On Existence, Uniqueness, and Convergence of Optimal Control Problems Governed by Parabolic Variational Inequalities - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Communication Dans Un Congrès Année : 2013

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

Résumé

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.
Fichier principal
Vignette du fichier
978-3-642-36062-6_8_Chapter.pdf (4 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

Dates et versions

hal-01347525 , version 1 (21-07-2016)

Licence

Paternité

Identifiants

Citer

Mahdi Boukrouche, Domingo A. Tarzia. On Existence, Uniqueness, and Convergence of Optimal Control Problems Governed by Parabolic Variational Inequalities. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. pp.76-84, ⟨10.1007/978-3-642-36062-6_8⟩. ⟨hal-01347525⟩
87 Consultations
77 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More