We are interested in reconstructing the initial condition of a wave equation in an unbounded domain configuration from measurements available in time on a subdomain. To solve this problem, we adopt an iterative strategy of reconstruction based on observers and time reversal adjoint formulations. We prove the convergence of our reconstruction algorithm with perfect measurements and its robustness to noise. Moreover, we develop a complete strategy to practically solve this problem on a bounded domain using artificial transparent boundary conditions to account for the exterior domain. Our work then demonstrates that the consistency error introduced by the use of approximate transparent boundary conditions is compensated by the stabilization properties obtained from the use of the available measurements, hence allowing to still be able to reconstruct the unknown initial condition.
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DOI : 10.1051/cocv/2019026
Mots-clés : Wave equation in unbounded domain, back and forth observer, state estimation, data assimilation
@article{COCV_2020__26_1_A45_0,
author = {Imperiale, S. and Moireau, P. and Tonnoir, A.},
title = {Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {26},
year = {2020},
doi = {10.1051/cocv/2019026},
mrnumber = {4144109},
zbl = {1452.65207},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2019026/}
}
TY - JOUR AU - Imperiale, S. AU - Moireau, P. AU - Tonnoir, A. TI - Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2020 VL - 26 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2019026/ DO - 10.1051/cocv/2019026 LA - en ID - COCV_2020__26_1_A45_0 ER -
%0 Journal Article %A Imperiale, S. %A Moireau, P. %A Tonnoir, A. %T Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains %J ESAIM: Control, Optimisation and Calculus of Variations %D 2020 %V 26 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2019026/ %R 10.1051/cocv/2019026 %G en %F COCV_2020__26_1_A45_0
Imperiale, S.; Moireau, P.; Tonnoir, A. Analysis of an observer strategy for initial state reconstruction of wave-like systems in unbounded domains. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 45. doi : 10.1051/cocv/2019026. http://www.numdam.org/articles/10.1051/cocv/2019026/
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