  The case of time minimization for affine control systems with control on the
disk is studied. After recalling the standard sufficient conditions for local
optimality in the smooth case, the analysis focusses on the specific type of
singularities encountered when the control is prescribed to the disk. Using a
suitable stratification, the regularity of the flow is analyzed, which helps to
devise verifiable sufficient conditions in terms of left and right limits of
Jacobi fields at a switching point. Under the appropriate assumptions,
piecewise regularity of the field of extremals is obtained.
