Exact controllability of the wave equation in perforated domains

The wave equation with Dirichlet boundary controls in perforated domains with small holes is considered. It is well known that the wave equation in a perforated domain is not exactly controllable in all of H¹₀(Ω)×L² (Ω) with L² controls supported on the exterior boundary. However, by using Lions Hilbert uniqueness method (HUM) a Hilbert space can be constructed such that initial data belonging to its dual space may be controlled by L² boundary controls supported on the exterior boundary. Under suitable assumptions on the geometry and the asymptotic size of the holes, it is proven that this Hilbert space (constructed by HUM) converges to the whole H^-1(Ω)×L²(Ω) in a suitable sense as the measure of the holes goes to zero. The method of proof combines HUM and multiplier and homogenization techniques