Optimal Synthesis for the Minimum Time Control Problems of Fed-Batch Bioprocesses for Growth Functions with Two Maxima

Abstract

International audienceWe 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. Recently, the optimal synthesis (optimal feeding strategy) has been obtained in systems in which the microorganisms involved are represented by increasing growth functions or growth functions with one maxima, with either Monod or Haldane functions, respectively (widely used in bioprocesses modeling). In the present work, we allow impulsive controls corresponding to instantaneous dilutions, and we assume that the growth function of the microorganism present in the process has exactly two local maxima. This problem has been considered from a numerical point of view in [15] without impulsive controls. In this article, we introduce two singular arc feeding strategies, and we define explicit regions of initial conditions in which the optimal strategy is either the first singular arc strategy or the second strategy

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]…”

“…The situation is more delicate for the target T 1 since there exist two singular strategies associated to SAS s1 and SAS s2 which are candidates to reach the target. Such competition between two singular arcs have already appeared in the minimal time control problem of fed-batch bioreactor for growth function with two maxima (see [1,7]). Following [1], we infer that an optimal control cannot switch from a singular arc to another one, and that the following conjecture holds true.…”

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

“…By differentiating, we haveρ = βρ which gives x(t)v(t) = x 0 v 0 e β(t−t0) . Now we havev = αxv = αx 0 v 0 e β(t−t0) , and by integrating, we obtain v = v 0 + Next, we assume the following condition that will ensure the controllability of the singular arc with r = 1 for the problem with mortality (see also [2,12]): Hypothesis 3.1 Initial conditions in E are such that:…”

“…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.…”

Minimal time problem for a fed-batch bioreactor with a non admissible singular arc

“…One often encounters singular trajectories which appear when the switching function of the system is vanishing on a sub-interval. In order to find an issue to an optimal control problem governed by (1), studies often require that the singular arc is controllable which means that the singular control u s allowing the trajectory to stay on the singular arc is supposed to verify the inequality:…”