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Pointwise and spectral control of plate vibrations. (English) Zbl 0778.73045

The problem of controlling pointwise (by means of a time-dependent Dirac measure supported by a given point) the motion of a vibrating plate is considered. 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, 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, is shown. 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 \(\mathbb{R}^ 2\), the problem of internal spectral control is solved, that 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 \((0,T)\times\omega\).

MSC:

74M05 Control, switches and devices (“smart materials”) in solid mechanics
74H45 Vibrations in dynamical problems in solid mechanics
74K20 Plates
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