We consider an optimal control problem for the three-dimensional non-linear Primitive Equations of the ocean in a vertically bounded and horizontally periodic domain. We aim to reconstruct the initial state of the ocean from lagrangian observations. This inverse problem is formulated as an optimal control problem which consists in minimizing a cost function representing the least square error between lagrangian observations and their model counterpart, plus a regularization term. This paper proves the existence of an optimal control for the regularized problem. To this end, we also prove new energy estimates for the Primitive Equations, thanks to well-chosen functional spaces, which distinguish the vertical dimension from the horizontal ones. We illustrate the result with a numerical experiment.
Mots-clés : optimal control, partial differential equations, primitive equations, numerical simulation
@article{COCV_2010__16_2_400_0,
author = {Nodet, Ma\"elle},
title = {Optimal control of the primitive equations of the ocean with lagrangian observations},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {400--419},
publisher = {EDP-Sciences},
volume = {16},
number = {2},
year = {2010},
doi = {10.1051/cocv/2009003},
mrnumber = {2654200},
zbl = {1189.35376},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2009003/}
}
TY - JOUR AU - Nodet, Maëlle TI - Optimal control of the primitive equations of the ocean with lagrangian observations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2010 SP - 400 EP - 419 VL - 16 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2009003/ DO - 10.1051/cocv/2009003 LA - en ID - COCV_2010__16_2_400_0 ER -
%0 Journal Article %A Nodet, Maëlle %T Optimal control of the primitive equations of the ocean with lagrangian observations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2010 %P 400-419 %V 16 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2009003/ %R 10.1051/cocv/2009003 %G en %F COCV_2010__16_2_400_0
Nodet, Maëlle. Optimal control of the primitive equations of the ocean with lagrangian observations. ESAIM: Control, Optimisation and Calculus of Variations, Tome 16 (2010) no. 2, pp. 400-419. doi : 10.1051/cocv/2009003. http://www.numdam.org/articles/10.1051/cocv/2009003/
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