Abstract
We consider an optimal control problem for the obstacle problem with an elliptic variational inequality. The obstacle function which is the control function is assumed in H2. We use an approximate technique to introduce a family of problems governed by variational equations. We prove optimal solutions existence and give necessary optimality conditions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.A. Adams, J.F. Fournier, Sobolev spaces. 2nd edn, Academic Press (2003).
D.R. Adams, S. Lenhart, An obstacle control problem with a source term, Appl. Math. Optim. 47 (2002), 79–95.
D.R. Adams, S. Lenhart, J. Yong, Optimal control of the obstacle for an elliptic variational inequality, Appl. Math. Optim. 38 (1998), 121–140.
V. Barbu, P. Korman, Approximating optimal controls for elliptic obstacle problem by monotone iteration schems, Num. Funct. Anal. Optim. 12 (1991), 429–442.
V. Barbu, Optimal control of varitional inequalities, Pitman, (1984).
M. Bergounioux, S. Lenhart, Optimal control of the bilateral obstacle problems, SIAM J. Control Optim. 43 (2004), 249–255.
M. Bergounioux, S. Lenhart, Optimal control of the obstacle in semilinear variational inequalities, Positivity 8 (2004), 229–242.
M. Bergounioux, Optimal control of semilinear elliptic obstacle problems, J Nonlin. Convex Anal. 3 (2002), 25–39.
H. Brezis, G. Stampacchia, Sur La régularité de la solution d’inéquations elliptiques, Bull. de la Soc. Math. de France 96 (1968), 153–180.
A. Friedman, Variational principles and free boundary problems, John Wiley, (1982).
D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, 2nd edn, Springer, (1983).
K. Ito, K. Kunisch, Optimal control of obstacle problems by H1—obstacles, Appl. Math. Optim. 56 (2007), 1–17.
J.L. Lions, E. Magens, Problèmes aux limites non homogènes et applivations, vol. 1, Dunod, (1968).
H. Lou, On the regularity of an obstacle control problem, J Math. Anal. Appl. 258 (2001), 32–51.
F. Mignot, Contrôle dans lesinéquations variationelles elliptiques, J. Funct. Anal. 22 (1976), 466–476.
F. Mignot, J.P. Puel, Optimal control in some variational inequalities, SIAM J Control Optim. 22 (1984), 466–476.
Y. Ye, Q. Chen, Optimal control of the obstacle in a quasilinear elliptic variational inequality, J. Math. Anal. Appl 294 (2004), 258–272.
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is grateful to Prof. M. Bergounioux for her instructive suggestions.
Rights and permissions
About this article
Cite this article
Ghanem, R. Optimal control of unilateral obstacle problem with a source term. Positivity 13, 321–338 (2009). https://doi.org/10.1007/s11117-008-2241-8
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-008-2241-8