Optimality of affine control system of several species in competition on a Sequential Batch Reactor

J.C. Rodriguez (1) , Hector Ramirez (2, 3, 4) , Pedro Gajardo (5, 4) , Alain Rapaport (6, 7)

1 Universidad de Valparaiso [Chile]
2 CMM - Center for Mathematical Modeling
3 DIM - Departamento de Ingeniería Matemática [Santiago]
4 Inria Chile
5 UTFSM - Universidad Tecnica Federico Santa Maria [Valparaiso]
6 MODEMIC - Modelling and Optimisation of the Dynamics of Ecosystems with MICro-organisme
7 MISTEA - Mathématiques, Informatique et STatistique pour l'Environnement et l'Agronomie

In this paper we analyze the optimalty of affine control system of several species in competition for a single substrate on a Sequential Batch Reactors (SBR), with the objective being to reach a given (low) level of the substrate. We allow controls to be bounded measurable functions of time plus possible impulses. A suitable modification of the dynamics leads to a slightly different optimal control problem, without impulsive controls, for which we apply different optimality conditions de-rived from Pontryagin principle and the Hamilton-Jacobi-Bellman equation. We thus characterize the singular trajectories of our problem as the extremal trajecto-ries keeping the substrate at a constant level. We also establish conditions for which a Immediate One impulse (IOI) strategy is optimal. Some numerical experiences are then included in order to illustrate our study and show that those conditions are also necessary to ensure the optimality of the IOI strategy.

Mots clés: affine control systems; pontryagin maximum principle; sequential batch reactors; optimal control problem; singular trajectories

Domaines: Optimisation et contrôle [math.OC]

J.C. Rodriguez, Hector Ramirez, Pedro Gajardo, Alain Rapaport. Optimality of affine control system of several species in competition on a Sequential Batch Reactor. International Journal of Control, 2014, 87 (9), pp.1877-1885. ⟨10.1080/00207179.2014.891360⟩. ⟨hal-01088224⟩