An explicit feedback controller is proposed for stabilization of linear parabolic equations, with a time-dependent reaction–convection operator. The range of the feedback controller is finite-dimensional, and is typically modeled by indicator functions of small subdomains. Its dimension depends polynomially on a suitable norm of the reaction–convection operator. A sufficient condition for stabilizability is given, which involves the asymptotic behavior of the eigenvalues of the (time-independent) diffusion operator, the norm of the reaction–convection operator, and the norm of the nonorthogonal projection onto the controller’s range along a suitable infinite-dimensional (higher-modes) eigenspace. To construct the explicit feedback, the essential step consists in computing the nonorthogonal projection. Numerical simulations are presented, in 1D and 2D, showing the practicability of the controller and its response to measurement errors, where the actuators are indicator functions of suitable small subsets.
Accepté le :
DOI : 10.1051/cocv/2018054
Mots-clés : Exponential stabilization, nonautonomous parabolic systems, finite-dimensional controller, explicit feedback
@article{COCV_2019__25__A67_0,
author = {Kunisch, Karl and Rodrigues, S\'ergio S.},
title = {Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018054},
zbl = {1436.93106},
mrnumber = {4027692},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018054/}
}
TY - JOUR AU - Kunisch, Karl AU - Rodrigues, Sérgio S. TI - Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018054/ DO - 10.1051/cocv/2018054 LA - en ID - COCV_2019__25__A67_0 ER -
%0 Journal Article %A Kunisch, Karl %A Rodrigues, Sérgio S. %T Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018054/ %R 10.1051/cocv/2018054 %G en %F COCV_2019__25__A67_0
Kunisch, Karl; Rodrigues, Sérgio S. Explicit exponential stabilization of nonautonomous linear parabolic-like systems by a finite number of internal actuators. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 67. doi : 10.1051/cocv/2018054. http://www.numdam.org/articles/10.1051/cocv/2018054/
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