  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.
