A Fredholm transformation for the rapid stabilization of a degenerate parabolic equation

TL;DR

This paper introduces a Fredholm transformation approach using backstepping to achieve rapid stabilization of a degenerate parabolic equation with Dirichlet control, ensuring exponential decay at a desired rate.

Contribution

It demonstrates that the Fredholm transformation is well-defined, continuous, and invertible in the natural energy space for this class of equations.

Findings

Fredholm transformation effectively stabilizes the degenerate parabolic equation.

The transformation guarantees exponential decay with an arbitrarily large rate.

The approach extends backstepping methods to degenerate PDEs.

Abstract

This paper deals with the rapid stabilization of a degenerate parabolic equation with a right Dirich-let control. Our strategy consists in applying a backstepping strategy, which seeks to find an invertible transformation mapping the degenerate parabolic equation to stabilize into an exponentially stable system whose decay rate is known and as large as we desire. The transformation under consideration in this paper is Fredholm. It involves a kernel solving itself another PDE, at least formally. The main goal of the paper is to prove that the Fredholm transformation is well-defined, continuous and invertible in the natural energy space. It allows us to deduce the rapid stabilization.