On Existence, Uniqueness, and Convergence of Optimal Control Problems Governed by Parabolic Variational Inequalities

Abstract : I) We consider a system governed by a free boundary problem with Tresca condition on a part of the boundary of a material domain with a source term g through a parabolic variational inequality of the second kind. We prove the existence and uniqueness results to a family of distributed optimal control problems over g for each parameter h > 0, associated to the Newton law (Robin boundary condition), and of another distributed optimal control problem associated to a Dirichlet boundary condition. We generalize for parabolic variational inequalities of the second kind the Mignot’s inequality obtained for elliptic variational inequalities (Mignot, J. Funct. Anal., 22 (1976), 130-185), and we obtain the strictly convexity of a quadratic cost functional through the regularization method for the non-differentiable term in the parabolic variational inequality for each parameter h. We also prove, when h → + ∞, the strong convergence of the optimal controls and states associated to this family of optimal control problems with the Newton law to that of the optimal control problem associated to a Dirichlet boundary condition.II) Moreover, if we consider a parabolic obstacle problem as a system governed by a parabolic variational inequalities of the first kind then we can also obtain the same results of Part I for the existence, uniqueness and convergence for the corresponding distributed optimal control problems.III) If we consider, in the problem given in Part I, a flux on a part of the boundary of a material domain as a control variable (Neumann boundary optimal control problem) for a system governed by a parabolic variational inequality of second kind then we can also obtain the existence and uniqueness results for Neumann boundary optimal control problems for each parameter h > 0, but in this case the convergence when h → + ∞ is still an open problem.
Type de document :
Communication dans un congrès
Dietmar Hömberg; Fredi Tröltzsch. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. Springer, IFIP Advances in Information and Communication Technology, AICT-391, pp.76-84, 2013, System Modeling and Optimization. 〈10.1007/978-3-642-36062-6_8〉
Liste complète des métadonnées

Littérature citée [20 références]  Voir  Masquer  Télécharger

https://hal.inria.fr/hal-01347525
Contributeur : Hal Ifip <>
Soumis le : jeudi 21 juillet 2016 - 11:15:18
Dernière modification le : lundi 16 octobre 2017 - 01:14:29

Fichier

978-3-642-36062-6_8_Chapter.pd...
Fichiers produits par l'(les) auteur(s)

Licence


Distributed under a Creative Commons Paternité 4.0 International License

Identifiants

Citation

Mahdi Boukrouche, Domingo Tarzia. On Existence, Uniqueness, and Convergence of Optimal Control Problems Governed by Parabolic Variational Inequalities. Dietmar Hömberg; Fredi Tröltzsch. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. Springer, IFIP Advances in Information and Communication Technology, AICT-391, pp.76-84, 2013, System Modeling and Optimization. 〈10.1007/978-3-642-36062-6_8〉. 〈hal-01347525〉

Partager

Métriques

Consultations de la notice

103

Téléchargements de fichiers

35