Singular perturbation for the Dirichlet boundary control of elliptic problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 5, p. 833-850

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.

DOI : https://doi.org/10.1051/m2an:2003057
Classification:  49N05,  49N10,  34D15
Keywords: boundary control problems, non-smooth Dirichlet condition, Robin penalization, singularly perturbed problem
@article{M2AN_2003__37_5_833_0,
     author = {Belgacem, Faker Ben and Fekih, Henda El and Metoui, Hejer},
     title = {Singular perturbation for the Dirichlet boundary control of elliptic problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {5},
     year = {2003},
     pages = {833-850},
     doi = {10.1051/m2an:2003057},
     zbl = {1051.49012},
     mrnumber = {2020866},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2003__37_5_833_0}
}
Belgacem, Faker Ben; Fekih, Henda El; Metoui, Hejer. Singular perturbation for the Dirichlet boundary control of elliptic problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 5, pp. 833-850. doi : 10.1051/m2an:2003057. http://www.numdam.org/item/M2AN_2003__37_5_833_0/

[1] D.A. Adams, Sobolev Spaces. Academic Press, New York (1975). | MR 450957 | Zbl 0314.46030

[2] N. Arada, H. El Fekih and J.-P. Raymond, Asymptotic analysis of some control problems. Asymptot. Anal. 24 (2000) 343-366. | Zbl 0979.49020

[3] I. Babuška, The finite element method with penalty. Math. Comp. 27 (1973) 221-228. | Zbl 0299.65057

[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. | Zbl 1043.35014

[5] M. Bergounioux and K. Kunisch, Augmented Lagrangian techniques for elliptic state constrained optimal control problems. SIAM J. Control Optim. 35 (1997) 1524-1543. | Zbl 0897.49001

[6] A. Bossavit, Approximation régularisée d'un problème aux limites non homogène. Séminaire J.-L. Lions 12 (Avril 1969). | Zbl 0204.48403

[7] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002

[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. | Zbl 0266.35024

[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. | Zbl 0293.49016

[10] M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem. Commun. Partial Differential Equations 21 1919-1949 (1996). | Zbl 0879.35017

[11] M. Dauge, Elliptic boundary value problems on corner domains. Smoothness and asymptotics of solutions. Springer-Verlag, Lecture Notes in Math. 1341 (1988). | MR 961439 | Zbl 0668.35001

[12] P. Grisvard, Singularities in boundary value problems. Masson (1992). | MR 1173209 | Zbl 0766.35001

[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. | Zbl 0917.49003

[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. | Zbl 0952.93036

[15] A. Kirsch, The Robin problem for the Helmholtz equation as a singular perturbation problem. Numer. Funct. Anal. Optim. 8 (1985) 1-20. | Zbl 0622.65107

[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. | Zbl 0704.35085

[17] J.-L. Lions, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod (1968). | MR 244606 | Zbl 0179.41801

[18] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vols. 1 and 2. Dunod, Paris (1968). | MR 247243 | Zbl 0165.10801

[19] T. Masrour, Contrôlabilité et observabilité des sytèmes distribués, problèmes et méthodes. Thesis, École Nationale des Ponts et Chaussées. Paris (1995).