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Controllability of the one-dimensional wave equation with rapidly oscillating density. (Contrôlabilité de l’équation des ondes à densité rapidement oscillante à une dimension d’espace.) (French. Abridged English version) Zbl 1007.93036

Summary: We consider the one-dimensional wave equation with periodic density of period \(\epsilon\to 0\). By a counterexample due to Avellaneda, Bardos, and Rauch, we know that the boundary controllability property does not hold uniformly as \(\epsilon\to 0\). We prove that the control remains uniformly bounded if we control the projection of the solution over the subspace generated by the eigenfunctions associated with the eigenvalues \(\lambda\leq C\epsilon^{-2}\), \(C>0\) being small enough. This result is sharp in the sense that the control diverges when the projection over the eigenfunctions such that \(\lambda\sim C\epsilon^{-2}\), with \(C\) large, is controlled. We use the classical WKB asymptotic development, which provides sharp results on the convergence of the spectrum and the theory of non-harmonic Fourier series.

MSC:

93C20 Control/observation systems governed by partial differential equations
35L05 Wave equation
93B05 Controllability
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