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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

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