Optimization Letters

, Volume 8, Issue 7, pp 2041–2051 | Cite as

State-constrained optimal control of nonlinear elliptic variational inequalities

Original Paper
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Abstract

An optimization control problem for systems described by abstract variational inequalities with state constraints is considered. The solvability of this problem is proved. The problem is approximated by the penalty method. The convergence of this method is proved. Necessary conditions of optimality for the approximation problem are obtained. Its solution is an approximate optimal control of the initial problem.

Keywords

Optimal control Variational inequality State constraints Penalty method 
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References

  1. 1.
    Duvaux, G., Lions, J.L.: Inequalities in Mechanics and Physics. Springer, Berlin (1976)Google Scholar
  2. 2.
    Kinderlehrer, D., Stampacchia, G.: An Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980)MATHGoogle Scholar
  3. 3.
    Lions, J.L.: Quelques méthods de résolution des problèmes aux limites non linéaires. Dunod, Gautier-Villars, Paris (1969)Google Scholar
  4. 4.
    Barbu V.: Optimal control of variational inequalities. Research Notes in Mathematics, vol. 100. Pitman, Boston (1984)Google Scholar
  5. 5.
    He, Z.-X.: State constrained control problems governed by variational inequalities. SIAM J. Control Optim. 25, 1119–1145 (1987)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Tiba, D.: Optimal control of nonsmooth distributed parameter systems. Lecture Notes in Mathematics, vol. 1459. Springer, Berlin (1990)Google Scholar
  7. 7.
    Bonnans, J., Casas, E.: An extension of Pontryagin’s principle for state-constrained optimal control of semilinear elliptic equations and variational inequalities. SIAM J. Control Optim. 33, 274–298 (1995)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Wang, G., Zhao, Y., Li, W.: Some optimal control problems governed by elliptic variational inequalities with control and state constraint on the boundary. J. Optim. Theory Appl. 106, 627–655 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen, Q., Chu, D., Tan, R.C.E.: Optimal control of obstacle for quasi-linear elliptic variational bilateral problems. SIAM J. Control Optim. 44, 1067–1080 (2005)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Zhou, Y.Y., Yang, X.Q., Teo, K.L.: The existence results for optimal control problems governed by a variational inequality. J. Math. Anal. Appl. 321, 595–608 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Neittaanmaki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems. Theory, Algorithms, and Applications. Marcel Dekker, New York (1994)Google Scholar
  12. 12.
    Adams, D.R., Lenhart, S.: Optimal control of the obstacle for a parabolic variational inequality. J. Math. Anal. Appl. 268, 602–614 (2002)Google Scholar
  13. 13.
    Lions, J.L.: Contrôle de Systèmes Distribués Singuliers. Gauthier-Villars, Paris (1983)Google Scholar
  14. 14.
    Warga, J.: Optimal Control of Differential and Functional Equations. Academic Press, New York (1972)MATHGoogle Scholar
  15. 15.
    Ioffe, A.D., Tihomirov, V.M.: Extremal Problems Theory. Nauka, Moscow (1974)Google Scholar
  16. 16.
    Matveev, A.S., Yakubovich, V.A.: Abstract Optimization Control Theory. S-P. University, Sankt-Petersburg (1994)Google Scholar
  17. 17.
    Madani, K., Mastroeni, G., Moldovan, A.: Constrained extremum problems with infinite-dimensional image: selection and necessary conditions. J. Optim. Theory Appl. 135, 37–53 (2007)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Casas, E.: Boundary control of semilinear elliptic equations with pointwise state constraints. SIAM J. Control Optim. 33, 993–1006 (1993)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Bonnans, J.F., Hermant, A.: Conditions d’optimalité du second ordre nécessaires ou suffisantes pour les problèmes de commande optimale avec une contrainte sur l’état et une commande scalaires. C. R. Acad. Sci. Paris Ser. I. 343, 473–478 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Rösch, A., Tröltzsch, F.: On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints. SIAM J. Control Optim. 46, 1098–1115 (2007)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Casas, E., Reyes, J.C., Tröltzsch, F.: Sufficient second-order optimality conditions for semilinear control problems with pointwise state constraints. SIAM J. Optim. 19, 616–643 (2008)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Floudas, C., Pardalos, P. (eds.) Encyclopedia of Optimization. Springer, New York (2008)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Al-Farabi Kazakh National UniversityAlmatyKazakhstan

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