# 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

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topAkkouchi, 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

top- M. Akkouchi, A. Bounabat, M. Goebel, Smooth and Nonsmooth Lipschitz Controls for a Class of Nonlinear Ordinary Differential Equations of second Order, (1998) Zbl0990.49018
- 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
- V. Barbu, K. Kunisch, Identification of Nonlinear Elliptic Equations, Appl. Math. Optim. 33 (1996), 139-167 Zbl0865.35139MR1365132
- V. Barbu, K. Kunisch, Identification of Nonlinear Parabolic Equations, (1996) Zbl0865.35139MR1424370
- V. Barbu, K. Kunisch, W. Ring, Control and estimation of the boundary heat transfer function in Stefan problems, Mathematical Modelling and Numerical Analysis 30 (6) (1996), 671-710 Zbl0865.65070MR1419934
- F. H. Clarke, Optimization and nonsmooth analysis, (1983), Canadian Math. Soc. Series of Monographs and Advanced Texts. John Wiley & sons Inc., New York Zbl0582.49001MR709590
- F. Colonius, K. Kunisch, Stability for parameter estimation in two point boundary value problems, J. Reine Angewandte Math. 370 (1986), 1-29 Zbl0584.34009MR852507
- I. Ekland, Nonconvex Minimization Problems, Bull. Am. Math. Soc. (N. S.) 1 (3) (1979), 443-474 Zbl0441.49011MR526967
- M. Goebel, On Smooth and Nonsmooth Lipschitz Controls, (1997)
- 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
- 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
- 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
- R. Kluge, Zur Parameterbestimmung in nichtlineraren Problemen, (1985), volume 81 of Teubner-Texte zur Matematik. BSB B. G. Teubner Verlagsgesellschaft, Leipzig Zbl0588.35083MR833967
- 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
- M. M. Mäkelä, P. Neittaanmäki, Nonsmooth optimization, (1992), World Scientific Publishing Co., New Jersey, London, Hong Kong MR1177832

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