Résumé
This paper is devoted to the study of an optimal control problem for a fed-batch bioreactor with one species and one substrate. Our objective is to obtain an optimal feedback control, steering the system in minimal time to a given target defined by conditions on the substrate concentration and the volume of the reactor. The novelty in this work is that a mortality rate for the biomass and hydrolysis of dead biomass are included in the model. The optimal synthesis (optimal feeding strategy) has been obtained by Moreno (1999) when both mortality and hydrolysis are considered negligible. Whenever the model includes these effects, the total mass of the system is no longer conserved, and it is not possible to reduce the dimension of the system. Thanks to the Pontryagin maximum principle and the Hamilton-Jacobi equation we overcome this difficulty and provide an optimal synthesis of the problem in the impulsive framework.