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Contrôle par les coefficients dans le modèle elrod-adams

Published online by Cambridge University Press:  15 August 2002

Mohamed El Alaoui Talibi  and
Abdellah El Kacimi
Mohamed El Alaoui Talibi
Affiliation:
Faculté des Sciences Semalila, Département de Mathématiques, BP. 2930 Marrakech, Maroc ; elalaoui@ucam.ac.ma.
Abdellah El Kacimi
Affiliation:
Faculté des Sciences Semalila, Département de Mathématiques, BP. 2930 Marrakech, Maroc ; elalaoui@ucam.ac.ma.
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Abstract

The purpose of this paper is to study a control by coefficients problem issued from the elastohydrodynamic lubrication. The control variable is the film thickness.The cavitation phenomenon takes place and described by the Elrod-Adams model, suggested in preference to the classical variational inequality due to its ability to describe input and output flow. The idea is to use the penalization in the state equation  by approximating the Heaviside graph whith a sequence of monotone and regular functions. We derive a necessary condition for the regularized problem,  then we establish estimates of the state and the adjoint state in the one dimensional case. Next we pass to the limit.

Keywords

Contrôle optimalidentification des coefficientslubrificationproblème elliptique non linéairerégularisation.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 97 - 118
Copyright
© EDP Sciences, SMAI, 2001

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References

C. Alvarez, Problemas de frontiera libre en teoría de lubrificación. Ph.D. Thesis, Complutense University of Madrid (1986).
Barbu, V., Necessary conditions for nonconvex distributed control problems governed by elliptic variational inequalities. J. Math. Anal. Appl. 80 (1981) 566-598. CrossRef
Barbu, V., Necessary conditions for distributed control problems governed by parabolic variational inequalities. SIAM. J. Control Optim. 19 (1981) 64-86. CrossRef
Bayada, G. et Chambat, M., Sur quelques modélisation de la zone de cavitation en lubrification hydrodynamique. J. Méc. Théor. Appl. 5 (1986) 703-729.
Bayada, G. et Chambat, M., Existence and uniqueness for a lubrification problem with non regular conditions on the free boundary. Boll. Un Math. Ital. 6 (1984) 543-547.
Bayada, G. et El Alaoui Talibi, M., Control by coefficients in a variational inequality: The inverse elastohydrodynamic lubrication problem. Nonlinear Analysis: Real World Applications 1 (2000) 315-328. CrossRef
Bayada, G. et El Alaoui Talibi, M., Une méthode du type caractérisitique pour la résolution d'un problème de lubrification hydrodynamique en régime transitoire. ESAIM: M2AN 25 (1991) 395-423.
A. Bensoussan, J.L. Lions et G. Papanicolau, Asymptotic analysis for periodic structures. North-Holland, Amsterdam (1978).
H. Brezis, Analyse fonctionnelle Théorie et Application. Masson, Paris (1983).
A. Cameron, Basic Lubrication Theory. John Whiley & Sons (1981).
Casas, E. et Bonnans, F., An extension of pontryagin's principle for state-constrainted optimal control of semilinear elliptic equations and variational inequalities. SIAM J. Control Optim. 33 (1995) 274-298. CrossRef
Casas, E. et Bonnans, F., Optimal control of semilinear multistate systems with state constraints. SIAM J. Control Optim. 27 (1989) 446-455.
Casas, E., Kavian, O. et Puel, J.P., Optimal control of an ill-posed elliptic semilinear equation whith an exponential non linearity. ESAIM: COCV 3 (1998) 361-380. CrossRef
G. Elrod H. et M.L. Adams, A computer program for cavitation, in st LEEDS LYON symposium on cavitation and related phenomena in lubrication, I.M.E. (1974).
D. Gilbarg et N.S. Trudinger, Elliptic Partial Differential Equations of second Order. Springer-Verlag (1983).
O.A. Ladyzhenskaya et N.N. Ural'tseva, Linear and quasilinear elliptic equations. Academic Press (1968).
M.H. Meurisse, Solution of the inverse problem in hydrodynamic lubrication, in Proc. of the X Lyon Leeds International Symposium (1983) 104-107.
J.F. Rodrigues, Obstacle problems in mathematical physics. North-Holland, Amsterdam (1978).
G. Stampachia et D. Kinderleher, An introduction to variational inequalities and applications. Academic Press (1980).