Tools for improving feeding strategies in a SBR with several species

This paper analyzes feeding strategies in a sequential batch reactor (SBR) with the objective of reaching a given (low) substrate level as quickly as possible for a given volume of water. Inside the SBR, several species compete for a single substrate, which leads to a minimal time control problem in which the control variable is the feeding rate. Following Gajardo et al. (2008) SIAM J Control Optim 47(6):2827-2856, we allow the control variable to be a bounded measurable function of time combined with possible impulses associated with instantaneous dilutions. For this problem, the extremal trajectories of the singular arc type are characterized as the strategies used to maintain the substrate at a constant level. Since this optimization problem is difficult to solve, this characterization provides a valuable tool for investigating the optimality of various feeding strategies. Our aim is thus to illustrate the use of this tool by proposing potential optimal feeding strategies, which may then be compared with other more intuitive strategies. This aim was accomplished via several numerical experiments in which two specific strategies are compared.