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Issue ESAIM: COCV
Volume 3, 1998
Page(s) 361 - 380
DOI 10.1051/cocv:1998116

ESAIM: COCV, Vol. 3, pp. 361-380
DOI: 10.1051/cocv:1998116

Optimal control of an ill-posed elliptic semilinear equation with an exponential non linearity

E. Casas1, O. Kavian2 and J.-P. Puel3

1  (casas@etsiso.macc.unican.es)
2  (kavian@math.uvsq.fr) 
3  (jppuel@cmapx.polytechnique.fr) 


Abstract
We study here an optimal control problem for a semilinear elliptic equation with an exponential nonlinearity, such that we cannot expect to have a solution of the state equation for any given control. We then have to speak of pairs (control, state). After having defined a suitable functional class in which we look for solutions, we prove existence of an optimal pair for a large class of cost functions using a non standard compactness argument. Then, we derive a first order optimality system assuming the optimal pair is slightly more regular.


Résumé
On étudie ici un problème de contrôle optimal pour une équation elliptique semilinéaire avec terme non linéaire exponentiel, tel que nous ne pouvons pas attendre que l'équation d'état admette une solution pour tout contrôle donné. Nous devons alors parler de couple (contrôle, état). Après avoir défini une classe fonctionnelle adaptée dans laquelle nous cherchons les solutions, nous montrons l'existence d'un couple optimal pour une grande classe de fonctions coût en utilisant un argument de compacité non standard. Ensuite, en supposant que le couple optimal soit quelque peu plus régulier, nous obtenons un système d'optimalité du premier ordre.


Key words: Optimal control, semilinear equation, exponential term, optimality conditions.


© EDP Sciences, SMAI 1998


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