Journal of Optimization Theory and Applications

, Volume 178, Issue 2, pp 627–659 | Cite as

Analysis of an Optimal Control Problem Related to the Anaerobic Digestion Process

  • Terence BayenEmail author
  • Olivier Cots
  • Pedro Gajardo


Our aim in this work is to synthesize optimal feeding strategies that maximize, over a time period, the biogas production in a continuously filled bioreactor controlled by its dilution rate. Such an anaerobic process is described by a four-dimensional dynamical system. Instead of modeling the optimization of the biogas production as a Lagrange-type optimal control problem, we propose a slightly different optimal control approach in this paper: We study the minimal time control problem to reach a target point, which is chosen in such a way that it maximizes the biogas production at steady state. Thanks to the Pontryagin maximum principle and the geometric control theory, we provide an optimal feedback control for the minimal time control problem, when the initial conditions are taken within the invariant and attractive manifold of the system. The optimal synthesis exhibits turnpike and anti-turnpike singular arcs and a cut locus.


Geometric control Pontryagin maximum principle Optimal feedback Chemostat model 

Mathematics Subject Classification

49J15 49M30 93A30 



The authors are grateful to Jérôme Harmand, Alain Rapaport and Victor Riquelme for their helpful discussions on this subject and to Aurélien Binet for contributing to the determination of the cut locus. The first author would like to thank INRA Montpellier and the UMR MISTEA for providing him a half-year delegation during the 2017-2018 academic year. This research benefited from the support of the FONDECYT grant (Chile) N 1160567 and Proyecto Redes 150011 (Chile) and from the LABEX NUMEV Montpellier. The third author was also partially supported by the Basal Project CMM Universidad de Chile.


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Authors and Affiliations

  1. 1.IMAGUniv Montpellier, CNRSMontpellierFrance
  2. 2.MISTEAUniv Montpellier, INRA, Montpellier SupAgroMontpellierFrance
  3. 3.Toulouse Univ. INP-ENSEEIHT, IRIT and CNRSToulouseFrance
  4. 4.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaisoChile

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