Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Claudia Alvarez-Latuz, Terence Bayen, Jerome Coville

TL;DR

The first main result proves the well-posedness of the control-to-state mapping and shows the existence of an optimal control for the minimum time problem associated with reaching the target set.

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.

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

TL;DR

The first main result proves the well-posedness of the control-to-state mapping and shows the existence of an optimal control for the minimum time problem associated with reaching the target set.

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.
Paper Structure (29 sections, 15 theorems, 224 equations, 3 figures)

This paper contains 29 sections, 15 theorems, 224 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General context
  3. Related models
  4. A minimum time control problem
  5. Notation and assumptions
  6. Overview of the contribution and organization
  7. Existence results for \ref{['main']}
  8. Stabilization results
  9. Target controllability results
  10. Minimum time results
  11. Existence of solutions to (\ref{['main']})
  12. Reformulation of the state equation
  13. Proof of Theorem \ref{['th-existence']}
  14. Stabilization by auxostat-type controls
  15. Control of the form (i)
  16. ...and 14 more sections

Key Result

Theorem 1.1

Suppose that $\alpha>0$ and that $\mu$ satisfies Hypotheses hyp0-hyp1. Then, for every admissible control $u$, and for every initial data $s_0\in (0,s_{in})$ and $f_0\in L^1({\Omega})$ such that $f_0\ge 0$, there is a unique pair of positive functions $(s,f)$ with $s\in C^{1}({\mathbb{R}}^*_{+},[0,s

Figures (3)

  • Figure 1: Behavior of the functional ${\mathcal{K}}[{f_\alpha(t,\cdot)}]$ w.r.t. time for different values of the parameter $\alpha$, where $f_\alpha$ is the solution of \ref{['main']} with $u(\cdot)$ given by (iv).
  • Figure 2:
  • Figure 3:

Theorems & Definitions (33)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1
  • Theorem 1.3
  • Remark 2
  • Remark 3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Theorem 1.7
  • ...and 23 more