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Issue ESAIM: Proc.
Volume 17, 2007
CSVAA 2004 - Control Set-Valued Analysis and Applications
Page(s) 50 - 66
DOI http://dx.doi.org/10.1051/proc:071705
Published online 26 April 2007

ESAIM: Proc., April 2007, Vol. 17, pp. 50-66
DOI: 10.1051/proc:071705

On the control of ill-posed distributed parameter systems

R. Dorville1, O. Nakoulima2 and A. Omrane3

1  Université Antilles-Guyane, Laboratoire d'Analyse, Optimisation et Contrôle (AOC), UFR Sciences Exactes et Naturelles, Campus de Fouillole, 97159 Pointe à Pitre. E-mail: rene.dor@wanadoo.fr
2  onakouli@univ-ag.fr
3  aomrane@univ-ag.fr


(Published online: 26 April 2007)

Abstract
We show that the so-called low-regret (or least-regret) control by J. L. Lions [8] fits on the control of ill-posed problems. At each time, we give the characterization of the so-called no-regret control by means of singular optimality systems. For the backward heat ill-posed problem, no Slater hypothesis is assumed on the admissible set of controls ${{\cal U}_{\mbox{\tiny ad}}}$. This work is two pieces, and two methods are considered : the regularization method and the null-controllability method. For the first method, a zero order corrector is used, while for the second method, the passage to the limit is easy. The results presented here generalize the works in [2,3] to the no-regret control.


Résumé
On montre que le contrôle à moindres regrets de J. L. Lions [8] est bien adapté pour le contrôle des des problèmes mal posés. A chaque fois, on donne une caractérisation du contrôle sans regret par le moyen de systèmes d'optimalité singuliers. Pour le problème de chaleur rétrograde mal posé, aucune hypothèse de type Slater sur l'ensemble des contrôles admissibles ${{\cal U}_{\mbox{\tiny ad}}}$ n'est nécessaire. Ce travail est divisé en deux parties, et deux méthodes sont considérées: la méthode de régularisation et celle de la contrôlabilité à zéro. Pour la première méthode, un correcteur d'ordre zéro est utilisé, alors que pour la seconde méthode le passage à la limite est simple. Les résultats présentés ici généralisent les travaux dans [2,3] au contrôle sans regret.


Mathematics Subject Classification. 49K40, 39B35, 35K55, 90C30

Key words: Ill-posed problems, Heat equation, backward heat problem, elliptic regularization, low-regret control, problems of incomplete data, null-controllability, cost function


© EDP Sciences, ESAIM 2007


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