  We consider scalar-input control systems in the vicinity of an equilibrium,
at which the linearized systems are not controllable. For finite dimensional
control systems, the authors recently classified the possible quadratic
behaviors. Quadratic terms introduce coercive drifts in the dynamics,
quantified by integer negative Sobolev norms, which are linked to Lie brackets
and which prevent smooth small-time local controllability for the full
nonlinear system.
  In the context of nonlinear parabolic equations, we prove that the same
obstructions persist. More importantly, we prove that two new behaviors occur,
which are impossible in finite dimension. First, there exists a continuous
family of quadratic obstructions quantified by fractional negative Sobolev
norms or by weighted variations of them. Second, and more strikingly,
small-time local null controllability can sometimes be recovered from the
quadratic expansion. We also construct a system for which an infinite number of
directions are recovered using a quadratic expansion.
  As in the finite dimensional case, the relation between the regularity of the
controls and the strength of the possible quadratic obstructions plays a key
role in our analysis.
