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In this article, an optimal control problem (OCP) for a system governed by a class of double obstacle elliptic variational inequalities on a mixed boundary is proposed. An approximating optimal control problemis then defined to approximate the original problem (OCP) by a penalty method. Under some assumptions, the existence result for the problemis proved. The sequence of solutions to the problemconverges to a solution of the original problem (OCP) as the penalty parameter tends to zero. Next, the necessary optimality conditions are obtained for the solution of the problemwith a given concrete penalty function. At last, an algorithm is used to solve a given concrete optimal control problem. Numerical experimental results are presented to show that the approximation method is effective and practical.

An Approximating Approach to an Optimal Control Problem for an Elliptic Variational Inequality on a Mixed Boundary ⋆