Singular perturbation for the Dirichlet boundary control of elliptic problems

Faker Ben Belgacem; Henda El Fekih; Hejer Metoui

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 37, Issue: 5, page 833-850
  • ISSN: 0764-583X

Abstract

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

How to cite

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Ben Belgacem, Faker, El Fekih, Henda, and Metoui, Hejer. "Singular perturbation for the Dirichlet boundary control of elliptic problems." ESAIM: Mathematical Modelling and Numerical Analysis 37.5 (2010): 833-850. <http://eudml.org/doc/194193>.

@article{BenBelgacem2010,
abstract = { A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small penalization parameter. Some numerical results are reported on to highlight the reliability of such an approach. },
author = {Ben Belgacem, Faker, El Fekih, Henda, Metoui, Hejer},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Boundary control problems; non-smooth Dirichlet condition; Robin penalization; singularly perturbed problem.; boundary control problems; non-smooth Dirichlet boundary condition; Robin penalization; singularly perturbed problem; optimal control; regularity},
language = {eng},
month = {3},
number = {5},
pages = {833-850},
publisher = {EDP Sciences},
title = {Singular perturbation for the Dirichlet boundary control of elliptic problems},
url = {http://eudml.org/doc/194193},
volume = {37},
year = {2010},
}

TY - JOUR
AU - Ben Belgacem, Faker
AU - El Fekih, Henda
AU - Metoui, Hejer
TI - Singular perturbation for the Dirichlet boundary control of elliptic problems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 37
IS - 5
SP - 833
EP - 850
AB - A current procedure that takes into account the Dirichlet boundary condition with non-smooth data is to change it into a Robin type condition by introducing a penalization term; a major effect of this procedure is an easy implementation of the boundary condition. In this work, we deal with an optimal control problem where the control variable is the Dirichlet data. We describe the Robin penalization, and we bound the gap between the penalized and the non-penalized boundary controls for the small penalization parameter. Some numerical results are reported on to highlight the reliability of such an approach.
LA - eng
KW - Boundary control problems; non-smooth Dirichlet condition; Robin penalization; singularly perturbed problem.; boundary control problems; non-smooth Dirichlet boundary condition; Robin penalization; singularly perturbed problem; optimal control; regularity
UR - http://eudml.org/doc/194193
ER -

References

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  1. D.A. Adams, Sobolev Spaces. Academic Press, New York (1975).  
  2. N. Arada, H. El Fekih and J.-P. Raymond, Asymptotic analysis of some control problems. Asymptot. Anal.24 (2000) 343-366.  Zbl0979.49020
  3. I. Babuska, The finite element method with penalty. Math. Comp.27 (1973) 221-228.  Zbl0299.65057
  4. F. Ben Belgacem, H. El Fekih and J.-P. Raymond, A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions. Asymptot. Anal.34 (2003) 121-136.  Zbl1043.35014
  5. M. Bergounioux and K. Kunisch, Augmented Lagrangian techniques for elliptic state constrained optimal control problems. SIAM J. Control Optim.35 (1997) 1524-1543.  Zbl0897.49001
  6. A. Bossavit, Approximation régularisée d'un problème aux limites non homogène. Séminaire J.-L. Lions12 (Avril 1969).  Zbl0204.48403
  7. F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991).  Zbl0788.73002
  8. P. Colli Franzoni, 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.  Zbl0266.35024
  9. 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.  
  10. M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem. Commun. Partial Differential Equations21 1919-1949 (1996).  Zbl0879.35017
  11. M. Dauge, Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions. Springer-Verlag, Lecture Notes in Math. 1341 (1988).  Zbl0668.35001
  12. P. Grisvard, Singularities in boundary value problems. Masson (1992).  Zbl0766.35001
  13. L.S. Hou and S.S. Ravindran, 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.  Zbl0917.49003
  14. L.S. Hou and S.S. Ravindran, Numerical approximation of optimal flow control problems by a penalty method: error estimates and numerical results. SIAM J. Sci. Comput.20 (1999) 1753-1777.  Zbl0952.93036
  15. A. Kirsch, The Robin problem for the Helmholtz equation as a singular perturbation problem. Numer. Funct. Anal. Optim.8 (1985) 1-20.  Zbl0622.65107
  16. I. Lasiecka and J. Sokolowski, Semidiscrete approximation of hyperbolic boundary value problem with nonhomogeneous Dirichlet boundary conditions. SIAM J. Math. Anal.20 (1989) 1366-1387.  Zbl0704.35085
  17. J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod (1968).  Zbl0179.41801
  18. J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vols. 1 and 2. Dunod, Paris (1968).  Zbl0165.10801
  19. T. Masrour, Contrôlabilité et observabilité des sytèmes distribués, problèmes et méthodes. Thesis, École Nationale des Ponts et Chaussées. Paris (1995).  

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