A singular controllability problem with vanishing viscosity

Ioan Florin Bugariu; Sorin Micu

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 1, page 116-140
  • ISSN: 1292-8119

Abstract

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The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term, being multiplied by a small parameter ε devoted to tend to zero, vanishes in the limit. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to ε, under the assumption that α∈[0,1){½. It follows that, under this assumption, our starting question has a positive answer.

How to cite

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Bugariu, Ioan Florin, and Micu, Sorin. "A singular controllability problem with vanishing viscosity." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 116-140. <http://eudml.org/doc/272929>.

@article{Bugariu2014,
abstract = {The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term, being multiplied by a small parameter ε devoted to tend to zero, vanishes in the limit. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to ε, under the assumption that α∈[0,1)\{½. It follows that, under this assumption, our starting question has a positive answer.},
author = {Bugariu, Ioan Florin, Micu, Sorin},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {wave equation; null-controllability; vanishing viscosity; moment problem; biorthogonals},
language = {eng},
number = {1},
pages = {116-140},
publisher = {EDP-Sciences},
title = {A singular controllability problem with vanishing viscosity},
url = {http://eudml.org/doc/272929},
volume = {20},
year = {2014},
}

TY - JOUR
AU - Bugariu, Ioan Florin
AU - Micu, Sorin
TI - A singular controllability problem with vanishing viscosity
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2014
PB - EDP-Sciences
VL - 20
IS - 1
SP - 116
EP - 140
AB - The aim of this paper is to answer the question: Do the controls of a vanishing viscosity approximation of the one dimensional linear wave equation converge to a control of the conservative limit equation? The characteristic of our viscous term is that it contains the fractional power α of the Dirichlet Laplace operator. Through the parameter α we may increase or decrease the strength of the high frequencies damping which allows us to cover a large class of dissipative mechanisms. The viscous term, being multiplied by a small parameter ε devoted to tend to zero, vanishes in the limit. Our analysis, based on moment problems and biorthogonal sequences, enables us to evaluate the magnitude of the controls needed for each eigenmode and to show their uniform boundedness with respect to ε, under the assumption that α∈[0,1){½. It follows that, under this assumption, our starting question has a positive answer.
LA - eng
KW - wave equation; null-controllability; vanishing viscosity; moment problem; biorthogonals
UR - http://eudml.org/doc/272929
ER -

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