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

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.

DOI : https://doi.org/10.5802/ambp.173
Classification : 49K15,  49J15
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 = {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/}
}
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] M. Akkouchi; A. Bounabat; M. Goebel 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] M. Akkouchi; A. Bounabat; M. Goebel 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 1828369 | Zbl 0990.49018

[3] V. Barbu; K. Kunisch Identification of Nonlinear Elliptic Equations, Appl. Math. Optim., Volume 33 (1996), pp. 139-167 | Article | MR 1365132 | Zbl 0865.35139

[4] V. Barbu; K. Kunisch Identification of Nonlinear Parabolic Equations (1996) (Preprint, Tech. Univ. Graz, Inst. Math.) | MR 1424370

[5] V. Barbu; K. Kunisch; W. Ring 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 1419934 | Zbl 0865.65070

[6] F. H. Clarke Optimization and nonsmooth analysis, Canadian Math. Soc. Series of Monographs and Advanced Texts. John Wiley & sons Inc., New York, 1983 | MR 709590 | Zbl 0582.49001

[7] F. Colonius; K. Kunisch Stability for parameter estimation in two point boundary value problems, J. Reine Angewandte Math., Volume 370 (1986), pp. 1-29 | Article | MR 852507 | Zbl 0584.34009

[8] I. Ekland Nonconvex Minimization Problems, Bull. Am. Math. Soc. (N. S.), Volume 1 (3) (1979), pp. 443-474 | Article | MR 526967 | Zbl 0441.49011

[9] M. Goebel On Smooth and Nonsmooth Lipschitz Controls (1997) (Martin-Luther-Univ. Halle-Wittenberg, FB Math. / Inf. Rep. No. 39)

[10] M. Goebel Smooth and Nonsmooth Optimal Lipschitz Control - a Model Problem, Variational Calculus, Optimal Control and Applications (W. H. Schmidt et al., ed.), Birkhäuser Verl. Basel, 1998, pp. 53-60 | MR 1728771 | Zbl 0927.49016

[11] M. Goebel; D. Oestreich Optimal Control of a Nonlinear Singular Integral Equation Arising in Electrochemical Machining, Z. Anal. Anwend., Volume 10 (1) (1991), pp. 73-82 | MR 1155357 | Zbl 0754.49016

[12] M. Goebel; U. Raitums On necessary optimality conditions for systems governed by a two point boundary value problem I, Optimization, Volume 20 (5) (1989), pp. 671-685 | Article | MR 1015435 | Zbl 0683.49004

[13] R. Kluge Zur Parameterbestimmung in nichtlineraren Problemen, volume 81 of Teubner-Texte zur Matematik. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, 1985 | MR 833967 | Zbl 0588.35083

[14] K. Kunisch; L. W. White 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 | Article | MR 872454 | Zbl 0612.93013

[15] M. M. Mäkelä; P. Neittaanmäki Nonsmooth optimization, World Scientific Publishing Co., New Jersey, London, Hong Kong, 1992 | MR 1177832