Stability of Integral Delay Equations and Stabilization of Age-Structured Models

TL;DR

This paper develops new output feedback laws for stabilizing age-structured PDE models like chemostats, using innovative control Lyapunov functionals and stability results for integral delay equations, without requiring parametric knowledge.

Contribution

It introduces non-standard, observer-free feedback stabilization methods for positive-valued PDE models, transforming them into ODEs and delay equations, with novel CLFs and stability results.

Findings

Global stabilization achieved in positive orthant

Development of unconventional Control Lyapunov Functionals

Explicit stability results for integral delay equations

Abstract

We present bounded dynamic (but observer-free) output feedback laws that achieve global stabilization of equilibrium profiles of the partial differential equation (PDE) model of a simplified, age-structured chemostat model. The chemostat PDE state is positive-valued, which means that our global stabilization is established in the positive orthant of a particular function space-a rather non-standard situation, for which we develop non-standard tools. Our feedback laws do not employ any of the (distributed) parametric knowledge of the model. Moreover, we provide a family of highly unconventional Control Lyapunov Functionals (CLFs) for the age-structured chemostat PDE model. Two kinds of feedback stabilizers are provided: stabilizers with continuously adjusted input and sampled-data stabilizers. The results are based on the transformation of the first-order hyperbolic partial differential equation to an ordinary differential equation (one-dimensional) and an integral delay equation (infinite-dimensional). Novel stability results for integral delay equations are also provided; the results are of independent interest and allow the explicit construction of the CLF for the age-structured chemostat model.