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Convergence of distributed optimal control problems governed by elliptic variational inequalities

Author

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  • Mahdi Boukrouche

    ()

  • Domingo Tarzia

    ()

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

Suggested Citation

  • Mahdi Boukrouche & Domingo Tarzia, 2012. "Convergence of distributed optimal control problems governed by elliptic variational inequalities," Computational Optimization and Applications, Springer, vol. 53(2), pages 375-393, October.
  • Handle: RePEc:spr:coopap:v:53:y:2012:i:2:p:375-393
    DOI: 10.1007/s10589-011-9438-7
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