Further results on stabilization of periodic trajectories for a chemostat with two species

Frédéric Mazenc 1 Michael Malisoff 2 Jérôme Harmand 3
1 MERE - Water Resource Modeling
CRISAM - Inria Sophia Antipolis - Méditerranée , INRA - Institut National de la Recherche Agronomique : UMR0729
2 LSU Mathematics - Department of Mathematics [Baton Rouge]
3 LBE - Laboratoire de Biotechnologie de l'Environnement [Narbonne]
Abstract : We discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model. We provide new tracking results for chemostats with two species and one limiting substrate, based on Lyapunov function methods. In particular, we use a linear feedback control of the dilution rate and an appropriate time-varying substrate input concentration to produce a locally exponentially stable oscillatory behavior. This means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast. We also obtain a globally stable oscillatory reference trajectory for the species concentrations, using a nonlinear feedback control depending on the dilution rate and the substrate input concentration. This guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory. Finally, we construct an explicit Lyapunov function for the corresponding global error dynamics. We demonstrate the efficacy of our method in a simulation.
Keywords : FEEDBACK CHEMOSTATS LYAPUNOV FUNCTIONS STABILIZATION SYSTEMS BIOLOGY TRACKING BIOLOGIE DES POPULATIONS ECOLOGIE STABILISATION REPERAGE RETROACTION DIVERGENTE COEXISTENCE DES ESPECES
Type de document :
Article dans une revue
IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2008, 53 (Spécial issue), pp.66-74. 〈10.1109/tac.2007.911315〉
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https://hal.inria.fr/hal-00857806
Contributeur : Alain Rapaport <>
Soumis le : mercredi 4 septembre 2013 - 10:02:37
Dernière modification le : vendredi 27 avril 2018 - 14:42:02

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Frédéric Mazenc, Michael Malisoff, Jérôme Harmand. Further results on stabilization of periodic trajectories for a chemostat with two species. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2008, 53 (Spécial issue), pp.66-74. 〈10.1109/tac.2007.911315〉. 〈hal-00857806〉

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