In this paper, we prove a Carleman estimate for a time-discrete parabolic operator under some condition relating the large Carleman parameter to the time step of the discretization scheme. This estimate is then used to obtain relaxed observability estimates that yield, by duality, some controllability results for linear and semi-linear time-discrete parabolic equations. We also discuss the application of this Carleman estimate to the controllability of time-discrete coupled parabolic systems.
Mots-clés : Carleman estimates, time-discrete heat equation, observability, null controllability
@article{COCV_2020__26_1_A12_0,
author = {Boyer, Franck and Hern\'andez-Santamar{\'\i}a, V{\'\i}ctor},
title = {Carleman estimates for time-discrete parabolic equations and applications to controllability},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2019072},
mrnumber = {4064474},
zbl = {1442.93006},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2019072/}
}
TY - JOUR AU - Boyer, Franck AU - Hernández-Santamaría, Víctor TI - Carleman estimates for time-discrete parabolic equations and applications to controllability JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019072/ DO - 10.1051/cocv/2019072 LA - en ID - COCV_2020__26_1_A12_0 ER -
%0 Journal Article %A Boyer, Franck %A Hernández-Santamaría, Víctor %T Carleman estimates for time-discrete parabolic equations and applications to controllability %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019072/ %R 10.1051/cocv/2019072 %G en %F COCV_2020__26_1_A12_0
Boyer, Franck; Hernández-Santamaría, Víctor. Carleman estimates for time-discrete parabolic equations and applications to controllability. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 12. doi : 10.1051/cocv/2019072. http://www.numdam.org/articles/10.1051/cocv/2019072/
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Cité par Sources :
The work of the second author was supported by the Labex CIMI (Centre International de Mathématiques et d’Informatique), ANR-11-LABX-0040-CIMI.
