Minimal time problem for a chemostat model with growth rate of Haldane type

In this work, we consider an optimal control problem for a system describing a chemostat with one species and one substrate. Our objective is to find an optimal feedback control in order to reach in minimal time a target point. This problem has been addressed in the case where the growth rate is of Monod type. Here, we suppose that the growth rate is of Haldane type, which implies the existence of a singular arc. Thanks to Pontryagin maximum principle, we provide an optimal synthesis (optimal feeding strategy) of the problem.