Optimal control of obstacle problems : existence of Lagrange multipliers

We study rst 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 point- wise constraint a multiplier to get a \good" optimality system? We give positive and negative answers for the nite and innite dimensional cases. These results are compared with the previous ones got by penalization or dierentiation. R esum e .O netudie le probl eme des syst emes d'optimalit e du premier ordre pour le contr^ ole des syst emes gouvern es par une in equation variationnelle, et qui prennent en compte les multiplicateurs de Lagrange que l'on peut associer a chaque contrainte ponctuelle. Le probl eme est etudi e en dimension nie puis innie : suivant les cas des r eponses positives ou n egatives sont donn ees. Ces r esultats sont compar es a ceux obtenus par p enalisation ou di erentiation. AMS Subject Classication. 49J40, 49K20, 49K35.

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