Approximate controllability of linear parabolic equations in perforated domains
ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 21-38.

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.

Classification : 35K05, 49A50, 93C20
Mots-clés : linear parabolic equation, approximate controlability, homogenization 
@article{COCV_2001__6__21_0,
     author = {Donato, Patrizia and Nabil, A{\"\i}ssam},
     title = {Approximate controllability of linear parabolic equations in perforated domains},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {21--38},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2001},
     mrnumber = {1804496},
     zbl = {0964.35015},
     language = {en},
     url = {https://www.numdam.org/item/COCV_2001__6__21_0/}
}
TY  - JOUR
AU  - Donato, Patrizia
AU  - Nabil, Aïssam
TI  - Approximate controllability of linear parabolic equations in perforated domains
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2001
SP  - 21
EP  - 38
VL  - 6
PB  - EDP-Sciences
UR  - https://www.numdam.org/item/COCV_2001__6__21_0/
LA  - en
ID  - COCV_2001__6__21_0
ER  - 
%0 Journal Article
%A Donato, Patrizia
%A Nabil, Aïssam
%T Approximate controllability of linear parabolic equations in perforated domains
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2001
%P 21-38
%V 6
%I EDP-Sciences
%U https://www.numdam.org/item/COCV_2001__6__21_0/
%G en
%F COCV_2001__6__21_0
Donato, Patrizia; Nabil, Aïssam. Approximate controllability of linear parabolic equations in perforated domains. ESAIM: Control, Optimisation and Calculus of Variations, Tome 6 (2001), pp. 21-38. https://www.numdam.org/item/COCV_2001__6__21_0/

[1] C. Brizzi and J.P. Chalot, Homogénéisation dans des ouverts à frontière fortement oscillante. Thèse à l'Université de Nice (1978).

[2] D. Cioranescu and P. Donato, Exact internal controllability in perforated domains. J. Math. Pures Appl. 319 (1989) 185-213. | Zbl

[3] D. Cioranescu and P. Donato, An introduction to Homogenization. Oxford University Press (1999). | MR | Zbl

[4] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl. 319 (1979) 509-607. | Zbl

[5] R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et Techniques. Masson, Tome 3, Paris (1985). | Zbl

[6] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell'area. Rend. Mat. 4 (1975) 277-294. | Zbl

[7] E. De Giorgi and T. Franzoni, Su un tipo di convergenza variazionale. Atti. Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (8) 58 (1975) 842-850. | Zbl

[8] P. Donato and A. Nabil, 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. | Zbl

[9] P. Donato and A. Nabil, Homogenization and correctors for heat equation in perforated domains. Ricerche di Matematica (to appear). | MR | Zbl

[10] C. Fabre, J.P. Puel and E. Zuazua, Contrôlabilité approchée de l'équation de la chaleur semilinéaire. C. R. Acad. Sci. Paris Sér. I Math. 314 (1992) 807-812. | Zbl

[11] 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. | Zbl

[12] 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. | Zbl

[13] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118-139. | Zbl

[14] E. Zuazua, Approximate controllability for linear parabolic equations with rapidly oscillating coefficients. Control Cybernet. 23 (1994) 1-8. | Zbl