Abstract

In this paper we investigate optimal control problems governed by variational inequalities. We give optimality conditions under classical assumptions using a dual regularized functional to interpret the Variational Inequality.

Author information

Contact details are reproduced from the original publication and may be historical.

Maïtine Bergounioux

Département de Mathématiques, UMR. 6628 - Université d’Orléans, B.P. 6759 - 45067 Orléans Cedex 2, France.

Helmut Dietrich

Institute of Optimization and Stochastics, Martin-Luther-University of Halle-Wittenberg, 06099 Halle, Germany

Keywords

  • Variational Inequalities
  • Optimal Control
  • Dual Methods
  • Penalization Methods

References

  1. J.-P. Aubin, H. Frankowska: Set-valued Analysis, Birkhäuser, Basel, 1990.Google Scholar
  2. V. Barbu: Optimal Control of Variational Inequalities, Research Notes in Mathematics 100, Pitman, Boston, 1984.Google Scholar
  3. V. Barbu: Analysis and Control of Non Linear Infinite Dimensional Systems, Math. in Science and Engineering 190, Academic Press, 1993.Google Scholar
  4. C. Berge: Espaces Topologiques et Fonctions Multivoques, Dunod, Paris, 1959.Google Scholar
  5. M. Bergounioux: Optimal control of an obstacle problem, Applied Mathematics and Optimization 36 (1997) 147-172.https://doi.org/10.1007/s002459900058Google Scholar
  6. M. Bergounioux: Optimal control of problems governed by abstract elliptic variational inequalities with state constraints, Siam Journal on Control and Optimization 36(1) (1998) 273-289.https://doi.org/10.1137/S0363012996302615Google Scholar
  7. F. Clarke: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics 5, SIAM, 1990.https://doi.org/10.1137/1.9781611971309Google Scholar
  8. I. Ekeland, R. Temam: Analyse Convexe et Problèmes Variationnels, Dunod-Gauthier-Villars, Paris, 1974.Google Scholar
  9. F. Mignot, J. P. Puel: Optimal control in some variational inequalities, SIAM Journal on Control and Optimization 22(3) (1984) 466-476.https://doi.org/10.1137/0322028Google Scholar
  10. J. P. Penot: Preservation of persistence and stability under intersections and operations, Part 1: Persistence, Journal of Optimization Theory and Applications 79(3) (1993) 525-550.https://doi.org/10.1007/BF00940557Google Scholar
  11. S. Shi: Optimal control of strongly monotone variational inequalities, SIAM Journal on Control and Optimization 25 (1988) 274-290.https://doi.org/10.1137/0326016Google Scholar
  12. S. Shi: Erratum: Optimal control of strongly monotone variational inequalities, SIAM Journal on Control and Optimization 28 (1990) 243-249.https://doi.org/10.1137/0328012Google Scholar
  13. L. Wenbin, J. E. Rubio: Optimality conditions for strongly monotone variational inequalities, Applied Mathematics and Optimization 27(3) (1993) 291-312.https://doi.org/10.1007/BF01314820Google Scholar
  14. E. Zeidler: Nonlinear Functional Analysis and Its Applications, Vol. IIB: Nonlinear Monotone Operators, Springer, Berlin, 1992.Google Scholar
  15. J. Zowe, S. Kurcyusz: Regularity and stability for the mathematical programming problem in Banach spaces, Applied Mathematics and Optimization 5 (1979) 49-62.https://doi.org/10.1007/BF01442543Google Scholar

Suggested citation

M. Bergounioux, H. Dietrich. “Optimal Control of Problems Governed by Obstacle Type Variational Inequalities: a Dual Regularization-Penalization Approach.” Journal of Convex Analysis 5 (1998), No. 2, 329–351. https://doi.org/10.68381/jca05-22

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Published by Heldermann Verlag, 1998. Rights now held by Banach Press.