Flatness for a Strongly Degenerate 1-D Parabolic Equation

TL;DR

This paper addresses controlling a strongly degenerate 1-D parabolic equation using flatness techniques to explicitly steer solutions to zero within any positive time, despite degeneracy challenges.

Contribution

It introduces a flatness-based control method for a strongly degenerate parabolic PDE, providing explicit controls in Gevrey classes for null controllability.

Findings

Explicit control functions constructed in Gevrey classes

Achieved null controllability for any initial data in finite time

Demonstrated effectiveness despite strong degeneracy

Abstract

We consider the degenerate equation $$\partial\_t f(t,x) - \partial\_x \left( x^{\alpha} \partial\_x f \right)(t,x) =0,$$ on the unit interval $x\in(0,1)$, in the strongly degenerate case $\alpha \in [1,2)$ with adapted boundary conditions at $x=0$ and boundary control at $x=1$. We use the flatness approach to construct explicit controls in some Gevrey classes steering the solution from any initial datum $f\_0 \in L^2(0,1)$ to zero in any time $T\textgreater{}0$.