# Pointwise and spectral control of plate vibrations.

• Volume: 7, Issue: 1, page 1-24
• ISSN: 0213-2230

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## Abstract

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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. Under general boundary conditions, including the special cases of simply supported or clamped plates, but of course excluding the cases where multiple eigenvalues exist for the biharmonic operator, we show the controlability of finite linear combinations of the eigenfunctions at any point of Ω where no eigenfunction vanishes at any time greater than half of the plate area. This result is optimal since no finite linear combination of the functions other than 0 is pointwise controllable at a time smaller than the plate's area. Under the same condition on the time, but for any domain Ω in R2, we solve the problem of internal spectral control, which means that for any open disk ω ⊂ Ω, any finite linear combination of eigenfunctions can be set to equilibrium by means of a control function h ∈ D((0,T) x Ω) supported in (0,T) x ω.

## How to cite

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Haraux, Alain, and Jaffard, Stéphane. "Pointwise and spectral control of plate vibrations.." Revista Matemática Iberoamericana 7.1 (1991): 1-24. <http://eudml.org/doc/39409>.

@article{Haraux1991,
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. Under general boundary conditions, including the special cases of simply supported or clamped plates, but of course excluding the cases where multiple eigenvalues exist for the biharmonic operator, we show the controlability of finite linear combinations of the eigenfunctions at any point of Ω where no eigenfunction vanishes at any time greater than half of the plate area. This result is optimal since no finite linear combination of the functions other than 0 is pointwise controllable at a time smaller than the plate's area. Under the same condition on the time, but for any domain Ω in R2, we solve the problem of internal spectral control, which means that for any open disk ω ⊂ Ω, any finite linear combination of eigenfunctions can be set to equilibrium by means of a control function h ∈ D((0,T) x Ω) supported in (0,T) x ω.},
author = {Haraux, Alain, Jaffard, Stéphane},
journal = {Revista Matemática Iberoamericana},
keywords = {Vibraciones; Placas; Métodos fisicomatemáticos; Autofunciones; Control; Espectro de vibración; Masas de Dirac; Punto fijo; time-dependent Dirac measure; biharmonic operator; finite linear combinations of the eigenfunctions; internal spectral control},
language = {eng},
number = {1},
pages = {1-24},
title = {Pointwise and spectral control of plate vibrations.},
url = {http://eudml.org/doc/39409},
volume = {7},
year = {1991},
}

TY - JOUR
AU - Haraux, Alain
AU - Jaffard, Stéphane
TI - Pointwise and spectral control of plate vibrations.
JO - Revista Matemática Iberoamericana
PY - 1991
VL - 7
IS - 1
SP - 1
EP - 24
AB - 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. Under general boundary conditions, including the special cases of simply supported or clamped plates, but of course excluding the cases where multiple eigenvalues exist for the biharmonic operator, we show the controlability of finite linear combinations of the eigenfunctions at any point of Ω where no eigenfunction vanishes at any time greater than half of the plate area. This result is optimal since no finite linear combination of the functions other than 0 is pointwise controllable at a time smaller than the plate's area. Under the same condition on the time, but for any domain Ω in R2, we solve the problem of internal spectral control, which means that for any open disk ω ⊂ Ω, any finite linear combination of eigenfunctions can be set to equilibrium by means of a control function h ∈ D((0,T) x Ω) supported in (0,T) x ω.
LA - eng
KW - Vibraciones; Placas; Métodos fisicomatemáticos; Autofunciones; Control; Espectro de vibración; Masas de Dirac; Punto fijo; time-dependent Dirac measure; biharmonic operator; finite linear combinations of the eigenfunctions; internal spectral control
UR - http://eudml.org/doc/39409
ER -

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