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

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

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

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topLeón, 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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