  We consider an optimal control problem $\cQ$ governed by an elliptic
quasivariational inequality with unilateral constraints. The existence of
optimal pairs of the problem is a well known result, see \cite{SS}, for
instance. We associate to $\cQ$ a new optimal control problem $\wQ$, obtained
by perturbing the state inequality (including the set of constraints and the
nonlinear operator) and the cost functional, as well. Then, we provide
sufficient conditions which guarantee the convergence of solutions of Problem
$\wQ$ to a solution of Problem $\cQ$. The proofs are based on convergence
results for elliptic quasivariational inequalities, obtained by using arguments
of compactness, lower semicontinuity, monotonicity, penalty and various
estimates. Finally, we illustrate the use of the abstract convergence results
in the study of optimal control associated with two boundary value problems.
The first one describes the equilibrium of an elastic body in frictional
contact with an obstacle, the so-called foundation. The process is static and
the contact is modeled with normal compliance and unilateral constraint,
associated to a version of Coulomb's law of dry friction. The second one
describes a stationary heat transfer problem with unilateral constraints. For
the two problems we prove existence, uniqueness and convergence results
together with the corresponding physical interpretation.
