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.


Hemivariational inequality Optimality system Necessary optimality condition Obstacle 

Mathematics Subject Classification

47J20 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.


  1. 1.
    Tiba, D.: Optimal Control of Nonsmooth Distributed Parameter Systems. Lecture Notes in Mathematics, vol. 1459. Springer, Berlin (1990)CrossRefMATHGoogle Scholar
  2. 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
  3. 3.
    Barbu, V.: Analysis and Control of Nonlinear Infinite Dimensional Systems. Academic Press, San Diego (1993)MATHGoogle Scholar
  4. 4.
    Ito, K., Kunisch, K.: Optimal control of elliptic variational inequalities. Appl. Math. Optim. 41, 343–364 (2000)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bergounioux, M.: Optimal control of problems governed by abstract elliptic variational inequalities with state constraints. SIAM J. Control Optim. 36, 273–289 (1998)MathSciNetCrossRefMATHGoogle Scholar
  6. 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
  7. 7.
    Ito, K., Kunisch, K.: Optimal control of parabolic variational inequalities. J. Math. Pures Appl. 93, 329–360 (2010)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Rindler, F.: Optimal control for nonconvex rate-independent evolution processes. SIAM J. Control Optim. 47, 2773–2794 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications: Convex and Nonconvex Energy Functions. Birkhäuser, Basel (1985)CrossRefMATHGoogle Scholar
  10. 10.
    Panagiotopoulos, P.D.: Hemivariational Inequalities: Applications in Mechanics and Engineering. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  11. 11.
    Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker, New York (1995)MATHGoogle Scholar
  12. 12.
    Carl, S., Le, V.K., Motreanu, D.: Nonsmooth Variational Problems and Their Inequalities: Comparison Principles and Applications. Springer, New York (2007)CrossRefMATHGoogle Scholar
  13. 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
  14. 14.
    Peng, Z.: Existence and regularity results for doubly nonlinear inclusions with nonmonotone perturbation. Nonlinear Anal. 115, 71–88 (2015)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gasiński, L., Liu, Z., Migórski, S., Ochal, A., Peng, Z.: Hemivariational inequality approach to evolutionary constrained problems on star-shaped sets. J. Optim. Theory Appl. 164, 514–533 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Lahmdani, A., Chadli, O., Yao, J.C.: Existence of solutions for noncoercive hemivariational inequalities by an equilibrium approach under pseudomonotone perturbation. J. Optim. Theory Appl. 160, 49–66 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Peng, Z., Liu, Z., Liu, X.: Boundary hemivariational inequality problems with doubly nonlinear operators. Math. Ann. 356, 1339–1358 (2013)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Miettinen, M., Haslinger, J.: Approximation of optimal control problems of hemivariational inequalities. Numer. funct. Anal. Optimiz. 13, 43–68 (1992)MathSciNetCrossRefMATHGoogle Scholar
  19. 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. 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. 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. 22.
    Miettinen, M.: Approximation of Hemivariational Inequalities and Optimal Control Problems. University of Jyväskylä Department of Mathematics Report, vol. 59 (1993)Google Scholar
  23. 23.
    Gasiński, L.: Optimal shape design problems for a class of systems described by parabolic hemivariational inequality. J. Global Optim. 12, 299–317 (1998)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Denkowski, Z., Migórski, S.: Optimal shape design problems for a class of systems described by hemivariational inequalities. J. Global Optim. 12, 37–59 (1998)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Migórski, S., Ochal, A.: Optimal control of parabolic hemivariational inequalities. J. Global Optim. 17, 285–300 (2000)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Migorski, S.: Evolution hemivariational inequalities in infinite dimension and their control. Nonlinear Anal. 47, 101–112 (2001)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Liu, Z., Zeng, B.: Optimal control of generalized quasi-variational hemivariational inequalities and its applications. Appl. Math. Optim. 72, 305–323 (2015)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Park, J.Y., Jeong, J.U.: Optimal control of hemivariational inequalities with delays. Taiwan. J. Math. 15, 433–447 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Denkowski, Z., Migórski, S., Ochal, A.: A class of optimal control problems for piezoelectric frictional contact models. Nonlinear Anal. Real World Appl. 12, 1883–1895 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 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
  31. 31.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)MATHGoogle Scholar
  32. 32.
    Zeidler, E.: Nonlinear Functional Analysis and Applications. II A/B. Springer, New York (1990)MATHGoogle Scholar
  33. 33.
    Chang, K.C.: Variational methods for non-differentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Goeleven, D., Motreanu, D., Dumont, Y., Rochdi, M.: Variational and Hemivariational Inequalities: Theory Methods and Applications. Kluwer, Boston (2003)MATHGoogle Scholar

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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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