Approximate Controllability of linear parabolic equations in perforated domains

Patrizia Donato; Aïssam Nabil

ESAIM: Control, Optimisation and Calculus of Variations (2010)

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

Abstract

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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

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Donato, Patrizia, and Nabil, Aïssam. "Approximate Controllability of linear parabolic equations in perforated domains." ESAIM: Control, Optimisation and Calculus of Variations 6 (2010): 21-38. <http://eudml.org/doc/197310>.

@article{Donato2010,
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. },
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},
month = {3},
pages = {21-38},
publisher = {EDP Sciences},
title = {Approximate Controllability of linear parabolic equations in perforated domains},
url = {http://eudml.org/doc/197310},
volume = {6},
year = {2010},
}

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
DA - 2010/3//
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 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.
LA - eng
KW - Linear parabolic equation; approximate controlability; homogenization.; rapidly oscillating coefficients; periodically perforated domain
UR - http://eudml.org/doc/197310
ER -

References

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  11. C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability for the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A125 (1995) 31-61.  Zbl0818.93032
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  14. E. Zuazua, Approximate controllability for linear parabolic equations with rapidly oscillating coefficients. Control Cybernet.23 (1994) 1-8.  Zbl0815.93041

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