Minimal Time Sequential Batch Reactors with Bounded and Impulse Controls for One or More Species

Abstract

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.

Optimization of the separation of two species in a chemostat

“…The next assumption is standard and means that the maximum value of the input flow rate can be larger than the growth of microorganisms (see e.g. [1,8]…”

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

“…Our main result is Theorem 3.4 which extends the optimality results of [11] to the case where both mortality and hydrolysis effects are taken into account. We prove that for Monod type kinetics, the optimal strategy is of type bang-bang.…”

“…We now consider the minimal time problem with an extension of (2.1) allowing impulsive controls, see [12,11]. From a practical point of view, this assumption corresponds to a maximum dilution rate u m sup s∈[0,sin] µ(s).…”

“…These results have been generalized in the impulsive framework in [11] for multi-species and in [12] for growth functions with two local maxima. In this paper, our objective is to extend the results above to the case where the model includes mortality and hydrolysis coefficients.…”

“…One way to tackle this problem is to follow the approach in [12,11] and to consider a more general problem where the input flow rate is considered in the class of impulsive controls. This corresponds to an instantaneous addition of wasted water in the reactor (that is, an instantaneous dilution).…”