Cioranescu, Doina; Donato, Patrizia Exact internal controllability in performed domains. (English) Zbl 0627.35057 J. Math. Pures Appl., IX. Sér. 68, No. 2, 185-213 (1989). We consider a system described by a generalized wave equation in the cylinder \(\Omega _{\epsilon}\times (0,T)\) where \(\Omega _{\epsilon}\) is obtained by removing a number \(N_{\epsilon}\) of holes periodically distributed of size \(\epsilon\), from a fixed domain \(\Omega\). Hilbert uniqueness method (HUM) gives an exact control \(v_{\epsilon}\) for every fixed \(\epsilon\). This means we can act on the system to drive it to rest after a finite time T, arbitrarily chosen. If \(\epsilon\) \(\to 0\) (i.e. \(N_{\epsilon}\to \infty)\) the main questions are: does the control \(v_{\epsilon}\) converge to a limit v, and if so, does v control a limit system? In this work we give a positive answer to these questions in the case where \(v_{\epsilon}\) is an internal control, i.e., \(v_{\epsilon}\) is applied on the whole \(\Omega _{\epsilon}\). Let \(\epsilon\) \(\to 0\) be a homogenization process: the initial state equation with rapidly oscillating coefficients defined on a domain depending on \(\epsilon\), is replaced by a homogenized equation with constant coefficients, given on the whole domain \(\Omega\). We prove that \(v_{\epsilon}\) converges to a function f which is an exact control for the homogenized equation. In order to prove this result we give preliminary homogenization theorems for the wave equation in perforated domain. Cited in 2 ReviewsCited in 18 Documents MSC: 35L20 Initial-boundary value problems for second-order hyperbolic equations 93B05 Controllability 35B40 Asymptotic behavior of solutions to PDEs 35B20 Perturbations in context of PDEs Keywords:generalized wave equation; Hilbert uniqueness method; exact control; limit system; internal control; homogenization process; initial state equation; rapidly oscillating coefficients; constant coefficients PDFBibTeX XMLCite \textit{D. Cioranescu} and \textit{P. Donato}, J. Math. Pures Appl. (9) 68, No. 2, 185--213 (1989; Zbl 0627.35057)