Approximate controllability of linear parabolic equations in perforated domains

Patrizia Donato; Aïssam Nabil

ESAIM: Control, Optimisation and Calculus of Variations (2001)

  • Volume: 6, page 21-38
  • ISSN: 1292-8119

Abstract

top
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.

How to cite

top

Donato, Patrizia, and Nabil, Aïssam. "Approximate controllability of linear parabolic equations in perforated domains." ESAIM: Control, Optimisation and Calculus of Variations 6 (2001): 21-38. <http://eudml.org/doc/90592>.

@article{Donato2001,
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 $\varepsilon $-periodic and of size $\varepsilon $. We show that, as $\varepsilon \rightarrow 0$, 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 $\varepsilon \rightarrow 0$, as that of the problem posed in the perforated domain having as rigth-hand side the (fixed) control of the limit problem.},
author = {Donato, Patrizia, Nabil, Aïssam},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {linear parabolic equation; approximate controlability; homogenization; rapidly oscillating coefficients; periodically perforated domain},
language = {eng},
pages = {21-38},
publisher = {EDP-Sciences},
title = {Approximate controllability of linear parabolic equations in perforated domains},
url = {http://eudml.org/doc/90592},
volume = {6},
year = {2001},
}

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
PB - EDP-Sciences
VL - 6
SP - 21
EP - 38
AB - 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 $\varepsilon $-periodic and of size $\varepsilon $. We show that, as $\varepsilon \rightarrow 0$, 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 $\varepsilon \rightarrow 0$, as that of the problem posed in the perforated domain having as rigth-hand side the (fixed) control of the limit problem.
LA - eng
KW - linear parabolic equation; approximate controlability; homogenization; rapidly oscillating coefficients; periodically perforated domain
UR - http://eudml.org/doc/90592
ER -

References

top
  1. [1] C. Brizzi and J.P. Chalot, Homogénéisation dans des ouverts à frontière fortement oscillante. Thèse à l’Université de Nice (1978). 
  2. [2] D. Cioranescu and P. Donato, Exact internal controllability in perforated domains. J. Math. Pures Appl. 319 (1989) 185–213. Zbl0627.35057
  3. [3] D. Cioranescu and P. Donato, An introduction to Homogenization. Oxford University Press (1999). Zbl0939.35001MR1765047
  4. [4] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl. 319 (1979) 509–607. Zbl0427.35073
  5. [5] R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérique pour les Sciences et Techniques. Masson, Tome 3, Paris (1985). Zbl0642.35001
  6. [6] E. De Giorgi, Sulla convergenza di alcune successioni di integrali del tipo dell’area. Rend. Mat. 4 (1975) 277–294. Zbl0316.35036
  7. [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. Zbl0339.49005
  8. [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. Zbl0877.35014
  9. [9] P. Donato and A. Nabil, Homogenization and correctors for heat equation in perforated domains. Ricerche di Matematica (to appear). Zbl1102.35305MR1941824
  10. [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. Zbl0770.35009
  11. [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. Zbl0818.93032
  12. [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. Zbl0752.93037
  13. [13] J.-C. Saut and B. Scheurer, Unique continuation for some evolution equations. J. Differential Equations 66 (1987) 118–139. Zbl0631.35044
  14. [14] E. Zuazua, Approximate controllability for linear parabolic equations with rapidly oscillating coefficients. Control Cybernet. 23 (1994) 1–8. Zbl0815.93041

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.