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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