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On optimality conditions in control of elliptic variational inequalities. (English) Zbl 1211.49036

Summary: In the paper we consider optimal control of a class of strongly monotone variational inequalities, whose solution map is directionally differentiable in the control variable. This property is used to derive sharp pointwise necessary optimality conditions provided we do not impose any control or state constraints. In presence of such constraints we make use of the generalized differential calculus and derive, under a mild constraint qualification, optimality conditions in a “fuzzy” form. For strings, these conditions may serve as an intermediate step toward pointwise conditions of limiting (Mordukhovich) type and in the case of membranes they lead to a variant of Clarke stationarity conditions.

MSC:

49K27 Optimality conditions for problems in abstract spaces
90C33 Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
49J40 Variational inequalities
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