  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.
