Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-17T03:00:24.317Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal Control of Semilinear Parabolic Equations with State-Constraints of Bottleneck Type

Published online by Cambridge University Press:  15 August 2002

Maïtine Bergounioux  and
Fredi Tröltzsch
Maïtine Bergounioux
Affiliation:
Département de Mathématiques, UMR 6628, Université d'Orléans, BP. 6759, 45067 Orléans Cedex 2, France.
Fredi Tröltzsch
Affiliation:
Technical University of Chemnitz, Faculty of Mathematics, 09107 Chemnitz, Germany.
Get access

Abstract

We consider optimal distributed and boundary control problems for semilinear parabolic equations, where pointwise constraints on the control and pointwise mixed control-state constraints of bottleneck type are given. Our main result states the existence of regular Lagrange multipliers for the state-constraints. Under natural assumptions, we are able to show the existence of bounded and measurable Lagrange multipliers. The method is based on results from the theory of continuous linear programming problems.

Keywords

Parabolic equationoptimal controlpointwise state-constraintbottleneck problem.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 4 , 1999 , pp. 595 - 608
Copyright
© EDP Sciences, SMAI, 1999

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

Bergounioux, M. and Tröltzsch, F., Optimality Conditions and Generalized Bang-Bang Principle for a State Constrained Semilinear Parabolic Problem. Num. Funct. Anal. Opt. 15 (1996) 517-537. CrossRef
Bergounioux, M. and Tröltzsch, F., Optimal Control of Linear Bottleneck Problems. ESAIM: Cont. Optim. Cal. Var. 3 (1998) 235-250. CrossRef
Casas, E., Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35 (1997) 1297-1327. CrossRef
Raymond, J.P., Non Linear Boundary Control of Semilinear Parabolic Problems with Pointwise State Constraints. Discrete and Continuous Dynamical Systems 3 (1997) 341-370. CrossRef
J.P. Raymond and H. Zidani, Hamiltonian Pontryagin's Principles for Control Problems Governed by Semilinear Parabolic Equations. Appl. Math. Optim., to appear
F. Tröltzsch, Optimality conditions for parabolic control problems and applications, Teubner Texte zur Mathematik, Teubner, Leipzig (1984).
Zowe, J. and Kurcyusz, S., Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5 (1979) 49-62. CrossRef