Optimal Control of Elliptic Variational–Hemivariational Inequalities
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.
KeywordsHemivariational inequality Optimality system Necessary optimality condition Obstacle
Mathematics Subject Classification47J20 49J20 49J40 49K20
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.
- 2.Tröltzsch, F.: Optimal control of partial differential equations: theory, methods and applications. In: Graduate Studies in Mathematics, vol. 112. AMS Providence, Rhode Island (2010)Google Scholar
- 6.Ito, K., Kunisch, K.: On the lagrange multiplier approach to variational problems and applications. In: Monographs and Studies in Mathematics, vol. 24. SIAM, Philadelphia (2008)Google Scholar
- 13.Migórski, S., Ochal, A., Sofonea, M.: Nonlinear inclusions and hemivariational inequalities. Models and analysis of contact problems. In: Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)Google Scholar
- 19.Haslinger, J., Panagiotopoulous, P.D.: Optimal control of hemivariational inequalities: approximation results. In: ISNM 99, pp. 165–173. Birkhäuser, Basel (1991)Google Scholar
- 20.Panagiotopoulous, P.D.: Optimal control of systems governed by variational–hemivariational inequalities. In: ISNM 101, pp. 161–181. Birkhäuser, Basel (1991)Google Scholar
- 21.Panagiotopoulous, P.D.: Optimal control of systems governed by hemivariational inequalities. Necessary conditions. In: ISNM 95, pp. 207–228. Birkhäuser, Basel (1990)Google Scholar
- 22.Miettinen, M.: Approximation of Hemivariational Inequalities and Optimal Control Problems. University of Jyväskylä Department of Mathematics Report, vol. 59 (1993)Google Scholar
- 30.Sofonea, M.: Optimal control of a class of variational–hemivariational inequalities in reflexive Banach spaces. Appl. Math. Optim. (2017). https://doi.org/10.1007/s00245-017-9450-0