Minimal time control of the two tanks gradostat model under a cascade inputs constraint

We study the minimum time control problem of a series of two interconnected chemostats under the input constraint u₂ ≤ u₁, where u_i are the respective dilution rates in the tanks. This constraint brings controllability issues in the study of the optimal strategies. We encounter this difficulty by splitting the state domain into two sub-domains, one with no lack of controllability of the target, and its complementary is such that any optimal control fulfills u₁ = u₂. We explicit the complete optimal synthesis that depends on the position of the target with respect to a semi-permeable curve that passes through a steady-state singular point.