + \beta u(1-u^{\delta})(u^{\delta}-\gamma), $$ and a damped viscous Burgers (DVB) equation: $$ u_t = \nu u_{xx} - \alpha u u_x - \beta |u|^\vartheta u, $$ where $\nu, \mu, \alpha, \beta > 0$, $\delta \in \mathbb{N}$, $\gamma \in (0,1)$ and $\vartheta \in [1,\infty)$. We establish the well-posedness of these equations using monotonicity arguments, the Minty-Browder theorem, the Hartman-Stampacchia theorem, and the Crandall-Liggett theorem. Challenges posed by the third-order term in the Korteweg-de Vries-Burgers-Huxley equation are addressed by proving a modified version of the Minty-Browder theorem. We prove the $L ^2$ and $\H ^1$ stability properties of the closed-loop system with suitable feedback boundary controls under conditions on viscosity $\nu$. Using the Chebychev collocation method for spatial discretization with the backward Euler method as a temporal scheme, numerical findings are reported, validating and confirming the analytical results for both controlled and uncontrolled systems across the studied equations.