A Geometrical Approach for the Optimal Control of Sequencing Batch Bio-Reactors

  • Nahla Abdellatif Manouba University, Tunis El Manar University
  • Walid Bouhafs Tunis El Manar University,Jendouba University
  • Jérôme Harmand Montpellier University
  • Frédéric Jean Paris-Saclay University
Keywords: Sequencing batch reactor (SBR), Optimal control, Minimal time problem, Pontryagin's Maximum Principle, Optimal synthesis.


In this work, we consider an optimal control problem of a biological sequencing batch reactor (SBR) for thetreatment of pollutants in wastewater. This model includes two biological reactions, one being aerobic while the other is anoxic. The objective is to find an optimal oxygen-injecting strategy to reach, in minimal time and in a minimal time/energy compromise, a target where the pollutants concentrations must fulfill normative constraints. Using a geometrical approach, we solve a more general optimal control problem and thanks to Pontryagin’s Maximum Principle, we explicitly give the complete optimal strategy.


M. ALIANE, N. MOUSSOUNI, M. BENTOBACHE, Optimal control of a rectilinear motion of a rocket. Stat., Optim. Inf. Comput., Vol.8, pp 281-295, (2020).

M. BARDI, I. CAPUZZO-DOLCETTA. Optimal Control and Visosity Solutions of Hamilton-Jacobi- Bellman Equations, Birkhauser, (1997).

W. BOUHAFS, N. ABDELLATIF, F. JEAN, J. HARMAND. Commande optimale en temps minimal d’un proc´ed´e biologique d’´epuration de l’eau. Arima Journal, 18, pp 37-51, (2014).

W. BOUHAFS, N. ABDELLATIF, F. JEAN, J. HARMAND. Commande optimale en temps et en ´energie d’un proc´ed´e biologique d’´epuration de l’eau. International Joint Conference CB-WR-MED Conference 2nd AOP, Tunisia Conference for Sustainable Water Management, Tunis: April, 24-27, (2013).

M. HENZE, W. GUJER, T. MINO, M. VAN LOOSDRECHT. Activated sludge models ASM1, ASM2, ASM2d and ASM3, Edited by IWA Task group on mathematical modelling for design and operation of biological wastewater treatment, IWA Publishing, (2000).

O. KOSTYLENKO, H. S. RODRIGUES, D. F. M. TORRES. The Risk of Contagion Spreading and its Optimal Control in the Economy. Stat., Optim. Inf. Comput., Vol.7, pp 578-587, (2019).

D. MAZOUNI, Th`ese de doctorat – Mod´elisation et commande en temps minimum des r´eacteurs biologiques s´equentiels discontinus, Universit´e Claude Bernard-Lyon I, (2006).

D. MAZOUNI, J. HARMAND, A. RAPAPORT, H. HAMMOURI. Optimal time control of switching systems : application to biological Sequencing Batch Reactors, Optimal Control Application and Methods, Vol. 31, No.4, pp 289-301, (2010).

J. MORENO, Optimal time control of bioreactors for the wastewater treatment. Optim. Control Appl. Meth., No 20, pp 145-164, (1999).

L. PONTRYAGIN, V. BOLTYANSKII, R. GAMKRELIDZE, E. MISCHENKO, The Mathematical Theory of Optimal Processes. Wiley Interscience (1962), pp 159-160, Edition de l’´ecole polytechnique, (2006).

R. VINTER, Optimal Control. Modern Birkhauser Classics (2000).

How to Cite
Abdellatif, N., Bouhafs, W., Harmand, J., & Jean, F. (2020). A Geometrical Approach for the Optimal Control of Sequencing Batch Bio-Reactors. Statistics, Optimization & Information Computing, 9(2), 368-382. https://doi.org/10.19139/soic-2310-5070-868
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