Feedback stabilization of a 1D linear reaction-diffusion equation with delay boundary control

TL;DR

This paper develops a boundary control method with delay compensation for stabilizing a reaction-diffusion PDE that is unstable without control, using a decomposition approach and explicit prediction-based control design.

Contribution

It introduces an explicit prediction-based boundary control for delayed boundary stabilization of reaction-diffusion equations, decomposing the system into unstable and stable parts.

Findings

Successfully stabilizes an unstable reaction-diffusion PDE with boundary delay.

Provides an explicit control design using Artstein transformation and Lyapunov functions.

Validates the approach through numerical simulations.

Abstract

The goal of this work is to compute a boundary control of reaction-diffusion partial differential equation. The boundary control is subject to a constant delay, whereas the equation may be unstable without any control. For this system equivalent to a parabolic equation coupled with a transport equation, a prediction-based control is explicitly computed. To do that we decompose the infinite-dimensional system into two parts: one finite-dimensional unstable part, and one stable infinite-dimensional part. An finite-dimensional delay controller is computed for the unstable part, and it is shown that this controller succeeds in stabilizing the whole partial differential equation. The proof is based on a an explicit form of the classical Artstein transformation, and an appropriate Lyapunov function. A numerical simulation illustrate the constructive design method.