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.
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