Convergence of distributed optimal control problems governed by elliptic variational inequalities

First, let u <Subscript> g </Subscript> be the unique solution of an elliptic variational inequality with source term g. We establish, in the general case, the error estimate between <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$u_{3}(\mu)=\mu u_{g_{1}}+ (1-\mu)u_{g_{2}}$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$u_{4}(\mu)=u_{\mu g_{1}+ (1-\mu) g_{2}}$</EquationSource> </InlineEquation> 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 <Subscript>3</Subscript>(μ) and u <Subscript>4</Subscript>(μ) given in Mignot (J. Funct. Anal. 22:130–185, <CitationRef CitationID="CR29">1976</CitationRef>), 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 Gariboldi and Tarzia (Appl. Math. Optim. 47:213–230, <CitationRef CitationID="CR10">2003</CitationRef>), for optimal control problems governed by elliptic variational equalities. Copyright Springer Science+Business Media, LLC 2012