Finite-dimensional predictor-based feedback stabilization of a 1D linear reaction-diffusion equation with boundary input delay

TL;DR

This paper presents a simple, explicit predictor-based feedback control method to stabilize a 1D reaction-diffusion equation with boundary input delay, using finite-dimensional system design and pole-shifting techniques.

Contribution

It introduces a novel predictor-based feedback stabilization approach for delayed boundary control in reaction-diffusion systems, offering simplicity and efficiency over existing methods.

Findings

Successful stabilization in H1 norm demonstrated

Explicit predictor-based feedback designed from finite-dimensional subsystem

Comparison shows advantages over backstepping method

Abstract

We consider a one-dimensional controlled reaction-diffusion equation, where the control acts on the boundary and is subject to a constant delay. Such a model is a paradigm for more general parabolic systems coupled with a transport equation. We prove that this is possible to stabilize (in H 1 norm) this process by means of an explicit predictor-based feedback control that is designed from a finite-dimensional subsystem. The implementation is very simple and efficient and is based on standard tools of pole-shifting. Our feedback acts on the system as a finite-dimensional predictor. We compare our approach with the backstepping method.