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Global Stabilization of Chemostats with Nonzero Mortality and Substrate Dynamics

Iasson Karafyllis, Epiphane Loko, Miroslav Krstic, Antoine Chaillet

TL;DR

This work tackles the global stabilization of chemostat population models that include substrate dynamics and nonzero mortality. By augmenting classical feedback laws with a nonlinear, small-gain fortification and constructing explicit control Lyapunov functions, the authors derive globally stabilizing inputs for both a lumped two-state model and an age-structured, three-state model derived from an infinite-dimensional formulation. They prove global asymptotic stabilization under clear, physically meaningful conditions (notably, a mortality-dominated regime and, for Haldane kinetics, a growth rate at the inlet exceeding mortality), with local exponential stability in certain parameter regimes. The results are validated through illustrative simulations and extend the control toolbox for nonlinear biological systems where mortality cannot be neglected, providing a bridge toward more general infinite-dimensional models.

Abstract

In "chemostat"-type population models that incorporate substrate (nutrient) dynamics, the dependence of the birth (or growth) rate on the substrate concentration introduces nonlinear coupling that creates a challenge for stabilization that is global, namely, for all positive concentrations of the biomass and nutrients. This challenge for global stabilization has been overcome in the literature using relatively simple feedback when natural mortality of the biomass is absent. However, under natural mortality, it takes fortified, more complex feedback, outside of the existing nonlinear control design toolbox, to avoid biomass extinction from nutrient-depleted initial conditions. Such fortified feedback, the associated control Laypunov function design, and Lyapunov analysis of global stability are provided in this paper. We achieve global stabilization for two different chemostat models: (i) a lumped model, with two state variables, and (ii) a three-state model derived from an age-structured infinite-dimensional model. The proposed feedback stabilizers are explicit, applicable to both the lumped and the age-structured models, and coincide with simple feedback laws proposed in the literature when the mortality rate is zero. Global stabilization means subject to constraints: all positive biomass and nutrient concentrations are within the region of attraction of the desired equilibrium, and, additionally, this is achieved with a dilution input that is guaranteed to remain positive. For the lumped case with Haldane kinetics, we show that the reproduction rate dominating the mortality (excluding the reproduction and mortality being in balance) is not only sufficient but also necessary for global stabilization. The obtained results are illustrated with simple examples.

Global Stabilization of Chemostats with Nonzero Mortality and Substrate Dynamics

TL;DR

This work tackles the global stabilization of chemostat population models that include substrate dynamics and nonzero mortality. By augmenting classical feedback laws with a nonlinear, small-gain fortification and constructing explicit control Lyapunov functions, the authors derive globally stabilizing inputs for both a lumped two-state model and an age-structured, three-state model derived from an infinite-dimensional formulation. They prove global asymptotic stabilization under clear, physically meaningful conditions (notably, a mortality-dominated regime and, for Haldane kinetics, a growth rate at the inlet exceeding mortality), with local exponential stability in certain parameter regimes. The results are validated through illustrative simulations and extend the control toolbox for nonlinear biological systems where mortality cannot be neglected, providing a bridge toward more general infinite-dimensional models.

Abstract

In "chemostat"-type population models that incorporate substrate (nutrient) dynamics, the dependence of the birth (or growth) rate on the substrate concentration introduces nonlinear coupling that creates a challenge for stabilization that is global, namely, for all positive concentrations of the biomass and nutrients. This challenge for global stabilization has been overcome in the literature using relatively simple feedback when natural mortality of the biomass is absent. However, under natural mortality, it takes fortified, more complex feedback, outside of the existing nonlinear control design toolbox, to avoid biomass extinction from nutrient-depleted initial conditions. Such fortified feedback, the associated control Laypunov function design, and Lyapunov analysis of global stability are provided in this paper. We achieve global stabilization for two different chemostat models: (i) a lumped model, with two state variables, and (ii) a three-state model derived from an age-structured infinite-dimensional model. The proposed feedback stabilizers are explicit, applicable to both the lumped and the age-structured models, and coincide with simple feedback laws proposed in the literature when the mortality rate is zero. Global stabilization means subject to constraints: all positive biomass and nutrient concentrations are within the region of attraction of the desired equilibrium, and, additionally, this is achieved with a dilution input that is guaranteed to remain positive. For the lumped case with Haldane kinetics, we show that the reproduction rate dominating the mortality (excluding the reproduction and mortality being in balance) is not only sufficient but also necessary for global stabilization. The obtained results are illustrated with simple examples.

Paper Structure

This paper contains 17 sections, 3 theorems, 119 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Motivation.
  3. Literature on chemostat control.
  4. Contribution.
  5. Organization.
  6. A Simple Chemostat Model with Mortality
  7. An age-structured Chemostat Model
  8. Illustrative examples
  9. Proofs
  10. Proof of Theorem \ref{['Th1']}
  11. Proof of Theorem \ref{['Th2']}
  12. Proof of Theorem \ref{['Th3']}
  13. Conclusions
  14. Appendix
  15. Proof of Claim \ref{['Claim-Lipschitz']}
  16. ...and 2 more sections

Key Result

Theorem 1

Consider system systeme-simple2 under assumption (A). Then for every $\delta>0$ and $\alpha \in [0,1)$ the feedback law defined as is locally Lipschitz and achieves global asymptotic stabilization of $0\in \mathbb{R}^2$. Moreover, if $\alpha >0$ then the feedback law Feedback1 also achieves local exponential stabilization of $0\in \mathbb{R}^2$. $\square$

Figures (6)

  • Figure 1: The phase diagram for the open-loop chemostat model \ref{['system-simple1']} with $D=D^*=\frac{9}{10}$.
  • Figure 2: The phase diagram for the closed-loop chemostat model \ref{['system-simple1']} with \ref{['Feedback-origin']}.
  • Figure 3: The component $S(t)$ of the solution of the closed-loop system \ref{['system-simple1']} with \ref{['Feedback-origin']}, $\alpha=0.5$ and initial condition $(X(0),S(0))=(1,1).$ Red line: $\delta=100.$ Blue line: $\delta=1.$
  • Figure 4: The phase diagram for the chemostat model \ref{['systeme-ageODE']}.
  • Figure 5: The phase diagram of the closed-loop system \ref{['systeme-ageODE']} with \ref{['D-example']}
  • ...and 1 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Example 1
  • Example 2