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.