An Algorithm for Maximizing the Biogas Production in a Chemostat

  • Antoine HaddonEmail author
  • Cristopher Hermosilla


In this work, we deal with the optimal control problem of maximizing biogas production in a chemostat. The dilution rate is the controlled variable, and we study the problem over a fixed finite horizon, for positive initial conditions. We consider the single reaction model and work with a broad class of growth rate functions. With the Pontryagin maximum principle, we construct a one-parameter family of extremal controls of type bang-singular arc. The parameter of these extremal controls is the constant value of the Hamiltonian. Using the Hamilton–Jacobi–Bellman equation, we identify the optimal control as the extremal associated with the value of the Hamiltonian, which satisfies a fixed point equation. We then propose a numerical algorithm to compute the optimal control by solving this fixed point equation. We illustrate this method with the two major types of growth functions of Monod and Haldane.


Optimal control Chemostat model Pontryagin maximum principle Hamilton–Jacobi–Bellman equations Optimal synthesis 

Mathematics Subject Classification

49M05 49J15 93A30 



A. Haddon was supported by a doctoral fellowship CONICYT –PFCHA/Doctorado Nacional/2017-21170249. The first author was also supported by FONDECYT Grant 1160567 and by Basal Program CMM-AFB 170001 from CONICYT–Chile. C. Hermosilla was supported by CONICYT-Chile through FONDECYT Grant Number 3170485.


