In this paper, we consider the infinite time horizon LQR optimal control problem for the linearized Boussinesq system. The goal is to justify the approximation by penalization of the free divergence condition in this context. We establish convergence results for optimal controls, optimal solutions and Riccati operators when the penalization parameter goes to zero. These results are obtained under two different assumptions. The first one treats the linearization around a sufficiently small stationary state and an arbitrary control operator (possibly of finite rank), while the second one does no longer require the smallness of the stationary state but requires to consider controls distributed in a subdomain and depending on the space variable.
Keywords: Linear quadratic optimal control, Riccati theory, Boussinesq system, penalty method, controllability
@article{COCV_2021__27_1_A19_0,
author = {Le Balc{\textquoteright}h, K\'evin and Tucsnak, Marius},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {A penalty approach to the infinite horizon {LQR} optimal control problem for the linearized {Boussinesq} system},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021008},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021008/}
}
TY - JOUR AU - Le Balc’h, Kévin AU - Tucsnak, Marius ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - A penalty approach to the infinite horizon LQR optimal control problem for the linearized Boussinesq system JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021008/ DO - 10.1051/cocv/2021008 LA - en ID - COCV_2021__27_1_A19_0 ER -
%0 Journal Article %A Le Balc’h, Kévin %A Tucsnak, Marius %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T A penalty approach to the infinite horizon LQR optimal control problem for the linearized Boussinesq system %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021008/ %R 10.1051/cocv/2021008 %G en %F COCV_2021__27_1_A19_0
Le Balc’h, Kévin; Tucsnak, Marius. A penalty approach to the infinite horizon LQR optimal control problem for the linearized Boussinesq system. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 17. doi: 10.1051/cocv/2021008
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