Journal of Optimization Theory and Applications

, Volume 158, Issue 2, pp 521–553 | Cite as

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



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. 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 tackled from a numerical point of view 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.


Optimal control Minimal time problem Impulsive control Pontryagin maximum principle Bioreactor 



We would like to thank the referees for meticulous reading of the manuscript and for several suggestions that improved the presentation.


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Université Montpellier 2Montpellier cedex 5France
  2. 2.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile

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