Minimal time bioremediation of water resources with two patches

Hector Ramirez (1) , Alain Rapaport (2, 3) , Victor Riquelme (1, 3)

1 CMM - Center for Mathematical Modeling
2 MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie
3 MODEMIC - Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme

This paper studies the bioremediation, in minimal time, of a water resource or reservoir using a single continuous bioreactor. The bioreactor is connected to two pumps, at different locations in the reservoir, that pump polluted water and inject back sufficiently clean water with the same flow rate. This leads to a minimal-time optimal control problem where the control variables are related to the inflow rates of both pumps. We obtain a non-convex problem for which it is not possible to directly prove the existence of its solutions. We overcome this difficulty and fully solve the studied problem by applying Pontryagin's principle to the associated generalized control problem. We also obtain explicit bounds on its value function via Hamilton-Jacobi-Bellman techniques.

Mots clés: Minimal-time control; feedback synthesis; Pontryagin's maximum principle; Hamilton-Jacobi-Bellman equation; water resource; bioreactor

Domaines: Optimisation et contrôle [math.OC]

Hector Ramirez, Alain Rapaport, Victor Riquelme. Minimal time bioremediation of water resources with two patches. 2014. ⟨hal-01069581v1⟩