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Issue ESAIM: Proc.
Volume 13, 2003
Proceedings of 2003 MODE-SMAI Conference
Page(s) 18 - 30
DOI http://dx.doi.org/10.1051/proc:2003001

ESAIM: Proc., December 2003, Vol. 13, pp. 18-30
DOI: 10.1051/proc:2003001

Necessary and Sufficient Optimality Conditions for Optimization Problems in Function Spaces and Applications to Control Theory

E. Casas1, Mariano Mateos2 and Fredi Tröltzsch3

1  Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain ,
2  Departamento de Matemáticas, EUIT Industriales, Universidad de Oviedo., Spain,
3  Fakultät für Mathematik, Technische Universität Chemnitz, D-09107 Chemnitz, Germany,

eduardo.casas@unican.es
mmateos@orion.ciencias.uniovi.es
f.troeltzsch@mathematik.tu-chemnitz.de

Abstract
We consider an abstract formulation for optimization problems in some Lp spaces. The variables are restricted by pointwise upper and lower bounds and by finitely many equality and inequality constraints of functional type. Second-order necessary and sufficient optimality conditions are established, where the cone of critical directions is arbitrarily close to the form which is expected from the optimization in finite dimensional spaces. The results are applied to an optimal control problem governed by a partial differential equation. Finally we compare the conditions obtained by applying this abstract procedure and those ones derived by using the methods adapted to the optimal control problem.

Résumé
Nous considérons des problèmes d'optimisation formulés de manière abstraite dans des espaces de type Lp. Les variables doivent vérifier des contraintes ponctuelles de borne supérieure et inférieure, ainsi qu'un nombre fini de contraintes d'égalité et d'inégalité fonctionnelles. Nous établissons des conditions nécessaires et des conditions suffisantes d'optimalité, dans lesquelles le cône des directions critiques est arbitrairement proche de la forme dérivée des résultats d'optimisation en dimension finie.Les résultats sont appliqués à des problèmes de contrôle optimal gouvernés par des équations aux dérivées partielles. Finalement nous comparons ces conditions avec celles obtenues par des méthodes adaptées à la forme spécifique des problèmes de contrôle.


Mathematics Subject Classification. 49K20, 35J25, 90C45, 90C48

Key words: Necessary and sufficient optimality conditions, control of partial differential equations, state constraints


© EDP Sciences, ESAIM 2003


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