Ergodic Theorem for Stabilization of a Hyperbolic PDE Inspired by Age-Structured Chemostat

TL;DR

This paper develops a feedback stabilization method for a hyperbolic PDE inspired by age-structured chemostats, using sparse sampling and without needing detailed state measurements or exact model knowledge.

Contribution

It introduces a novel sampled-data feedback control approach for hyperbolic PDEs with multiplicative input, applicable to chemostat models.

Findings

Ensures stabilization with arbitrarily sparse sampling.

Does not require measurement of the age profile.

Operates without exact model knowledge.

Abstract

We study a feedback stabilization problem for a first-order hyperbolic partial differential equation. The problem is inspired by the stabilization of equilibrium age profiles for an age-structured chemostat, using the dilution rate as the control. Two distinguishing features of the problem are that (a) the PDE has a multiplicative (instead of an additive) input and (b) the state is fed back to the inlet boundary. We provide a sampled-data feedback that ensures stabilization under arbitrarily sparse sampling and that satisfies input constraints. Our chemostat feedback does not require measurement of the age profile, nor does it require exact knowledge of the model.