Null Controllability of One-dimensional Parabolic Equations by the Flatness Approach

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 provides explicitly 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 $a(x)>0$ 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 $\mu\ge 1/4$.