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 (µ) = µu g1 + (1 − µ)u g2 and u 4 (µ) = u µg1+(1−µ)g2 for µ ∈ [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 (µ) and u 4 (µ) 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.Key words: Elliptic variational inequalities, convex combinations of the solutions, distributed optimal control problems, convergence of the optimal controls, obstacle problem, free boundary problems.
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