  In this paper, we revisit the $L_2$-norm error estimate for $C^0$-interior
penalty analysis of Dirichlet boundary control problem governed by biharmonic
operator. In this work, we have relaxed the interior angle condition of the
domain from $120$ degrees to $180$ degrees, therefore this analysis can be
carried out for any convex domain. The theoretical findings are illustrated by
numerical experiments. Moreover, we propose a new analysis to derive the error
estimates for the biharmonic equation with Cahn-Hilliard type boundary
condition under minimal regularity assumption.
