Buttazzo, G. ; Casas, E. ; de Teresa, L. ; Glowinski, R. ; Leugering, G. ; Trélat, E. ; Zhang, X.(éd.)
We propose a numerical method to approximate the exact averaged boundary control of a family of wave equations depending on an unknown parameter σ. More precisely the control, independent of σ, that drives an initial data to a family of final states at time t = T, whose average in σ is given. The idea is to project the control problem in the finite dimensional space generated by the first N eigenfunctions of the Laplace operator. When applied to a single (nonparametric) wave equation, the resulting discrete control problem turns out to be equivalent to the Galerkin approximation proposed by F. Bourquin et al. [C.R. Acad. Sci. Paris 313 I (1991) 757–760]. We give a convergence result of the discrete controls to the continuous one. The method is illustrated with several examples in 1-d and 2-d in a square domain and allows us to give some conjectures on the averaged controllability for the continuous problem.
Keywords: Exact control, numerical approximation, averaged control, projection method
@article{COCV_2021__27_1_A66_0,
author = {Abdelli, Mouna and Castro, Carlos},
editor = {Buttazzo, G. and Casas, E. and de Teresa, L. and Glowinski, R. and Leugering, G. and Tr\'elat, E. and Zhang, X.},
title = {Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021060},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021060/}
}
TY - JOUR AU - Abdelli, Mouna AU - Castro, Carlos ED - Buttazzo, G. ED - Casas, E. ED - de Teresa, L. ED - Glowinski, R. ED - Leugering, G. ED - Trélat, E. ED - Zhang, X. TI - Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021060/ DO - 10.1051/cocv/2021060 LA - en ID - COCV_2021__27_1_A66_0 ER -
%0 Journal Article %A Abdelli, Mouna %A Castro, Carlos %E Buttazzo, G. %E Casas, E. %E de Teresa, L. %E Glowinski, R. %E Leugering, G. %E Trélat, E. %E Zhang, X. %T Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021060/ %R 10.1051/cocv/2021060 %G en %F COCV_2021__27_1_A66_0
Abdelli, Mouna; Castro, Carlos. Numerical approximation of the averaged controllability for the wave equation with unknown velocity of propagation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 64. doi: 10.1051/cocv/2021060
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