Optimality Conditions for a Nonlinear Boundary Value Problem Using Nonsmooth Analysis
Annales mathématiques Blaise Pascal, Volume 10 (2003) no. 2, pp. 181-194.

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.

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.

DOI: 10.5802/ambp.173
Classification: 49K15,  49J15
Keywords: Semilinear second order ordinary differential equation. Optimality conditions. Nemytskij operator. Clarke’s generalized directional derivative.
Mohamed Akkouchi 1; Abdellah Bounabat 1; Manfred Goebel 2

1 Faculté des Sciences-Semlalia Départment de Mathématiques Univ. Cadi Ayyad, B.P. 2390, Av. du Prince My. Abdellah Marrakech MAROC (MOROCCO).
2 Martin-Luther-Universität Halle-Wittenberg FB Mathematik und Informatik Theodor-Lieser-Str. 5 D-06099 Halle (Saale) GERMANY.
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Mohamed Akkouchi; Abdellah Bounabat; Manfred Goebel. Optimality Conditions for a Nonlinear Boundary Value Problem Using Nonsmooth Analysis. Annales mathématiques Blaise Pascal, Volume 10 (2003) no. 2, pp. 181-194. doi : 10.5802/ambp.173. https://ambp.centre-mersenne.org/articles/10.5802/ambp.173/

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