Fredholm transformation on Laplacian and rapid stabilization for the heat equation

TL;DR

This paper develops a Fredholm transformation-based backstepping method for rapid stabilization of the heat equation on a 1D torus, establishing controllability conditions and extending results to the viscous Burgers equation.

Contribution

It introduces a novel Fredholm transformation approach for Laplace operators in control theory, providing sharp functional analysis framework and extending to nonlinear PDEs.

Findings

Two scalar controls are necessary and sufficient for controllability and stabilization.

The Fredholm transformation achieves local rapid stability for the viscous Burgers equation.

The method applies to Laplace operators in a rigorous functional setting.

Abstract

We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present the backstepping method with Fredholm transformations on Laplace operators in a sharp functional setting, which is the main objective of this work. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.