# Convergence of distributed optimal control problems governed by elliptic variational inequalities

## Author Info

• Mahdi Boukrouche

()

• Domingo Tarzia

()

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## Abstract

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 $u_{3}(\mu)=\mu u_{g_{1}}+ (1-\mu)u_{g_{2}}$ and $u_{4}(\mu)=u_{\mu g_{1}+ (1-\mu) g_{2}}$ 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 Mignot (J. Funct. Anal. 22:130–185, 1976 ), 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, 2003 ), for optimal control problems governed by elliptic variational equalities. Copyright Springer Science+Business Media, LLC 2012

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## Bibliographic Info

Article provided by Springer in its journal Computational Optimization and Applications.

Volume (Year): 53 (2012)
Issue (Month): 2 (October)
Pages: 375-393

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Handle: RePEc:spr:coopap:v:53:y:2012:i:2:p:375-393

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Web page: http://www.springer.com/math/journal/10589

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## Related research

Keywords: 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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