  The paper is devoted to the local classification of generic control-affine
systems on an n-dimensional manifold with scalar input for any n>3 or with two
inputs for n=4 and n=5, up to state-feedback transformations, preserving the
affine structure. First using the Poincare series of moduli numbers we
introduce the intrinsic numbers of functional moduli of each prescribed number
of variables on which a classification problem depends. In order to classify
affine systems with scalar input we associate with such a system the canonical
frame by normalizing some structural functions in a commutative relation of the
vector fields, which define our control system. Then, using this canonical
frame, we introduce the canonical coordinates and find a complete system of
state-feedback invariants of the system. It also gives automatically the
micro-local (i.e. local in state-input space) classification of the generic
non-affine n-dimensional control system with scalar input for n>2. Further we
show how the problem of feedback-equivalence of affine systems with
two-dimensional input in state space of dimensions 4 and 5 can be reduced to
the same problem for affine systems with scalar input. In order to make this
reduction we distinguish the subsystem of our control system, consisting of the
directions of all extremals in dimension 4 and all abnormal extremals in
dimension 5 of the time optimal problem, defined by the original control
system. In each classification problem under consideration we find the
intrinsic numbers of functional moduli of each prescribed number of variables
according to its Poincare series.
