On optimal strategies for feeding in minimal time a SBR with several species

In this paper we consider the optimal control problem consisting of feeding in minimal time a Sequential Batch Reactors (SBR) where several species compete for a single substrate, with the objective being to reach a given (low) level of the substrate. Following [8, Gajardo et al. Minimal Time Sequential Batch Reactors with Bounded and Impulse Controls for One or More Species. SIAM J. Control and Optimization, vol. 47, Issue 6, pp. 2827-2856, 2008], 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 derived from the Pontryagin principle and the Hamilton-Jacobi-Bellman equation. We thus characterize the singular arcs of our problem as the extremal trajectories 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 show that those conditions are also necessary to ensure the optimality of the IOI strategy.