# Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass

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

- Volume: 2, page 231-280
- ISSN: 1292-8119

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topCastro, C.. "Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass." ESAIM: Control, Optimisation and Calculus of Variations 2 (2010): 231-280. <http://eudml.org/doc/197331>.

@article{Castro2010,

abstract = {
We consider a hybrid, one-dimensional, linear system consisting
in two flexible strings connected by a point mass. It is known
that this system presents two interesting features. First, it is well
posed in an asymmetric space in which solutions have one more degree
of regularity to one side of the point mass. Second, that the spectral
gap vanishes asymptotically. We prove that the first property is a
consequence of the second one. We also consider a system in which the
point mass is replaced by a string of length 2ε and density 1/2ε. We
show that, as ε → 0, the
solutions of this system converge to those of the original one. We also
analyze the convergence of the spectrum and obtain the well-posedness
of the limit system in the asymmetric space as a consequence of non-standard
uniform bounds of solutions of the approximate problems. Finally we consider
the controllability problem. It is well known that the limit system with
L-controls on one end is exactly controllable in an asymmetric space. We
show how this result can be obtained as the limit when ε → 0 of partial controllability results for the approximate systems in
which the number of controlled frequencies converges to infinity as ε → 0. This is done by means of some new results on
non-harmonic Fourier series.
},

author = {Castro, C.},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Hybrid system / point mass / asymptotic analysis /
controllability.},

language = {eng},

month = {3},

pages = {231-280},

publisher = {EDP Sciences},

title = {Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass},

url = {http://eudml.org/doc/197331},

volume = {2},

year = {2010},

}

TY - JOUR

AU - Castro, C.

TI - Asymptotic analysis and control of a hybrid system composed by two vibrating strings connected by a point mass

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 2

SP - 231

EP - 280

AB -
We consider a hybrid, one-dimensional, linear system consisting
in two flexible strings connected by a point mass. It is known
that this system presents two interesting features. First, it is well
posed in an asymmetric space in which solutions have one more degree
of regularity to one side of the point mass. Second, that the spectral
gap vanishes asymptotically. We prove that the first property is a
consequence of the second one. We also consider a system in which the
point mass is replaced by a string of length 2ε and density 1/2ε. We
show that, as ε → 0, the
solutions of this system converge to those of the original one. We also
analyze the convergence of the spectrum and obtain the well-posedness
of the limit system in the asymmetric space as a consequence of non-standard
uniform bounds of solutions of the approximate problems. Finally we consider
the controllability problem. It is well known that the limit system with
L-controls on one end is exactly controllable in an asymmetric space. We
show how this result can be obtained as the limit when ε → 0 of partial controllability results for the approximate systems in
which the number of controlled frequencies converges to infinity as ε → 0. This is done by means of some new results on
non-harmonic Fourier series.

LA - eng

KW - Hybrid system / point mass / asymptotic analysis /
controllability.

UR - http://eudml.org/doc/197331

ER -

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