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.
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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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