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On Optimality Conditions in Control of Elliptic Variational Inequalities

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Abstract

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.

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Authors and Affiliations

  1. Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou věží 4, 182 08, Praha 8, Czech Republic

    Jiří Outrata

  2. Institute of Mathematics, Academy of Sciences of the Czech Republic, Žitná 25, 115 67, Praha 1, Czech Republic

    Jiří Jarušek

  3. Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 18600, Praha 8, Czech Republic

    Jana Stará

Authors
  1. Jiří Outrata
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  2. Jiří Jarušek
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  3. Jana Stará
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Corresponding author

Correspondence to Jiří Outrata.

Additional information

The work presented here was partially supported by the Czech Academy of Sciences under grant IAA 100750802 and under the Institutional research plan AVOZ 10190503. The third author was supported by the Czech Grant Agency under the grant GACR 201/09/0917 and by the research project MSM0021620839.

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Outrata, J., Jarušek, J. & Stará, J. On Optimality Conditions in Control of Elliptic Variational Inequalities. Set-Valued Anal 19, 23–42 (2011). https://doi.org/10.1007/s11228-010-0158-4

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  • DOI: https://doi.org/10.1007/s11228-010-0158-4

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