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The paper is devoted to the local classification of generic germs of control-affine systems on an $n$-dimensional manifold with scalar input for any $n\geq4$ or with two inputs for $n=4$ and $n=5$, up to state-feedback transformations, preserving the affine structure (in the $C^\infty$ category for $n=4$ and the $C^\omega$ category for $n\geq5$). First, using the Poincaré 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 generic germs of 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 automatically gives the local in state-input space classification of generic germs of nonaffine $n$-dimensional control systems with scalar input for $n\geq3$ (in the $C^\infty$ category for $n=3$ and in the $C^\omega$ category for $n\geq4$). Further, we show how the problem of feedback equivalence of generic germs 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 Poincaré series.

On Feedback Classification of Control-Affine Systems with One- and Two-Dimensional Inputs