Stabilization of Age-Structured Chemostat Hyperbolic PDE with Actuator Dynamics

TL;DR

This paper develops new feedback control strategies for age-structured population models governed by hyperbolic PDEs, accounting for realistic actuation dynamics and constraints, ensuring stable and positive population densities.

Contribution

It extends existing control laws to include actuation dynamics and bounds, providing multiple designs that guarantee stability and positivity in age-structured PDE systems.

Findings

Achieved global asymptotic stabilization of population density profiles.

Ensured control inputs remain within specified positive bounds.

Developed multiple control designs with varying measurement requirements.

Abstract

For population systems modeled by age-structured hyperbolic partial differential equations (PDEs), we redesign the existing feedback laws, designed under the assumption that the dilution input is directly actuated, to the more realistic case where dilution is governed by actuation dynamics (modeled simply by an integrator). In addition to the standard constraint that the population density must remain positive, the dilution dynamics introduce constraints of not only positivity of dilution, but possibly of given positive lower and upper bounds on dilution. We present several designs, of varying complexity, and with various measurement requirements, which not only ensure global asymptotic (and local exponential) stabilization of a desired positive population density profile from all positive initial conditions, but do so without violating the constraints on the dilution state. To develop the results, we exploit the relation between first-order hyperbolic PDEs and an equivalent representation in which a scalar input-driven mode is decoupled from input-free infinite-dimensional internal dynamics represented by an integral delay system.