Optimal feeding strategy for the minimal time problem of a fed-batch bioreactor with mortality rate

We address the problem of finding an optimal feedback control for feeding a fed-batch bioreactor with one species and one substrate, from a given initial condition to a given target value in a minimal amount of time. Mortality rate for the biomass and nutrient recycling are taken into account in this work. The optimal synthesis (optimal feeding strategy) has been obtained by Moreno in 1999 when both mortality and recycling are considered negligible, in the case of Monod and Haldane growth function. This problem has been recently revisited whenever the growth function has two local maxima. Our objective is to study the effect of mortality and recycling on the optimal synthesis. Taking into account these effects does not allow to reduce the controlled system into a two-dimensional one. We provide an optimal synthesis of the problem using both Hamilton-Jacobi equation and Pontryagin maximum principle in the impulsive framework (that is when the dilution rate is allowed to take large values). We also investigate the case where the singular arc is non-necessarily controllable.