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In this paper we study the null controllability property of the linear Kuramoto−Sivashinsky equation by means of either boundary or internal controls. In the Dirichlet boundary case, we use the moment theory to prove that the null controllability property holds with only one boundary control if and only if the anti-diffusion parameter of the equation does not belong to a critical set of parameters. Regarding the Neumann boundary case, we prove that the null controllability property does not hold with only one boundary control. However, it does always hold when either two boundary controls or an internal control are considered. The proof of the latter is based on the controllability-observability duality and a suitable Carleman estimate.
Cerpa, Eduardo 1 ; Guzmán, Patricio 1 ; Mercado, Alberto 1
@article{COCV_2017__23_1_165_0,
author = {Cerpa, Eduardo and Guzm\'an, Patricio and Mercado, Alberto},
title = {On the control of the linear {Kuramoto\ensuremath{-}Sivashinsky} equation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {165--194},
publisher = {EDP-Sciences},
volume = {23},
number = {1},
year = {2017},
doi = {10.1051/cocv/2015044},
mrnumber = {3601020},
zbl = {1364.35117},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2015044/}
}
TY - JOUR AU - Cerpa, Eduardo AU - Guzmán, Patricio AU - Mercado, Alberto TI - On the control of the linear Kuramoto−Sivashinsky equation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2017 SP - 165 EP - 194 VL - 23 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2015044/ DO - 10.1051/cocv/2015044 LA - en ID - COCV_2017__23_1_165_0 ER -
%0 Journal Article %A Cerpa, Eduardo %A Guzmán, Patricio %A Mercado, Alberto %T On the control of the linear Kuramoto−Sivashinsky equation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2017 %P 165-194 %V 23 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2015044/ %R 10.1051/cocv/2015044 %G en %F COCV_2017__23_1_165_0
Cerpa, Eduardo; Guzmán, Patricio; Mercado, Alberto. On the control of the linear Kuramoto−Sivashinsky equation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 1, pp. 165-194. doi : 10.1051/cocv/2015044. http://geodesic.mathdoc.fr/articles/10.1051/cocv/2015044/
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