Convergence of Distributed Optimal Control Problems Governed by Elliptic Variational Inequalities

First, let $u_{g}$ be the unique solution of an elliptic variational inequality with source term $g$. We establish, in the general case, the error estimate between $u_{3}(\mu)=\mu u_{g_{1}}+ (1-\mu)u_{g_{2}}$ %(the convex combination of two solutions) and $u_{4}(\mu)=u_{\mu g_{1}+ (1-\mu) g_{2}}$ %(the solution corresponding to the convex combination of two data) for $\mu\in [0, 1]$. Secondly, we consider a family of distributed optimal control problems governed by elliptic variational inequalities over the internal energy $g$ for each positive heat transfer coefficient $h$ given on a part of the boundary of the domain. For a given cost functional and using some monotony property between $u_{3}(\mu)$ and $u_{4}(\mu)$ given in F. Mignot, J. Funct. Anal., 22 (1976), 130-185, we prove the strong convergence of the optimal controls and states associated to this family of distributed optimal control problems governed by elliptic variational inequalities to a limit Dirichlet distributed optimal control problem, governed also by an elliptic variational inequality, when the parameter $h$ goes to infinity. We obtain this convergence without using the adjoint state problem (or the Mignot's conical differentiability) which is a great advantage with respect to the proof given in C.M. Gariboldi - D.A. Tarzia, Appl. Math. Optim., 47 (2003), 213-230, for optimal control problems governed by elliptic variational equalities.