  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 $\mu\ge 1/4$.

Analysis of PDEs