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Singular perturbation for the Dirichlet boundary control of elliptic problems

Published online by Cambridge University Press:  15 November 2003

Faker Ben Belgacem ,
Henda El Fekih  and
Hejer Metoui
Faker Ben Belgacem
Affiliation:
Mathématiques pour l'Industrie et la Physique, UMR 5640, Université Paul Sabatier, 31062 Toulouse Cedex 04, France. belgacem@mip.ups-tlse.fr.
Henda El Fekih
Affiliation:
LAMSIN, École Nationale d'Ingénieurs de Tunis, BP 37, 1002 Tunis-Belvédère, Tunisie. henda.elfekih@enit.rnu.tn.
Hejer Metoui
Affiliation:
LAMSIN, École Nationale d'Ingénieurs de Tunis, BP 37, 1002 Tunis-Belvédère, Tunisie. hajer.metoui@enit.rnu.tn.
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Abstract

A current procedure that takes into account the Dirichlet boundary conditionwith non-smooth data is to change it into aRobin type condition by introducing a penalization term; a major effect of thisprocedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem wherethe control variable is the Dirichlet data.We describe the Robin penalization,and we bound the gap between the penalized and the non-penalized boundary controlsfor the small penalization parameter. Some numerical results are reported on to highlightthe reliability of such an approach.

Keywords

Boundary control problemsnon-smooth Dirichlet conditionRobin penalizationsingularly perturbed problem.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 5 , September 2003 , pp. 833 - 850
Copyright
© EDP Sciences, SMAI, 2003

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References

D.A. Adams, Sobolev Spaces. Academic Press, New York (1975).
Arada, N., El Fekih, H. and Raymond, J.-P., Asymptotic analysis of some control problems. Asymptot. Anal. 24 (2000) 343-366.
Babuska, I., The finite element method with penalty. Math. Comp. 27 (1973) 221-228. CrossRef
Ben Belgacem, F., El Fekih, H. and Raymond, J.-P., A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions. Asymptot. Anal. 34 (2003) 121-136.
Bergounioux, M. and Kunisch, K., Augmented Lagrangian techniques for elliptic state constrained optimal control problems. SIAM J. Control Optim. 35 (1997) 1524-1543. CrossRef
A. Bossavit, Approximation régularisée d'un problème aux limites non homogène. Séminaire J.-L. Lions 12 (Avril 1969).
F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991).
Colli Franzoni, P., Approssimazione mediante il metodo de penalizazione de problemi misti di Dirichlet-Neumann per operatori lineari ellittici del secondo ordine. Boll. Un. Mat. Ital. A (7) 4 (1973) 229-250.
P. Colli Franzoni, Approximation of optimal control problems of systems described by boundary value mixed problems of Dirichlet-Neumann type, in 5th IFIP Conference on Optimization Techniques. Springer, Berlin, Lecture Notes in Computer Science 3 (1973) 152-162.
M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem. Commun. Partial Differential Equations 21 1919-1949 (1996).
M. Dauge, Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions. Springer-Verlag, Lecture Notes in Math. 1341 (1988).
P. Grisvard, Singularities in boundary value problems. Masson (1992).
Hou, L.S. and Ravindran, S.S., A penalized Neumann control approach for solving an optimal Dirichlet control problem for the Navier-Stokes equations. SIAM J. Control Optim. 20 (1998) 1795-1814. CrossRef
Hou, L.S. and Ravindran, S.S., Numerical approximation of optimal flow control problems by a penalty method: error estimates and numerical results. SIAM J. Sci. Comput. 20 (1999) 1753-1777. CrossRef
Kirsch, A., The Robin problem for the Helmholtz equation as a singular perturbation problem. Numer. Funct. Anal. Optim. 8 (1985) 1-20. CrossRef
Lasiecka, I. and Sokolowski, J., Semidiscrete approximation of hyperbolic boundary value problem with nonhomogeneous Dirichlet boundary conditions. SIAM J. Math. Anal. 20 (1989) 1366-1387. CrossRef
J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod (1968).
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vols. 1 and 2. Dunod, Paris (1968).
T. Masrour, Contrôlabilité et observabilité des sytèmes distribués, problèmes et méthodes. Thesis, École Nationale des Ponts et Chaussées. Paris (1995).