JournalsrmiVol. 7, No. 1pp. 1–24

Pointwise and Spectral Control of Plate Vibrations

  • Alain Haraux

    Université Pierre et Marie Curie, Paris, France
  • Stéphane Jaffard

    Université Paris Est, Créteil, France
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Abstract

We consider the problem of controlling pointwise (by means of a time dependent Dirac measure supported by a given point) the motion of a vibrating plate Ω\Omega. Under general boundary conditions, including the special cases of simply supported or clamped plates, but of course excluding the cases where some multiple eigenvalues exist for the biharmonic operator, we show the controllability of finite linear combinations of the eigenfunctions at any point of Ω\Omega wherewhere nono eigenfunctioneigenfunction vanishesvanishes at any time greater than half of the plate's area. This result is optimal since nono finite linear combination of the eigenfunctions other than 0 is pointwise controllable at a time smaller than half of the plate's area. Under the same condition on the time, but for an arbitraryarbitrary domain Ω\Omega in R2\mathbb R^2, we solve the problem of internalinternal spectral control, which means that for any open disk ωΩ\omega \subset \Omega, any finite linear combination of the eigenfunctions can be set to equilibrium by means of a control function hD((0,T)×Ω)h \in \mathcal D ((0, T) \times \Omega) supported in (0,T)×ω(0, T) \times \omega.

Cite this article

Alain Haraux, Stéphane Jaffard, Pointwise and Spectral Control of Plate Vibrations. Rev. Mat. Iberoam. 7 (1991), no. 1, pp. 1–24

DOI 10.4171/RMI/103