Approximate Boundary Controllability for the Wave Equation in Perforated Domains

Abstract

This paper considers the wave equation in a perforated domain with holes of size $r(\epsilon )$ distributed with $\epsilon $-periodicity, with the assumption that there exists a neighborhood of the exterior boundary without holes. The following question is asked: Is it possible to approximately control the wave equation in the perforated domain in such a way that when $\epsilon $ goes to zero the exact controllability of the limit system is obtained? Two main theorems give a positive answer to this question when $r(\epsilon )$ is the critical size that transforms at the limit the operator $({{\partial ^2 } / {\partial t^2 }}) - \Delta $ into $({{\partial ^2 } / {\partial t^2 }}) - \Delta + \mu $, where $\mu $ is a positive measure. In the first theorem; in a suitable sense, $L^2 (\Omega ) \times H^{ - 1} (\Omega )$ is approximated by the space of the admissible data for the exact controllability in the perforated domain. In the second, it is shown that the limit control, supported only by the exterior boundary of the perforated domain, is such that the related state in the perforated domain goes at the limit to the equilbrium state at the time T.