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Exact internal controllability in performed domains. (English) Zbl 0627.35057
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.

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
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