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

On $C^{0}$ Interior Penalty Method for Fourth Order Dirichlet Boundary  Control Problem and a New Error Analysis for Fourth Order Elliptic Equation  with Cahn-Hilliard Boundary Condition