  1. 1.
    Rehl, T., Müller, J.: Co\(_{2}\) abatement costs of greenhouse gas (ghg) mitigation by different biogas conversion pathways. J. Environ. Manag. 114, 13–25 (2013)CrossRefGoogle Scholar
  2. 2.
    Beddoes, J.C., Bracmort, K.S., Burns, R.T., Lazarus, W.F.: An analysis of energy production costs from anaerobic digestion systems on us livestock production facilities. USDA NRCS Technical Note (1) (2007)Google Scholar
  3. 3.
    Guwy, A., Hawkes, F., Wilcox, S., Hawkes, D.: Neural network and on-off control of bicarbonate alkalinity in a fluidised-bed anaerobic digester. Water Res. 31(8), 2019–2025 (1997)CrossRefGoogle Scholar
  4. 4.
    Nguyen, D., Gadhamshetty, V., Nitayavardhana, S., Khanal, S.K.: Automatic process control in anaerobic digestion technology: a critical review. Bioresour. Technol. 193, 513–522 (2015)CrossRefGoogle Scholar
  5. 5.
    García-Diéguez, C., Molina, F., Roca, E.: Multi-objective cascade controller for an anaerobic digester. Process Biochem. 46(4), 900–909 (2011)CrossRefGoogle Scholar
  6. 6.
    Rodríguez, J., Ruiz, G., Molina, F., Roca, E., Lema, J.: A hydrogen-based variable-gain controller for anaerobic digestion processes. Water Sci. Technol. 54(2), 57–62 (2006)CrossRefGoogle Scholar
  7. 7.
    Djatkov, D., Effenberger, M., Martinov, M.: Method for assessing and improving the efficiency of agricultural biogas plants based on fuzzy logic and expert systems. Appl. Energy 134, 163–175 (2014)CrossRefGoogle Scholar
  8. 8.
    Dimitrova, N., Krastanov, M.: Nonlinear adaptive stabilizing control of an anaerobic digestion model with unknown kinetics. Int. J. Robust Nonlinear Control 22(15), 1743–1752 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sbarciog, M., Loccufier, M., Wouwer, A.V.: An optimizing start-up strategy for a bio-methanator. Bioprocess Biosyst. Eng. 35(4), 565–578 (2012)CrossRefGoogle Scholar
  10. 10.
    Bayen, T., Cots, O., Gajardo, P.: Analysis of an optimal control problem related to the anaerobic digestion process. J. Optim. Theory Appl. 178, 627–659 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bernard, O., Chachuat, B., Hélias, A., Rodriguez, J.: Can we assess the model complexity for a bioprocess: theory and example of the anaerobic digestion process. Water Sci. Technol. 53(1), 85–92 (2006)CrossRefGoogle Scholar
  12. 12.
    Stamatelatou, K., Lyberatos, G., Tsiligiannis, C., Pavlou, S., Pullammanappallil, P., Svoronos, S.: Optimal and suboptimal control of anaerobic digesters. Environ. Model. Assess. 2(4), 355–363 (1997)CrossRefGoogle Scholar
  13. 13.
    Ghouali, A., Sari, T., Harmand, J.: Maximizing biogas production from the anaerobic digestion. J. Process Control 36, 79–88 (2015)CrossRefGoogle Scholar
  14. 14.
    Haddon, A., Harmand, J., Ramírez, H., Rapaport, A.: Guaranteed value strategy for the optimal control of biogas production in continuous bio-reactors. IFAC PapersOnLine 50(1), 8728–8733 (2017)CrossRefGoogle Scholar
  15. 15.
    Haddon, A., Ramírez, H., Rapaport, A.: First results of optimal control of average biogas production for the chemostat over an infinite horizon. IFAC PapersOnLine 51(2), 725–729 (2018)CrossRefGoogle Scholar
  16. 16.
    Team Commands, Inria Saclay: Bocop: an open source toolbox for optimal control (2017). Assessed 2018
  17. 17.
    Gerdts, M.: Optimal control and parameter identification with differential-algebraic equations of index 1: user’s guide. Tech. rep., Institut für Mathematik und Rechneranwendung, Universität der Bundeswehr München (2011)Google Scholar
  18. 18.
    Bokanowski, O., Desilles, A., Zidani, H.: ROC-HJ-Solver. a C++ Library for Solving HJ equations (2013). Assessed 2018
  19. 19.
    Yane (2018). Assessed 2018
  20. 20.
    Lobry, C., Rapaport, A., Sari, T., et al.: The Chemostat: Mathematical Theory of Microorganism Cultures. Wiley, New York (2017)Google Scholar
  21. 21.
    Bastin, G., Dochain, D.: On-line Estimation and Adaptive Control of Bioreactors. Elsevier, Amsterdam (1991)Google Scholar
  22. 22.
    Hermosilla, C.: Stratified discontinuous differential equations and sufficient conditions for robustness. Discrete Contin. Dyn. Syst.-Ser. A 35(9), 23 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Clarke, F.H., Ledyaev, Y.S., Stern, R.J., Wolenski, P.R.: Nonsmooth Analysis and Control Theory, vol. 178. Springer Science & Business Media, Berlin (2008)zbMATHGoogle Scholar
  24. 24.
    Clarke, F.: Functional Analysis, Calculus of Variations and Optimal Control, vol. 264. Springer Science & Business Media, Berlin (2013)zbMATHGoogle Scholar
  25. 25.
    Hermosilla, C., Zidani, H.: Infinite horizon problems on stratifiable state-constraints sets. J. Differ. Equ. 258(4), 1430–1460 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Springer Science & Business Media, Berlin (2008)zbMATHGoogle Scholar
  27. 27.
    Bernard, O., Hadj-Sadok, Z., Dochain, D., Genovesi, A., Steyer, J.P.: Dynamical model development and parameter identification for an anaerobic wastewater treatment process. Biotechnol. Bioeng. 75(4), 424–438 (2001)CrossRefGoogle Scholar
  28. 28.
    Bonnans Frederic, J., Giorgi, D., Grelard, V., Heymann, B., Maindrault, S., Martinon, P., Tissot, O., Liu, J.: Bocop–A collection of examples. Tech. rep., INRIA (2017). Assessed 2018

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematical Engineering Department and Center for Mathematical Modelling (CNRS UMI 2807)Universidad de ChileSantiagoChile
  2. 2.MISTEA, Université Montpellier, INRA, Montpellier SupAgroMontpellierFrance
  3. 3.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile

Personalised recommendations