In this paper we study the global controllability of families of the so called non-viscous and viscous Burgers-α systems by using boundary and space independent distributed controls. In these equations, the usual convective velocity of the Burgers equation is replaced by a regularized velocity, induced by a Helmholtz filter of characteristic wavelength α. First, we prove a global exact controllability result (uniform with respect to α) for the non-viscous Burgers-α system, using the return method and a fixed-point argument. Then, the global uniform exact controllability to constant states is deduced for the viscous equations. To this purpose, we first prove a local exact controllability property and, then, we establish a global approximate controllability result for smooth initial and target states.
Keywords: Burgers-$$ system, global exact controllability, return method
@article{COCV_2021__27_1_A80_0,
author = {Ara\'ujo, Raul K. C. and Fern\'andez-Cara, Enrique and Souza, Diego A.},
title = {On the uniform controllability for a family of non-viscous and viscous {Burgers-}$\alpha$ systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021073},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021073/}
}
TY - JOUR AU - Araújo, Raul K. C. AU - Fernández-Cara, Enrique AU - Souza, Diego A. TI - On the uniform controllability for a family of non-viscous and viscous Burgers-$\alpha$ systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021073/ DO - 10.1051/cocv/2021073 LA - en ID - COCV_2021__27_1_A80_0 ER -
%0 Journal Article %A Araújo, Raul K. C. %A Fernández-Cara, Enrique %A Souza, Diego A. %T On the uniform controllability for a family of non-viscous and viscous Burgers-$\alpha$ systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021073/ %R 10.1051/cocv/2021073 %G en %F COCV_2021__27_1_A80_0
Araújo, Raul K. C.; Fernández-Cara, Enrique; Souza, Diego A. On the uniform controllability for a family of non-viscous and viscous Burgers-$\alpha$ systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 78. doi: 10.1051/cocv/2021073
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