Pointwise and Spectral Control of Plate Vibrations

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 $\Omega$. 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 $\Omega$ $where$ $no$ $eigenfunction$ $vanishes$ at any time greater than half of the plate's area. This result is optimal since $no$ 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 $arbitrary$ domain $\Omega$ in ${\mathbb{R}}^2$, we solve the problem of $internal$ spectral control, which means that for any open disk $\omega \subset \Omega$, any finite linear combination of the eigenfunctions can be set to equilibrium by means of a control function $h\in \mathcal{D}((0,T)\times \Omega )$ supported in $(0,T)\times \omega$.