Boundary controllability of the finite-difference space semi-discretizations of the beam equation

Liliana León; Enrique Zuazua

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 827-862
  • ISSN: 1292-8119

Abstract

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We propose a finite difference semi-discrete scheme for the approximation of the boundary exact controllability problem of the 1-D beam equation modelling the transversal vibrations of a beam with fixed ends. First of all we show that, due to the high frequency spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural functional setting. We then prove that there are two ways of restoring the uniform controllability property: a) filtering the high frequencies, i.e. controlling projections on subspaces where the high frequencies have been filtered; b) adding an extra boundary control to kill the spurious high frequency oscillations. In both cases the convergence of controls and controlled solutions is proved in weak and strong topologies, under suitable assumptions on the convergence of the initial data.

How to cite

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León, Liliana, and Zuazua, Enrique. "Boundary controllability of the finite-difference space semi-discretizations of the beam equation." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 827-862. <http://eudml.org/doc/90673>.

@article{León2010,
abstract = { We propose a finite difference semi-discrete scheme for the approximation of the boundary exact controllability problem of the 1-D beam equation modelling the transversal vibrations of a beam with fixed ends. First of all we show that, due to the high frequency spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural functional setting. We then prove that there are two ways of restoring the uniform controllability property: a) filtering the high frequencies, i.e. controlling projections on subspaces where the high frequencies have been filtered; b) adding an extra boundary control to kill the spurious high frequency oscillations. In both cases the convergence of controls and controlled solutions is proved in weak and strong topologies, under suitable assumptions on the convergence of the initial data. },
author = {León, Liliana, Zuazua, Enrique},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Beam equation; finite difference semi-discretization; exact boundary controllability.; beam equation; exact boundary controllability},
language = {eng},
month = {3},
pages = {827-862},
publisher = {EDP Sciences},
title = {Boundary controllability of the finite-difference space semi-discretizations of the beam equation},
url = {http://eudml.org/doc/90673},
volume = {8},
year = {2010},
}

TY - JOUR
AU - León, Liliana
AU - Zuazua, Enrique
TI - Boundary controllability of the finite-difference space semi-discretizations of the beam equation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 827
EP - 862
AB - We propose a finite difference semi-discrete scheme for the approximation of the boundary exact controllability problem of the 1-D beam equation modelling the transversal vibrations of a beam with fixed ends. First of all we show that, due to the high frequency spurious oscillations, the uniform (with respect to the mesh-size) controllability property of the semi-discrete model fails in the natural functional setting. We then prove that there are two ways of restoring the uniform controllability property: a) filtering the high frequencies, i.e. controlling projections on subspaces where the high frequencies have been filtered; b) adding an extra boundary control to kill the spurious high frequency oscillations. In both cases the convergence of controls and controlled solutions is proved in weak and strong topologies, under suitable assumptions on the convergence of the initial data.
LA - eng
KW - Beam equation; finite difference semi-discretization; exact boundary controllability.; beam equation; exact boundary controllability
UR - http://eudml.org/doc/90673
ER -

References

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