Target controllability for a minimum time problem in a trait-structured chemostat model

TL;DR

This paper studies a minimum time control problem in a trait-structured chemostat model, establishing well-posedness, convergence of controls, reachability of target states, and existence of optimal controls, supported by numerical simulations.

Contribution

It introduces a rigorous analysis of control strategies for trait-structured chemostats, including well-posedness, convergence, reachability, and optimal control existence, with numerical validation.

Findings

Proved well-posedness of the control-to-state mapping.

Established convergence of auxostat-type controls to stationary states.

Demonstrated the existence of optimal controls for the minimum time problem.

Abstract

In this paper, we consider a minimum time control problem governed by a trait-structured chemostat model including mutation and one limiting substrate. Our first main result proves the well-posedness of the control-to-state mapping. We subsequently analyze the class of auxostat-type controls, feedback laws designed to regulate substrate concentration, and prove that the corresponding solutions converge to a stationary state of the system. These convergence results are used to show the reachability of a target set corresponding to the selection of a population with a low weighted averaged half-saturation constant. Finally, we show the existence of an optimal control for the minimum time problem associated with reaching the target set. These theoretical findings are completed by numerical simulations.