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Contrôlabilité exacte frontière de l’équation des ondes dans des domaines avec des petits trous. (Exact boundary controllability for the wave equation in domains with small holes). (French) Zbl 0694.49003
Let $$\Omega$$ be a bounded domain of $$R^ n$$ with smooth boundary $$\Gamma$$ and $$\Omega_{\epsilon}$$ the domain obtained by removing from $$\Omega$$ a set $$S_{\epsilon}$$ of closed smooth subsets of $$\Omega$$. It is assumed that the measure of $$S_{\epsilon}$$ goes to 0 as $$\epsilon$$ $$\to 0$$. Let $$y_{\epsilon}$$ be the solution of $y''_{\epsilon}- \Delta y_{\epsilon}=0\quad in\quad \Omega_{\epsilon}\times (0,T);\quad y_{\epsilon}=v_{\epsilon}\quad on\quad \Sigma_{\epsilon}=\partial \Omega_{\epsilon}\times (0,T),$ $y_{\epsilon}(0)=y^ 0_{\epsilon},\quad y'_{\epsilon}(0)=y^ 1_{\epsilon}\quad in\quad \Omega_{\epsilon}$ with $$\{y^ 0_{\epsilon},y^ 1_{\epsilon}\}=\{y^ 0,y^ 1\}\cdot \chi_{\Omega_{\epsilon}}$$, $$\{y^ 0,y^ 1\}\in L^ 2(\Omega)\times L^ 2(\Omega)$$. The existence of a uniform exact controllability time $$T_ 0>0$$ is known from the work of J. L. Lions [“Contrôlabilité exact, perturbations et stabilisation de systèmes distribués”, Tome 1: “Contrôlabilité exacte” (1988; Zbl 0653.93002)]. Namely, for every $$T>T_ 0$$ there exists a control $$v_{\epsilon}\in L^ 2(\Sigma_{\epsilon})$$ such that $$y_{\epsilon}(T)=y'_{\epsilon}(T)=y'_{\epsilon}(T)=0.$$
The author constructs a sequence of controls $$\{v_{\epsilon}\}$$ such that $$\tilde y_{\epsilon}$$ $$(=$$ the extension of $$y_{\epsilon}$$ by 0 to the whole of $$\Omega$$ $$\times (0,T))$$ converges to y in the weak* topology of $$L^{\infty}(0,T;L^ 2(\Omega))$$, where y is the solution of some related initial-boundary value problem satisfying $$y(T)=y'(T)=0$$. It is remarkable that the boundedness of $$\{y_{\epsilon}\}$$ is shown without knowing if $$\{v_{\epsilon}\}$$ is bounded.
Reviewer: H.Tanabe

##### MSC:
 49J10 Existence theories for free problems in two or more independent variables 93B03 Attainable sets, reachability 93B05 Controllability 93C20 Control/observation systems governed by partial differential equations 35L05 Wave equation
##### Keywords:
Hilbert uniqueness method