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Issue ESAIM: COCV
Volume 5, 2000
Page(s) 45 - 70
DOI 10.1051/cocv:2000101

DOI: 10.1051/cocv:2000101

ESAIM: COCV, January 2000, Vol. 5, p. 45-70

Optimal Control of Obstacle Problems:
Existence of Lagrange Multipliers

Contrôle optimal de problèmes d'obstacles :
existence de multiplicateurs de Lagrange

Maïtine Bergounioux
Département de Mathématiques, UMR 6628, Université d'Orléans, BP. 6759, 45067 Orléans Cedex 2, France; (Maitine.Bergounioux@labomath.univ-orleans.fr)

Fulbert Mignot
Laboratoire de Mathématique, bâtiment 425, Université Paris-Sud, 91405 Orsay, France; (Fulbert.Mignot@math.u-psud.fr)

Received March 30, 1999. Revised November 9, 1999.

Abstract: We study first order optimality systems for the control of a system governed by a variational inequality and deal with Lagrange multipliers: is it possible to associate to each pointwise constraint a multiplier to get a "good'' optimality system? We give positive and negative answers for the finite and infinite dimensional cases. These results are compared with the previous ones got by penalization or differentiation.

Résumé: On étudie le problème des systèmes d'optimalité du premier ordre pour le contrôle des systèmes gouvernés par une inéquation variationnelle, et qui prennent en compte les multiplicateurs de Lagrange que l'on peut associer à chaque contrainte ponctuelle. Le problème est étudié en dimension finie puis infinie : suivant les cas des réponses positives ou négatives sont données. Ces résultats sont comparés à ceux obtenus par pénalisation ou différentiation.

Keywords and phrases: Variational inequalities, optimal control, Lagrange multiplier, obstacle problem.

AMS Subject Classification: 49J40, 49K20, 49K35.

Article without figures.

Copyright EDP Sciences, SMAI



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