Optimal Control of Problems Governed by Obstacle

In this paper we investigate optimal control problems governed by variational inequalities. We give optimality conditions under classical assumptions using a dual regularized functional to interpret the Variational Inequality. 1. Introduction. In this paper we investigate optimal control problems governed by vari-ational inequalities of obstacle type. This problem has been widely studied during the last years by many authors. It is now known that one cannot obtain classical optimality systems (in the sense of Mathematical Programming) for such problems. This come essentially from the fact that the mapping S which associates the state y solution of a Variational Inequality to the control v, is not diierentiable as pointed it out Mignot 9] and one can only deene a conical derivative for S. In 10], Mignot and Puel obtain optimality conditions using the results of 9]. Diierent methods have been used to consider this problem. Barbu 2, 3] studies approximations of the Variational Inequality which lead to optimal control problems governed by variational equations. Then he gets existence results and optimality conditions using a passage to the limit in the approximation process. In 5, 6], the rst author has obtained classical optimality systems for suitable approximations of the original problem which can be easily used from the numerical point of view. On the other hand, Rubio and Wenbin 14] obtain results for strongly monotone variational inequalities of obstacle type introducing a dual penalization for the variational inequality on increasing radius balls. We have adopted these last authors point of view to interpret the 1