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Strong Stationarity for Optimal Control of Variational Inequalities of the Second Kind

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Book cover Non-Smooth and Complementarity-Based Distributed Parameter Systems

Part of the book series: International Series of Numerical Mathematics ((ISNM,volume 172))

Abstract

This chapter is concerned with necessary optimality conditions for optimal control problems governed by variational inequalities of the second kind. The so-called strong stationarity conditions are derived in an abstract framework. Strong stationarity conditions are regarded as the most rigorous ones, since they imply all other types of stationarity concepts and are equivalent to purely primal optimality conditions. The abstract framework is afterward applied to four application-driven examples.

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Acknowledgements

The research of this work was carried out in Project P16 (Optimal Control of Variational Inequalities of the Second Kind with Application to Yield Stress Fluids) within the DFG Priority Program SPP 1962 (Non-smooth and Complementarity-Based Distributed Parameter Systems: Simulation and Hierarchical Optimization). The support by the DFG is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, TU München, Garching bei München, Germany

    Constantin Christof

  2. Fakultät für Mathematik, TU Dortmund, Dortmund, Germany

    Christian Meyer, Ben Schweizer & Stefan Turek

Authors
  1. Constantin Christof
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  2. Christian Meyer
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  3. Ben Schweizer
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  4. Stefan Turek
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Corresponding author

Correspondence to Christian Meyer .

Editor information

Editors and Affiliations

  1. Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany

    Michael Hintermüller

  2. Institute for Applied Mathematics, University of Heidelberg, Heidelberg, Germany

    Roland Herzog

  3. Lehrstuhl für Mathematik VII, Julius-Maximilians-Universität Würzburg, Würzburg, Germany

    Christian Kanzow

  4. Department of Mathematics, Technical University of Munich, Garching b. München, Germany

    Michael Ulbrich

  5. Fachbereich Mathematik, Technische Universitaet Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

Appendix A: Auxiliary Results

Appendix A: Auxiliary Results

Lemma A.1

Under Assumption 2.3 , U is dense in .

Proof

Let us assume that U is not a dense subset of so that there exists a . Then, the strict separation theorem in combination with the reflexivity of implies the existence of a , v ≠ 0, such that

Since and the embedding is injective, this yields v = 0, which is a contradiction. □

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Christof, C., Meyer, C., Schweizer, B., Turek, S. (2022). Strong Stationarity for Optimal Control of Variational Inequalities of the Second Kind. In: Hintermüller, M., Herzog, R., Kanzow, C., Ulbrich, M., Ulbrich, S. (eds) Non-Smooth and Complementarity-Based Distributed Parameter Systems. International Series of Numerical Mathematics, vol 172. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-79393-7_12

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