Approximate Controllability of linear parabolic equations in perforated domains

¹ Université de Rouen, UFR des Sciences, UPRES-A 6085 du CNRS, Site Colbert, 76821 Mont-Saint-Aignan, France; donato@univ-rouen.fr. and Laboratoire d'Analyse Numérique, Université Paris VI, Case Postale 187, 4 place Jussieu, 75252 Paris Cedex 05, France; donato@ann.jussieu.fr.
² ENSA Agadir, Université Ibn Zohr, BP. 33/S, Agadir, Morocco.

Received: 2 March 1999
Revised: 7 January 2000
Revised: 10 July 2000
Revised: 8 September 2000

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.