  In this article, global stabilization results for the
Benjamin-Bona-Mahony-Burgers' (BBM-B) type equations are obtained using
nonlinear Neumann boundary feedback control laws. Based on the $C^0$-conforming
finite element method, global stabilization results for the semidiscrete
solution are also discussed. Optimal error estimates in $L^\infty(L^2)$,
$L^\infty(H^1)$ and $L^\infty(L^\infty)$-norms for the state variable are
derived, which preserve exponential stabilization property. Moreover, for the
first time in the literature, superconvergence results for the boundary
feedback control laws are established. Finally, several numerical experiments
are conducted to confirm our theoretical findings.
