Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-10T10:22:21.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Control of Obstacle Problems:Existence of LagrangeMultipliers

Published online by Cambridge University Press:  15 August 2002

Maïtine Bergounioux  and
Fulbert Mignot
Maïtine Bergounioux
Affiliation:
Département de Mathématiques, UMR 6628, Université d'Orléans, BP. 6759, 45067 Orléans Cedex 2, France; Maitine.Bergounioux@labomath.univ-orleans.fr.
Fulbert Mignot
Affiliation:
Laboratoire de Mathématique, bâtiment 425, Université Paris-Sud, 91405 Orsay, France; Fulbert.Mignot@math.u-psud.fr.
Get access

Abstract

We study first order optimality systems for the control of a systemgoverned by avariationalinequality and deal with Lagrange multipliers: isit possible to associate to each pointwise constraint a multiplier to get a “good” optimality system? We givepositive and negative answers for the finite and infinite dimensional cases.These results are compared withthe previous ones got by penalization or differentiation.

Keywords

Variational inequalitiesoptimal controlLagrange multiplierobstacle problem.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 5 , 2000 , pp. 45 - 70
Copyright
© EDP Sciences, SMAI, 2000

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adams, D.R., Lenhart, S. and Yong, J., Optimal control of variational inequalities. Appl. Math. Optim. 38 (1998) 121-140. CrossRef
V. Barbu, Optimal control of variational inequalities. Pitman, Boston, Res. Notes Math. 100 (1984).
Bergounioux, M., Optimal control of an obstacle problem. Appl. Math. Optim. 36 (1997) 147-172. CrossRef
Bergounioux, M., Optimal control of problems governed by abstract variational inequalities with state constraints. SIAM J. Control Optim. 36 (1998) 273-289. CrossRef
Bergounioux, M., Augmented lagrangian method for distributed optimal control problems with state constraints. J. Optim. Theory Appl. 78 (1993) 493-521. CrossRef
Bergounioux, M. and Dietrich, H., Optimal control of problems governed by obstacle type variational inequalities: A dual regularization-penalization approach. J. Convex Anal. 5 (1998) 329-351.
M. Bergounioux, M. Haddou, M. Hintermueller and K. Kunisch, A comparison of interior point methods and a Moreau-Yosida based active set strategy for constrained optimal control problems. Report 98-15 Université d'Orléans (1998).
Bergounioux, M. and Zidani, H., Pontryagin principle for problems governed by parabolic variational inequalities. SIAM J. Control Optim. 37 (1999) 1273-1290. CrossRef
Bermudez, A. and Saguez, C., Optimal control of variational inequalities. Optimality conditions and numerical methods. Collection Free Boundary Problems, Application and Theory, Vol. IV. Maubusson, Res. Notes Math. 121 (1984) 478-487.
Bermudez, A. and Saguez, C., Optimal control of a Signorini Problem. SIAM J. Control Optim. 25 (1987) 576-582. CrossRef
Ciarlet, P.G. and Raviart, P.A., Maximum principle and uniform convergence for the finite element method. Comput. Methods Appl. Mech. Engrg. 2 (1973) 17-31. CrossRef
W. Hackbusch, Elliptic Differential Equations, Theory and Numerical Treatment. Springer Verlag, Berlin, Ser. Comput. Math. 18 (1992).
Ito, K. and Kunisch, K., An augmented Lagrangian technics for variational inequalities. Appl. Math. Optim. 21 (1990) 223-241. CrossRef
K. Ito and K. Kunisch, Optimal control of elliptic variational inequalities, to appear.
Kurcyusz, S., On the existence and nonexistence of Lagrange multipliers in Banach spaces. J. Optim. Theory Appl. 5 (1976) 81-110. CrossRef
Mignot, F., Contrôle dans les inéquations variationnelles elliptiques. J. Funct. Anal. 22 (1976) 130-185. CrossRef
Mignot, F. and Puel, J.P., Optimal control in some variational inequalities. SIAM J. Control Optim. 22 (1984) 466-476. CrossRef
Mignot, F. and Puel, J.P., Contrôle optimal d'un système gouverné par une inéquation variationnelle parabolique. C. R. Acad. Sci. Paris Sér. I Math. 298 (1984) 277-280.
Tiba, D. and Tröltzsch, F., Error estimates for the discretization of state constrained convex control problems. Num. Funct. Anal. Optim. 17 (1996) 1005-1028. CrossRef
Wenbin, L. and Rubio, J.E., Optimality conditions for strongly monotone variational inequalities. Appl. Math. Optim. 27 (1993) 291-312.
Zowe, J. and Kurcyusz, S., Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5 (1979) 49-62. CrossRef