Applied Mathematics & Optimization

, Volume 79, Issue 3, pp 621–646 | Cite as

Optimal Control of a Class of Variational–Hemivariational Inequalities in Reflexive Banach Spaces

  • Mircea SofoneaEmail author


The present paper represents a continuation of Migórski et al. (J Elast 127:151–178, 2017). There, the analysis of a new class of elliptic variational–hemivariational inequalities in reflexive Banach spaces, including existence and convergence results, was provided. An inequality in the class is governed by a nonlinear operator, a convex set of constraints and two nondifferentiable functionals, among which at least one is convex. In the current paper we complete this study with new results, including a convergence result with respect the set of constraints. Then we formulate two optimal control problems for which we prove the existence of optimal pairs, together with some convergence results. Finally, we exemplify our results in the study of a one-dimensional mathematical model which describes the equilibrium of an elastic rod in unilateral contact with a foundation, under the action of a body force.


Variational–hemivariational inequality Clarke subdifferential Weak convergence Optimal pair Optimal control Elastic rod Contact problem 

Mathematics Subject Classification

47J20 47J22 49J53 74M15 


  1. 1.
    Baiocchi, C., Capelo, A.: Variational and Quasivariational Inequalities: Applications to Free-Boundary Problems. Wiley, Chichester (1984)zbMATHGoogle Scholar
  2. 2.
    Barbu, V.: Optimal Control of Variational Inequalities. Pitman, Boston (1984)zbMATHGoogle Scholar
  3. 3.
    Brézis, H.: Equations et inéquations non linéaires dans les espaces vectoriels en dualité. Ann. Inst. Fourier (Grenoble) 18, 115–175 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Brézis, H.: Problèmes unilatéraux. J. Math. Pures Appl. 51, 1–168 (1972)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Capatina, A.: Variational Inequalities Frictional Contact Problems, Advances in Mechanics and Mathematics, vol. 31. Springer, New York (2014)zbMATHGoogle Scholar
  6. 6.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  7. 7.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Theory. Kluwer Academic/Plenum Publishers, Boston/London (2003)CrossRefzbMATHGoogle Scholar
  8. 8.
    Denkowski, Z., Migórski, S., Papageorgiou, N.S.: An Introduction to Nonlinear Analysis: Applications. Kluwer Academic/Plenum Publishers, Boston/London (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    Eck, C., Jarušek, J., Krbeč, M.: Unilateral Contact Problems: Variational Methods and Existence Theorems, Pure and Applied Mathematics 270. Chapman/CRC Press, New York (2005)CrossRefzbMATHGoogle Scholar
  10. 10.
    Han, W., Sofonea, M.: Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics 30. Americal Mathematical Society/International Press, Providence/Somerville (2002)CrossRefzbMATHGoogle Scholar
  11. 11.
    Haslinger, J., Miettinen, M., Panagiotopoulos, P.D.: Finite Element Method for Hemivariational Inequalities. Methods and Applications. Kluwer Academic Publishers, Boston, Theory (1999)CrossRefzbMATHGoogle Scholar
  12. 12.
    Hlaváček, I., Haslinger, J., Necǎs, J., Lovíšek, J.: Solution of Variational Inequalities in Mechanics. Springer, New York (1988)CrossRefzbMATHGoogle Scholar
  13. 13.
    Kikuchi, N., Oden, J.T.: Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods. SIAM, Philadelphia (1988)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and their Applications, Classics in Applied Mathematics 31. SIAM, Philadelphia (2000)CrossRefzbMATHGoogle Scholar
  15. 15.
    Liu, Z.H., Zeng, B.: Optimal control of generalized quasi-variational hemivariational inequalities and its applications. Appl. Math. Optim. 72, 305–323 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Martins, J.A.C., Marques, M.D.P.M. (eds.): Contact Mechanics. Kluwer, Dordrecht (2002)Google Scholar
  17. 17.
    Matei, A., Micu, S.: Boundary optimal control for nonlinear antiplane problems. Nonlinear Anal. 74, 1641–1652 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Matei, A., Micu, S.: Boundary optimal control for a frictional contact problem with normal compliance. Appl. Math. Optim. 1–23 (2017)Google Scholar
  19. 19.
    Migórski, S., Ochal, A., Sofonea, M.: Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, vol. 26. Springer, New York (2013)Google Scholar
  20. 20.
    Migórski, S., Ochal, A., Sofonea, M.: History-dependent variational-hemivariational inequalities in contact mechanics. Nonlinear Anal. Real World Appl. 22, 604–618 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Migórski, S., Ochal, A., Sofonea, M.: A class of variational-hemivariational inequalities in reflexive banach spaces. J. Elast. 127, 151–178 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Naniewicz, Z., Panagiotopoulos, P.D.: Mathematical Theory of Hemivariational Inequalities and Applications. Marcel Dekker Inc, New York, Basel, Hong Kong (1995)zbMATHGoogle Scholar
  23. 23.
    Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications. Birkhäuser, Boston (1985)CrossRefzbMATHGoogle Scholar
  24. 24.
    Panagiotopoulos, P.D.: Hemivariational Inequalities. Applications in Mechanics and Engineering. Springer, Berlin (1993)CrossRefzbMATHGoogle Scholar
  25. 25.
    Shillor, M., Sofonea, M., Telega, I.J.: Models and Analysis of Quasistatic Contact. Lecture Notes in Physics. Springer, Berlin (2004)CrossRefzbMATHGoogle Scholar
  26. 26.
    Sofonea, M., Matei, A.: Mathematical Models in Contact Mechanics. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge (2012)CrossRefzbMATHGoogle Scholar
  27. 27.
    Sofonea, M., Migórski, S.: Variational-Hemivariational Inequalities with Applications. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2017)CrossRefzbMATHGoogle Scholar
  28. 28.
    Sofonea, M., Han, W., Migórski, S.: Numerical analysis of history-dependent variational inequalities with applications to contact problems. Eur. J. Appl. Math. 26, 427–452 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tiba, D.: Optimal Control of Nonsmooth Distributed Parameter Systems. Springer, Berlin (1990)CrossRefzbMATHGoogle Scholar
  30. 30.
    Zeidler, E.: Nonlinear Functional Analysis and Applications II A/B. Springer, New York (1990)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et PhysiqueUniversité de Perpignan Via DomitiaPerpignanFrance

Personalised recommendations