Optimal control of elliptic variational inequalities with bounded and unbounded operators

Livia Betz; Irwin Yousept

doi:10.3934/mcrf.2021009

Article Contents

2021, Volume 11, Issue 3: 479-498. Doi: 10.3934/mcrf.2021009

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Optimal control of elliptic variational inequalities with bounded and unbounded operators

Livia Betz and
Irwin Yousept^,

University of Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, Essen, 45127, Germany

^* Corresponding author: irwin.yousept@uni-due.de
^* Corresponding author: irwin.yousept@uni-due.de

Received: October 31, 2018

Revised: October 31, 2019

Early access: March 2021

Published: September 2021

This work was supported by the DFG under the grant YO 159/2-2 within the priority programme SPP 1962 'Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization'

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Abstract

This paper examines optimal control problems governed by elliptic variational inequalities of the second kind with bounded and unbounded operators. To tackle the bounded case, we employ the polyhedricity of the test set appearing in the dual formulation of the governing variational inequality. Based thereon, we are able to prove the directional differentiability of the associated solution operator, which leads to a strong stationary optimality system. The second part of the paper deals with the unbounded case. Due to the non-smoothness of the variational inequality and the unboundedness of the governing elliptic operator, the directional differentiability of the solution operator becomes difficult to handle. Our strategy is to apply the Yosida approximation to the unbounded operator, while the non-smoothness of the variational inequality is still preserved. Based on the developed strong stationary result for the bounded case, we are able to derive optimality conditions for the unbounded case by passing to the limit in the Yosida approximation. Finally, we apply the developed results to Maxwell-type variational inequalities arising in superconductivity.

Keywords:

Variational inequality of the second kind,
directional differentiability,
strong stationarity,
Yosida approximation,
Maxwell variational inequality.

Mathematics Subject Classification: 35J86.

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