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.
Mohamed Akkouchi 1 ; Abdellah Bounabat 1 ; Manfred Goebel 2
@article{AMBP_2003__10_2_181_0,
author = {Mohamed Akkouchi and Abdellah Bounabat and Manfred Goebel},
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},
mrnumber = {2031268},
zbl = {1048.49013},
language = {en},
url = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.173/}
}
TY - JOUR AU - Mohamed Akkouchi AU - Abdellah Bounabat AU - Manfred Goebel 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 - https://ambp.centre-mersenne.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 Mohamed Akkouchi %A Abdellah Bounabat %A Manfred Goebel %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 https://ambp.centre-mersenne.org/articles/10.5802/ambp.173/ %R 10.5802/ambp.173 %G en %F AMBP_2003__10_2_181_0
Mohamed Akkouchi; Abdellah Bounabat; Manfred Goebel. 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. https://ambp.centre-mersenne.org/articles/10.5802/ambp.173/
[1] Smooth and Nonsmooth Lipschitz Controls for a Class of Nonlinear Ordinary Differential Equations of second Order (1998) (Martin-Luther-Univ. Halle-Wittenberg, FB Math. / Inf. Rep. No. 33)
[2] Optimality conditions for controls acting as coefficients of a nonlinear Ordinary Differential Equations of second Order, Acta Mathematica Vietnamica, Volume 26 (1) (2001), pp. 115-124 | MR | Zbl
[3] Identification of Nonlinear Elliptic Equations, Appl. Math. Optim., Volume 33 (1996), pp. 139-167 | DOI | MR | Zbl
[4] Identification of Nonlinear Parabolic Equations (1996) (Preprint, Tech. Univ. Graz, Inst. Math.) | MR
[5] Control and estimation of the boundary heat transfer function in Stefan problems, Mathematical Modelling and Numerical Analysis, Volume 30 (6) (1996), pp. 671-710 | Numdam | MR | Zbl
[6] Optimization and nonsmooth analysis, Canadian Math. Soc. Series of Monographs and Advanced Texts. John Wiley & sons Inc., New York, 1983 | MR | Zbl
[7] Stability for parameter estimation in two point boundary value problems, J. Reine Angewandte Math., Volume 370 (1986), pp. 1-29 | DOI | MR | Zbl
[8] Nonconvex Minimization Problems, Bull. Am. Math. Soc. (N. S.), Volume 1 (3) (1979), pp. 443-474 | DOI | MR | Zbl
[9] On Smooth and Nonsmooth Lipschitz Controls (1997) (Martin-Luther-Univ. Halle-Wittenberg, FB Math. / Inf. Rep. No. 39)
[10] Smooth and Nonsmooth Optimal Lipschitz Control - a Model Problem, Variational Calculus, Optimal Control and Applications, Birkhäuser Verl. Basel, 1998, pp. 53-60 | MR | Zbl
[11] Optimal Control of a Nonlinear Singular Integral Equation Arising in Electrochemical Machining, Z. Anal. Anwend., Volume 10 (1) (1991), pp. 73-82 | MR | Zbl
[12] On necessary optimality conditions for systems governed by a two point boundary value problem I, Optimization, Volume 20 (5) (1989), pp. 671-685 | DOI | MR | Zbl
[13] Zur Parameterbestimmung in nichtlineraren Problemen, volume 81 of Teubner-Texte zur Matematik. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1985 | MR | Zbl
[14] Parameter estimation, regularity and the penalty method for a class of two point boundary value problems, SIAM J. Control Optim., Volume 25 (1) (1987), pp. 100-120 | DOI | MR | Zbl
[15] Nonsmooth optimization, World Scientific Publishing Co., New Jersey, London, Hong Kong, 1992 | MR
Cité par Sources :