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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.
@article{COCV_2014__20_1_116_0,
author = {Bugariu, Ioan Florin and Micu, Sorin},
title = {A singular controllability problem with vanishing viscosity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {116--140},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
doi = {10.1051/cocv/2013057},
mrnumber = {3182693},
zbl = {1282.93047},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013057/}
}
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 SP - 116 EP - 140 VL - 20 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013057/ DO - 10.1051/cocv/2013057 LA - en ID - COCV_2014__20_1_116_0 ER -
%0 Journal Article %A Bugariu, Ioan Florin %A Micu, Sorin %T A singular controllability problem with vanishing viscosity %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 116-140 %V 20 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013057/ %R 10.1051/cocv/2013057 %G en %F COCV_2014__20_1_116_0
Bugariu, Ioan Florin; Micu, Sorin. A singular controllability problem with vanishing viscosity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 116-140. doi : 10.1051/cocv/2013057. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013057/
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