Optimal Feedback Synthesis and Minimal Time Function for the Bioremediation of Water Resources with Two Patches

Abstract

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 nonconvex 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.

Modelling of Biological Decontamination of a Water Resource in Natural Environment and Related Feedback Strategies

“…Moreover, one can check that the usual Monod law given in (2.2) is a concave function, that leads to the following expression of the function S eq R : S eq R (S a ) = K 2 s + K s S a − K s . In [13], it is also shown that this feedback is robust w.r.t. the approximation of the quasi-steady state of the dynamics (2.1a) and (2.1b).…”

Controlling recirculation rate for minimal-time bioremediation of natural water resources

“…u 2 . As we shall see later in the analysis of the minimal time problem, the consideration of a second control u 2 changes significantly the optimal control problem, compared to the former works [3,9]). PROOF.…”

“…Therefore, one cannot guarantee the existence of an optimal trajectory with the usual Filippov's Theorem. Nevertheless, it exists among relaxed controls, which amounts to consider controls (u a 1 , u b 1 , p, u 2 ) in U 2 with the dynamicsṡ = pF (s, u a 1 ) + (1 − p)F (s, u b 1 ) + G(s)u 2 as we did in [9].…”

Modeling and control of in-situ decontamination of large water resources

“…The principle of the above proposition and its proof, is to establish that the optimal feedback S → S opt T (S) is the one that makes the time derivative of S the most negative at any time in (6). Observe that a necessary condition for a feedback S opt T (S) to be optimal is to verify…”