The Age-Structured Chemostat with Substrate Dynamics as a Control System

TL;DR

This paper models an age-structured chemostat with substrate dynamics as a control system, proving global existence, uniqueness of solutions, and establishing a well-defined control framework for the nonlinear PDE-ODE system.

Contribution

It introduces a novel control system formulation for an age-structured chemostat with substrate dynamics, including existence and uniqueness results under a weak solution framework.

Findings

Global existence and uniqueness of solutions for all initial conditions and controls.

The model defines a well-structured control system on a metric space.

Establishment of a weak solution framework for the nonlinear PDE-ODE system.

Abstract

In this work we study an age-structured chemostat model with a renewal boundary condition and a coupled substrate equation. The model is nonlinear and consists of a hyperbolic partial differential equation and an ordinary differential equation with nonlinear, nonlocal terms appearing both in the ordinary differential equation and the boundary condition. Both differential equations contain a non-negative control input, while the states of the model are required to be positive. Under an appropriate weak solution framework, we determine the state space and the input space for this model. We prove global existence and uniqueness of solutions for all admissible initial conditions and all allowable control inputs. To this purpose we employ a combination of Banach's fixed-point theorem with implicit solution formulas and useful solution estimates. Finally, we show that the age-structured chemostat model gives a well-defined control system on a metric space.