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Approximate Controllability of linear parabolic equationsin perforated domains

Published online by Cambridge University Press:  15 August 2002

Patrizia Donato  and
Aïssam Nabil
Patrizia Donato
Affiliation:
Université de Rouen, UFR des Sciences, UPRES-A 6085 du CNRS, Site Colbert, 76821 Mont-Saint-Aignan, France; [email protected]. and Laboratoire d'Analyse Numérique, Université Paris VI, Case Postale 187, 4 place Jussieu, 75252 Paris Cedex 05, France; [email protected].
Aïssam Nabil
Affiliation:
ENSA Agadir, Université Ibn Zohr, BP. 33/S, Agadir, Morocco.
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Abstract

In this paper we consider an approximate controllability problem for linear parabolic equations with rapidly oscillating coefficients in a periodically perforated domain. The holes are ε-periodic and of size ε. We show that, as ε → 0, the approximate control and the corresponding solution converge respectively to the approximate control and to the solution of the homogenized problem. In the limit problem, the approximation of the final state is alterated by a constant which depends on the proportion of material in the perforated domain and is equal to 1 when there are no holes. We also prove that the solution of the approximate controllability problem in the perforated domain behaves, as ε → 0, as that of the problem posed in the perforated domain having as rigth-hand side the (fixed) control of the limit problem.

Keywords

Linear parabolic equationapproximate controlabilityhomogenization.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 6 , 2001 , pp. 21 - 38
Copyright
© EDP Sciences, SMAI, 2001

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References

C. Brizzi and J.P. Chalot, Homogénéisation dans des ouverts à frontière fortement oscillante. Thèse à l'Université de Nice (1978).
Cioranescu, D. and Donato, P., Exact internal controllability in perforated domains. J. Math. Pures Appl. 319 (1989) 185-213.
D. Cioranescu and P. Donato, An introduction to Homogenization. Oxford University Press (1999).
Cioranescu, D. and Saint Jean Paulin, J., Homogenization in open sets with holes. J. Math. Anal. Appl. 319 (1979) 509-607.
R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et Techniques. Masson, Tome 3, Paris (1985).
De Giorgi, E., Sulla convergenza di alcune successioni di integrali del tipo dell'area. Rend. Mat. 4 (1975) 277-294.
De Giorgi, E. and Franzoni, T., Su un tipo di convergenza variazionale. Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8) 58 (1975) 842-850.
Donato, P. and Nabil, A., Homogénéisation et contrôlabilité approchée de l'équation de la chaleur dans des domaines perforés. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 789-794. CrossRef
P. Donato and A. Nabil, Homogenization and correctors for heat equation in perforated domains. Ricerche di Matematica (to appear).
Fabre, C., Puel, J.P. and Zuazua, E., Contrôlabilité approchée de l'équation de la chaleur semilinéaire. C. R. Acad. Sci. Paris Sér. I Math. 314 (1992) 807-812.
C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability for the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995) 31-61.
J.-L. Lions, Remarques sur la contrôlabilité approchée, in Jornadas Hispano-Francesas sobre Control de Sistemas Distribuidos, octubre 1990. Grupo de Análisis Matemático Aplicado de la University of Málaga, Spain (1991) 77-87.
Saut, J.-C. and Scheurer, B., Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118-139. CrossRef
Zuazua, E., Approximate controllability for linear parabolic equations with rapidly oscillating coefficients. Control Cybernet. 23 (1994) 1-8.