Optimality Conditions for a Nonlinear Boundary Value Problem Using Nonsmooth Analysis

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.

Annales mathématiques Blaise Pascal (2003)

  • Volume: 10, Issue: 2, page 181-194
  • ISSN: 1259-1734

Abstract

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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.

How to cite

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Akkouchi, Mohamed, Bounabat, Abdellah, and Goebel, Manfred. "Optimality Conditions for a Nonlinear Boundary Value Problem Using Nonsmooth Analysis." Annales mathématiques Blaise Pascal 10.2 (2003): 181-194. <http://eudml.org/doc/10486>.

@article{Akkouchi2003,
abstract = {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.},
affiliation = {Faculté des Sciences-Semlalia Départment de Mathématiques Univ. Cadi Ayyad, B.P. 2390, Av. du Prince My. Abdellah Marrakech MAROC (MOROCCO).; Faculté des Sciences-Semlalia Départment de Mathématiques Univ. Cadi Ayyad, B.P. 2390, Av. du Prince My. Abdellah Marrakech MAROC (MOROCCO).; Martin-Luther-Universität Halle-Wittenberg FB Mathematik und Informatik Theodor-Lieser-Str. 5 D-06099 Halle (Saale) GERMANY.},
author = {Akkouchi, Mohamed, Bounabat, Abdellah, Goebel, Manfred},
journal = {Annales mathématiques Blaise Pascal},
keywords = {Semilinear second order ordinary differential equation. Optimality conditions. Nemytskij operator. Clarke’s generalized directional derivative; semilinear second order ordinary differential equation; optimality conditions; Nemytskij operator; Clarke's generalized directional derivative; Lipschitz control},
language = {eng},
month = {7},
number = {2},
pages = {181-194},
publisher = {Annales mathématiques Blaise Pascal},
title = {Optimality Conditions for a Nonlinear Boundary Value Problem Using Nonsmooth Analysis},
url = {http://eudml.org/doc/10486},
volume = {10},
year = {2003},
}

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
DA - 2003/7//
PB - Annales mathématiques Blaise Pascal
VL - 10
IS - 2
SP - 181
EP - 194
AB - 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.
LA - eng
KW - Semilinear second order ordinary differential equation. Optimality conditions. Nemytskij operator. Clarke’s generalized directional derivative; semilinear second order ordinary differential equation; optimality conditions; Nemytskij operator; Clarke's generalized directional derivative; Lipschitz control
UR - http://eudml.org/doc/10486
ER -

References

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  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 26 (1) (2001), 115-124 Zbl0990.49018MR1828369
  3. V. Barbu, K. Kunisch, Identification of Nonlinear Elliptic Equations, Appl. Math. Optim. 33 (1996), 139-167 Zbl0865.35139MR1365132
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  8. I. Ekland, Nonconvex Minimization Problems, Bull. Am. Math. Soc. (N. S.) 1 (3) (1979), 443-474 Zbl0441.49011MR526967
  9. M. Goebel, On Smooth and Nonsmooth Lipschitz Controls, (1997) 
  10. M. Goebel, Smooth and Nonsmooth Optimal Lipschitz Control - a Model Problem, Variational Calculus, Optimal Control and Applications (1998), 53-60, et al.W. H. SchmidtW. H. S. Zbl0927.49016MR1728771
  11. M. Goebel, D. Oestreich, Optimal Control of a Nonlinear Singular Integral Equation Arising in Electrochemical Machining, Z. Anal. Anwend. 10 (1) (1991), 73-82 Zbl0754.49016MR1155357
  12. M. Goebel, U. Raitums, On necessary optimality conditions for systems governed by a two point boundary value problem I, Optimization 20 (5) (1989), 671-685 Zbl0683.49004MR1015435
  13. R. Kluge, Zur Parameterbestimmung in nichtlineraren Problemen, (1985), volume 81 of Teubner-Texte zur Matematik. BSB B. G. Teubner Verlagsgesellschaft, Leipzig Zbl0588.35083MR833967
  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. 25 (1) (1987), 100-120 Zbl0612.93013MR872454
  15. M. M. Mäkelä, P. Neittaanmäki, Nonsmooth optimization, (1992), World Scientific Publishing Co., New Jersey, London, Hong Kong MR1177832

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