# Optimal and Sub-optimal Feedback Controls for Biogas Production

## TL;DR

This paper investigates optimal and sub-optimal feedback control strategies to maximize biogas production in continuous bio-processes, providing explicit solutions and bounds for different horizon scenarios and illustrating results with specific growth functions.

## Contribution

It introduces explicit feedback controls for biogas production optimization over infinite and finite horizons, including bounds on sub-optimality and applications to specific growth models.

## Key findings

- Identified optimal controls for infinite horizon problems with averaged and discounted rewards.
- Developed explicit, time-independent feedback controls for finite horizon problems.
- Provided bounds on the sub-optimality of proposed controllers.

## Abstract

We revisit the optimal control problem of maximizing biogas production in continuous bio-processes in two directions: 1. over an infinite horizon, 2. with sub-optimal controllers independent of the time horizon. For the first point, we identify a set of optimal controls for the problems with an averaged reward and with a discounted reward when the discount factor goes to 0 and we show that the value functions of both problems are equal. For the finite horizon problem, our approach relies on a framing of the value function by considering a different reward for which the optimal solution has an explicit optimal feedback that is time-independent. In particular, we show that this technique allows us to provide explicit bounds on the sub-optimality of the proposed controllers. The various strategies are finally illustrated on Haldane and Contois growth functions.