We consider the optimal control problem of feeding in minimal time a tank where several species compete for a single resource, with the objective being to reach a given level of the resource. We allow controls to be bounded measurable functions of time plus possible impulses. For the one-species case, we show that the immediate one-impulse strategy (filling the whole reactor with one single impulse at the initial time) is optimal when the growth function is monotonic. For nonmonotonic growth functions with one maximum, we show that a particular singular arc strategy (precisely defined in section 3) is optimal. These results extend and improve former ones obtained for the class of measurable controls only. For the two-species case with monotonic growth functions, we give conditions under which the immediate one-impulse strategy is optimal. We also give optimality conditions for the singular arc strategy (at a level that depends on the initial condition) to be optimal. The possibility for the immediate one-impulse strategy to be nonoptimal while both growth functions are monotonic is a surprising result and is illustrated with the help of numerical simulations.

We study minimal time strategies for the treatment of pollution of large volumes, such as lakes or natural reservoirs, with the help of an autonomous bioreactor. The control consists in feeding the bioreactor from the resource, the clean output returning to the resource with the same flow rate. We first characterize the optimal policies among constant and feedback controls, under the assumption of a uniform concentration in the resource. In a second part, we study the influence of an inhomogeneity in the resource, considering two measurements points. With the help of the Maximum Principle, we show that the optimal control law is non-monotonic and terminates with a constant phase, contrary to the homogeneous case for which the optimal flow rate is decreasing with time. This study allows the decision makers to identify situations for which the benefit of using non-constant flow rates is significant.