  We consider nonlinear scalar-input differential control systems in the
vicinity of an equilibrium. When the linearized system at the equilibrium is
controllable, the nonlinear system is smoothly small-time locally controllable,
i.e., whatever $m>0$ and $T>0$, the state can reach a whole neighborhood of the
equilibrium at time $T$ with controls arbitrary small in $C^m$-norm. When the
linearized system is not controllable, we prove that small-time local
controllability cannot be recovered from the quadratic expansion and that the
following quadratic alternative holds.
  Either the state is constrained to live within a smooth strict invariant
manifold, up to a cubic residual, or the quadratic order adds a signed drift in
the evolution with respect to this manifold. In the second case, the quadratic
drift holds along an explicit Lie bracket of length $(2k+1)$, it is quantified
in terms of an $H^{-k}$-norm of the control, it holds for controls small in
$W^{2k,\infty}$-norm. These spaces are optimal for general nonlinear systems
and are slightly improved in the particular case of control-affine systems.
  Unlike other works based on Lie-series formalism, our proof is based on an
explicit computation of the quadratic terms by means of appropriate
transformations. In particular, it does not require that the vector fields
defining the dynamic are smooth. We prove that $C^3$ regularity is sufficient
for our alternative to hold.
  This work underlines the importance of the norm used in the smallness
assumption on the control: depending on this choice of functional setting, the
same system may or may not be small-time locally controllable, even though the
state lives within a finite dimensional space.

Optimization and Control;Classical Analysis and ODEs