# Approximate Controllability of linear parabolic equations in perforated domains

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

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

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topDonato, 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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- P. Donato and A. Nabil, Homogenization and correctors for heat equation in perforated domains. Ricerche di Matematica (to appear).
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- C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability for the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A125 (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.
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- E. Zuazua, Approximate controllability for linear parabolic equations with rapidly oscillating coefficients. Control Cybernet.23 (1994) 1-8.

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