Null controllability of one-dimensional parabolic equations by the flatness approach

TL;DR

This paper proves the null controllability of one-dimensional parabolic equations with possibly degenerate or singular coefficients using the flatness approach, explicitly constructing controls and trajectories with Gevrey regularity.

Contribution

It introduces a flatness-based method for null controllability of degenerate or singular parabolic equations with boundary control, providing explicit control and trajectory formulas.

Findings

Achieved null controllability for degenerate/singular heat equations.

Explicit control and trajectory series with Gevrey regularity.

Applicable to equations with inverse square potential.

Abstract

We consider linear one-dimensional parabolic equations with space dependent coefficients that are only measurable and that may be degenerate or singular.Considering generalized Robin-Neumann boundary conditions at both extremities, we prove the null controllability with one boundary control by following the flatness approach, which providesexplicitly the control and the associated trajectory as series. Both the control and the trajectory have a Gevrey regularity in time related to the $L^p$ class of the coefficient in front of $u\_t$.The approach applies in particular to the (possibly degenerate or singular) heat equation $(a(x)u\_x)\_x-u\_t=0$ with $a(x)\textgreater{}0$ for a.e. $x\in (0,1)$ and $a+1/a \in L^1(0,1)$, or to the heat equation with inverse square potential $u\_{xx}+(\mu / |x|^2)u-u\_t=0$with $\mu\ge 1/4$.