L’objet de cet article est d’étudier un problème de contrôle optimal gouverné par une équation différentielle ordinaire du second ordre sous conditions aux limites et contraintes sur l’état. Les contrôles sont Lipschitziens et agissent comme des coefficients pour cette équation. La partie non linéaire de cette équation est donnée par l’action d’un opérateur de composition (de Nemytskij) défini par une fonction Lipschitzienne mais non nécessairement régulière. Nous établissons l’existence des contrôles optimaux et trouvons des conditions nécessaires d’optimalité qui ressemblent au principe du maximum de Pontriaguine. Ces conditions utilisent des notions d’analyse non régulière telles que les notions de sous-différentiel et la derivée directionnelle généralisée de Clarke. Ainsi, ce travail complète notre article [2] qui traite le même problème mais dans le cas régulier avec des outils d’analyse classique. A la fin de ce travail, nous donnons un exemple d’applications.
We study in this paper a Lipschitz control problem associated to a semilinear second order ordinary differential equation with pointwise state constraints. The control acts as a coefficient of the state equation. The nonlinear part of the equation is governed by a Nemytskij operator defined by a Lipschitzian but possibly nonsmooth function. We prove the existence of optimal controls and obtain a necessary optimality conditions looking somehow to the Pontryagin’s maximum principle. These conditions utilize the notion of Clarke’s generalized directional derivative. We point out that this work provides complements to our previous paper [2], where a similar problem was studied but with tools only from classical analysis.
Mots clés : Semilinear second order ordinary differential equation. Optimality conditions. Nemytskij operator. Clarke’s generalized directional derivative.
@article{AMBP_2003__10_2_181_0,
author = {Akkouchi, Mohamed and Bounabat, Abdellah and Goebel, Manfred},
title = {Optimality {Conditions} for a {Nonlinear} {Boundary} {Value} {Problem} {Using} {Nonsmooth} {Analysis}},
journal = {Annales math\'ematiques Blaise Pascal},
pages = {181--194},
publisher = {Annales math\'ematiques Blaise Pascal},
volume = {10},
number = {2},
year = {2003},
doi = {10.5802/ambp.173},
zbl = {1048.49013},
mrnumber = {2031268},
language = {en},
url = {http://www.numdam.org/articles/10.5802/ambp.173/}
}
TY - JOUR AU - Akkouchi, Mohamed AU - Bounabat, Abdellah AU - Goebel, Manfred TI - Optimality Conditions for a Nonlinear Boundary Value Problem Using Nonsmooth Analysis JO - Annales mathématiques Blaise Pascal PY - 2003 SP - 181 EP - 194 VL - 10 IS - 2 PB - Annales mathématiques Blaise Pascal UR - http://www.numdam.org/articles/10.5802/ambp.173/ DO - 10.5802/ambp.173 LA - en ID - AMBP_2003__10_2_181_0 ER -
%0 Journal Article %A Akkouchi, Mohamed %A Bounabat, Abdellah %A Goebel, Manfred %T Optimality Conditions for a Nonlinear Boundary Value Problem Using Nonsmooth Analysis %J Annales mathématiques Blaise Pascal %D 2003 %P 181-194 %V 10 %N 2 %I Annales mathématiques Blaise Pascal %U http://www.numdam.org/articles/10.5802/ambp.173/ %R 10.5802/ambp.173 %G en %F AMBP_2003__10_2_181_0
Akkouchi, Mohamed; Bounabat, Abdellah; Goebel, Manfred. Optimality Conditions for a Nonlinear Boundary Value Problem Using Nonsmooth Analysis. Annales mathématiques Blaise Pascal, Tome 10 (2003) no. 2, pp. 181-194. doi : 10.5802/ambp.173. http://www.numdam.org/articles/10.5802/ambp.173/
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