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

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Abstract

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.

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References

  1. Lee, J., Lee, S.Y., Park, S., Middelberg, P.J.: Control of fed-batch fermentations. Biotechnol. Adv. 17(11), 29–48 (1999)

    Article  Google Scholar 

  2. Hong, J.: Optimal substrate feeding policy for fed batch fermentation with substrate and product inhibition kinetics. Biotechnol. Bioeng. 28, 1421–1431 (1986)

    Article  Google Scholar 

  3. Liu, C.: Optimal control for nonlinear dynamical system of microbial fed-batch culture. J. Comput. Appl. Math. 232(2), 252–261 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Moreno, J.: Optimal time control of bioreactors for the wastewater treatment. Opt. Control Appl. Methods 20(3), 145–164 (1999)

    Article  Google Scholar 

  5. Miele, A.: Application of Green’s theorem to the extremization of linear integrals. In: Symp. on Vehicle Systems Optimization. Garden City, New York (1961)

    Google Scholar 

  6. Monod, J.: Recherches sur la Croissance des Cultures Bactériennes. Hermann, Paris (1942)

    Google Scholar 

  7. Smith, H.L., Waltman, P.: The Theory of the Chemostat, Dynamics of Microbial Competition. Cambridge Studies in Mathematical Biology, vol. 13. Cambridge University Press, Cambridge (1995)

    Book  MATH  Google Scholar 

  8. Andrews, J.: A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioeng. 10, 707–723 (1968)

    Article  Google Scholar 

  9. Gajardo, P., Ramírez, H.C., Rapaport, A.: Minimal time sequential batch reactors with bounded and impulse controls for one or more species. SIAM J. Control Optim. 47(6), 2827–2856 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Betancur, M.J., Moreno, J.A., Moreno-Andrade, I., Buitrón, G.: Practical optimal control of fed-batch bioreactors for the waste water treatment. Int. J. Robust Nonlinear Control 16(3), 173–190 (2006)

    Article  MATH  Google Scholar 

  11. Cougnon, P., Dochain, D., Guay, M., Perrier, M.: On-line optimization of fedbatch bioreactors by adaptive extremum seeking control. J. Process Control 21(10), 1526–1532 (2011)

    Article  Google Scholar 

  12. Doran, P.M.: Bioprocess Engineering Principles. Academic Press, London (1995)

    Google Scholar 

  13. Bayen, T., Gajardo, P., Mairet, F.: Minimal time control of fed-batch bioreactor with product inhibition. In: 20th Mediterranean Conference on Control and Automation, Barcelona (2012)

    Google Scholar 

  14. Rapaport, A., Dochain, D.: Minimal time control of fed-batch processes with growth functions having several maxima. IEEE Trans. Automat. Control 56(11), 2671–2676 (2011)

    Article  MathSciNet  Google Scholar 

  15. Silva, C., Trélat, E.: Asymptotic approach on conjugate points for minimal time bang–bang controls. Syst. Control Lett. 59(11), 720–733 (2010)

    Article  MATH  Google Scholar 

  16. Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., Mishchenko, E.F.: The Mathematical Theory of Optimal Processes. Pergamon/Macmillan, New York (1964). Translated by D.E. Brown

    MATH  Google Scholar 

  17. Rishel, R.W.: An extended Pontryagin principle for control systems whose control laws contain measures. J. Soc. Ind. Appl. Math. Ser. A Control 3, 191–205 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  18. Cesari, L.: Optimization—Theory and Applications, Problems with Ordinary Differential Equations. Applications of Mathematics (New York), vol. 17. Springer, New York (1983)

    MATH  Google Scholar 

  19. Robbins, H.M.: A generalized Legendre–Clebsch condition for the singular cases of optimal control. IBM. J. Res. Dev. 11, 361–372 (1967)

    MATH  Google Scholar 

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Acknowledgements

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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Authors and Affiliations

  1. Université Montpellier 2, Case courrier 051, 34095, Montpellier cedex 5, France

    Térence Bayen

  2. Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile

    Pedro Gajardo & Francis Mairet

Authors
  1. Térence Bayen
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  2. Pedro Gajardo
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  3. Francis Mairet
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Corresponding author

Correspondence to Térence Bayen.

Additional information

Communicated by Mimmo Iannelli.

This work was supported by the Programa de Financiamiento Basal from the Center of Mathematical Modeling, Universidad de Chile and was developed in the context of DYMECOS INRIA associated team and of the program Stic-AmSud MOMARE. The first author thanks the CNRS and the team DYMECOS for financial support. P. Gajardo and F. Mairet were partially supported by the FONDECYT-Chile program (Nos. 1120239 and 3120117, respectively) and by the Communication and Information Research and Innovation Center (CIRIC).

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Bayen, T., Gajardo, P. & Mairet, F. Optimal Synthesis for the Minimum Time Control Problems of Fed-Batch Bioprocesses for Growth Functions with Two Maxima. J Optim Theory Appl 158, 521–553 (2013). https://doi.org/10.1007/s10957-012-0225-0

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  • DOI: https://doi.org/10.1007/s10957-012-0225-0

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