Optimization of the separation of two species in a chemostat

In this work, we study a two species chemostat model with one limiting substrate, and our aim is to optimize the selection of the species of interest. More precisely, the objective is to find an optimal feeding strategy in order to reach in minimal time a target where the concentration of the first species is significantly larger than the concentration of the other one. Thanks to the Pontryagin Maximum Principle, we introduce a singular feeding strategy which allows to reach the target, and we prove that the feedback control provided by this strategy is optimal whenever initial conditions are chosen in the invariant attractive manifold of the system. The optimal synthesis of the problem in presence of more than one singular arc is also investigated.