Approximate controllability of linear parabolic equations in perforated domains
ESAIM: Control, Optimisation and Calculus of Variations, Volume 6  (2001), p. 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 ε0, 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 ε0, 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
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {6},
     year = {2001},
     pages = {21-38},
     zbl = {0964.35015},
     mrnumber = {1804496},
     language = {en},
     url = {http://www.numdam.org/item/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, Volume 6 (2001) , pp. 21-38. http://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 0627.35057

[3] D. Cioranescu and P. Donato, An introduction to Homogenization. Oxford University Press (1999). | MR 1765047 | Zbl 0939.35001

[4] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl. 319 (1979) 509-607. | Zbl 0427.35073

[5] R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et Techniques. Masson, Tome 3, Paris (1985). | Zbl 0642.35001

[6] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell'area. Rend. Mat. 4 (1975) 277-294. | Zbl 0316.35036

[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 0339.49005

[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 0877.35014

[9] P. Donato and A. Nabil, Homogenization and correctors for heat equation in perforated domains. Ricerche di Matematica (to appear). | MR 1941824 | Zbl 1102.35305

[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 0770.35009

[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 0818.93032

[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 0752.93037

[13] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118-139. | Zbl 0631.35044

[14] E. Zuazua, Approximate controllability for linear parabolic equations with rapidly oscillating coefficients. Control Cybernet. 23 (1994) 1-8. | Zbl 0815.93041