Optimal Control of Elliptic Variational–Hemivariational Inequalities

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Abstract

This paper deals with the optimality system of an optimal control problem governed by a nonlinear elliptic inclusion and a nonsmooth cost functional. The system describing the state consists of a variational–hemivariational inequality, the solution mapping of which with respect to the control is proved to be weakly closed. Existence of optimal pairs for the optimal control problem is obtained. Approximation results and abstract necessary optimality conditions of first order are derived based on the adapted penalty method and nonsmooth analysis techniques. Moreover, the optimality system for a class of obstacle problems with nonmonotone perturbation is given.

Keywords

Hemivariational inequality Optimality system Necessary optimality condition Obstacle 

Mathematics Subject Classification

47J20 49J20 49J40 49K20 

Notes

Acknowledgements

The authors would like to thank the reviewers for their useful suggestions which improve the presentation of the manuscript. This work was carried out while Z. Peng was a visiting associate professor at the Institute for Mathematics and Scientific Computing, University of Graz, Austria. Z. Peng was supported by NNSF of China Grant 11561007 and the Special Funds of Guangxi Distinguished Experts Construction Engineering. K. Kunisch was supported by the ERC Advanced Grant 668998 (OCLOC) under the EUs H2020 research program.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Guangxi Key Laboratory of Universities Optimization Control and Engineering Calculation, and College of SciencesGuangxi University for NationalitiesNanningPeople’s Republic of China
  2. 2.Institut für MathematikKarl-Franzens-Universität GrazGrazAustria

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