Journal of Global Optimization

, Volume 47, Issue 3, pp 421–435 | Cite as

Optimal control of a quasi-variational obstacle problem

  • Samir AdlyEmail author
  • Maïtine Bergounioux
  • Mohamed Ait Mansour


We consider an optimal control where the state-control relation is given by a quasi-variational inequality, namely a generalized obstacle problem. We give an existence result for solutions to such a problem. The main tool is a stability result, based on the Mosco-convergence theory, that gives the weak closeness of the control-to-state operator. We end the paper with some examples.


Optimal control Quasi-variational inequalities Mosco convergence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adly S., Ait Mansour M., Scrimali L.: Sensitivity analysis of solutions to a class of quasi-variational inequalities. Bollettino U.M.I. 8-B(8), 767–771 (2005)Google Scholar
  2. 2.
    Aubin, J.-P., Frankowska, H.: Set-Valued Analysis. Birkauser (1990)Google Scholar
  3. 3.
    Aussel F., Luc D.T.: Existence conditions in general quasimonotone variational inequalities. Bull. Aust. Math. Soc. 71(2), 285–303 (2005)CrossRefGoogle Scholar
  4. 4.
    Baiocchi C., Capelo A.: Variational and Quasi-Variational Inequalities. Wiley, New York (1984)Google Scholar
  5. 5.
    Barbu, V.: Analysis and control of non linear infinite dimensional systems. Math. Sci. Eng. 190. Academic Press, San Diego (1993)Google Scholar
  6. 6.
    Bensoussan A., Lions J.L.: Contrle impulsionnel et inéquations quasi-variationnelles. Dunod, Paris (1982)Google Scholar
  7. 7.
    Bergounioux M., Lenhart S.: Optimal control of bilateral obstacle. Siam J. Control Optim. 43(1), 240–255 (2004)CrossRefGoogle Scholar
  8. 8.
    Borwein J., Zhu Q.J.: Techniques of Variational Analysis, An Introduction. Springer, New York (2004)Google Scholar
  9. 9.
    Dietrich H.: A smooth dual gap function to a class of quasivariational inequalities. JMAA 235, 380–393 (1999)Google Scholar
  10. 10.
    Dietrich H.: Optimal control problems for certain quasivariational inequalities. Optimization 49, 67–93 (2001)CrossRefGoogle Scholar
  11. 11.
    Dontchev, A., Zolezzi, T.: Well Posed Optimization Problems. Lecture Notes in Mathematics, 1543. Springer (1993)Google Scholar
  12. 12.
    Duvaut, G., Lions, J.L.: Les inéquations en Mécanique et en physique. Dunod (1972); Inequalities Mechanics and Physics. Springer, Berlin (1976)Google Scholar
  13. 13.
    Flores Bazán F.: Existence theorems for generalized noncoercive equilibrium problems: The quasi-convex case. SIAM J. Optim. 3, 675–690 (2000)Google Scholar
  14. 14.
    Gianessi F., Maugeri A.: Variational Inequalities and Network Equilibrium Problems Applications. Plenum Press, New York (1995)Google Scholar
  15. 15.
    Granas A., Dugundji J.: Fixed Point Theory. Springer Monographs in Mathematics. Springer-Verlag, New York (2003)Google Scholar
  16. 16.
    Joly J.-L., Mosco U.: A propos de l’existence et de la régularité des solutions de certaines inéquations quasi-variationnelles. J. Funct. Anal. 34, 107–137 (1979)CrossRefGoogle Scholar
  17. 17.
    Lignola M.B.: Well posedness and L-wellposedness for quasivariational inequalities. J. Optim. Theory Appl. 128(1), 119–138 (2006)CrossRefGoogle Scholar
  18. 18.
    Lignola M.B., Morgan J.: Convergence of Solutions of Quasi-Variational Inequalities and Applications. Topol. Methods Nonlinear Anal. 10, 375–385 (1997)Google Scholar
  19. 19.
    Lions J.L.: Contrle optimal de systmes gouvernés par des équations aux dérivées partielles, vol. 1. Gauthier-Villars, Paris (1968)Google Scholar
  20. 20.
    Lions P.L.: Two remarks on the convergence of convex functions and monotone operators. Nonlinear Anal. 2, 553–562 (1978)CrossRefGoogle Scholar
  21. 21.
    Lions J.L., Magenes E.: Problmes aux limites non homognes et applications, vol. 1. Dunod, Paris (1968)Google Scholar
  22. 22.
    Sonntag, Y.: Convergence au sens de U. Mosco: théorie et application à l’approximation des solutions d’inéquations. Thèse, Université de Provence (1982)Google Scholar
  23. 23.
    Zeidler E.: Nonlinear Functional Analysis and Its Applications. Springer, Berlin (1992)Google Scholar

Further reading

  1. 24.
    Hukuhara, M.: Sur lexistence des points invariants dune transformation dans lespace fonctionnel. Jpn. J. Math. 20, 14(1950)Google Scholar
  2. 25.
    Kinderlehrer D., Stampacchia G.: An introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)Google Scholar
  3. 26.
    Kunze, M., Rodrigues, J.F.: An elliptic quasi-variational inequality with gradient constraints and some of its applications, Math. Meth. Appl. Sci. 897-908 (2000)Google Scholar
  4. 27.
    Mosco U.: Convergence of convex sets and of solutions of variational inequalities. Adv. Math. 3, 510–585 (1969)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  • Samir Adly
    • 1
    Email author
  • Maïtine Bergounioux
    • 2
  • Mohamed Ait Mansour
    • 3
  1. 1.XLIM UMR-CNRS 6172Université de LimogesLimogesFrance
  2. 2.UMR 6628-MAPMO, Fédération Denis PoissonUniversité d’OrléansOrléans Cedex 2France
  3. 3.Faculté Poly-DisciplinaireSafiMorocco

Personalised recommendations