An optimal control problem of contaminated underground water is considered. The spatio-temporal objective takes into account the economic trade off between the pollutant use –for instance fertilizer– and the cleaning costs. It is constrained by a hydrogeological model for the spread of the pollution in the aquifer. We consider a broad range of reaction kinetics. The aim of the paper is two-fold. On the one hand, we rigorously derive, by asymptotic analysis, the effective optimal control problem for contaminant species that are slightly concentrated in the aquifer. On the other hand, the mathematical analysis of the optimal control problems is performed and we prove in particular that the latter effective problem is well-posed. Furthermore, a stability property of the optimal control process is provided: any optimal solution of the properly scaled problem tends to the optimal solution of the effective problem as the characteristic pollutant concentration decreases.
Mots-clés : Optimal control problem, hydrogeological state equations, nonlinearly coupled problem, parabolic and elliptic PDEs, asymptotic analysis, well-posedness
@article{COCV_2019__25__A53_0,
author = {Augeraud-V\'eron, Emmanuelle and Choquet, Catherine and Comte, \'Elo{\"\i}se},
title = {Existence, uniqueness and asymptotic analysis of optimal control problems for a model of groundwater pollution},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {25},
year = {2019},
doi = {10.1051/cocv/2018041},
zbl = {1437.49032},
mrnumber = {4019756},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018041/}
}
TY - JOUR AU - Augeraud-Véron, Emmanuelle AU - Choquet, Catherine AU - Comte, Éloïse TI - Existence, uniqueness and asymptotic analysis of optimal control problems for a model of groundwater pollution JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2019 VL - 25 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018041/ DO - 10.1051/cocv/2018041 LA - en ID - COCV_2019__25__A53_0 ER -
%0 Journal Article %A Augeraud-Véron, Emmanuelle %A Choquet, Catherine %A Comte, Éloïse %T Existence, uniqueness and asymptotic analysis of optimal control problems for a model of groundwater pollution %J ESAIM: Control, Optimisation and Calculus of Variations %D 2019 %V 25 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018041/ %R 10.1051/cocv/2018041 %G en %F COCV_2019__25__A53_0
Augeraud-Véron, Emmanuelle; Choquet, Catherine; Comte, Éloïse. Existence, uniqueness and asymptotic analysis of optimal control problems for a model of groundwater pollution. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 53. doi : 10.1051/cocv/2018041. http://www.numdam.org/articles/10.1051/cocv/2018041/
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