Volume 25, 2019
|Number of page(s)||20|
|Published online||18 October 2019|
Existence, uniqueness and asymptotic analysis of optimal control problems for a model of groundwater pollution
Université de La Rochelle, Laboratoire MIA, 23 Avenue A. Einstein,
La Rochelle, France.
* Corresponding author: firstname.lastname@example.org
Accepted: 29 June 2018
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.
Mathematics Subject Classification: 49A20 / 49A50 / 37N40 / 76R99 / 37N35
Key words: Optimal control problem / hydrogeological state equations / nonlinearly coupled problem / parabolic and elliptic PDEs / asymptotic analysis / well-posedness
© EDP Sciences, SMAI 2019
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