Mathematics

First we will discuss how to determine intrinsically the number of parameters in a classification problem, containing functional moduli. For this we analyze the Poincare series of moduli numbers of spaces of jets. Secondly, we show how the problem of feedback-equivalence of affine systems with two inputs in state space of dimension $4$ and $5$ can be reduced to the same problem for affine systems with one input, which was treated before. 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. Finally, for generic four-dimensional affine system with two inputs we construct feedback invariants, which are obstacles for their $1$-flatness. Namely, the system is $1$-flat if an only if these invariants vanish. The construction is based on the use of the coordinates of the Engel normal form for rank 2 distribution, corresponding to the affine system, and on the existence of the canonical projective structure on each abnormal extremal path of this distribution. Some results of the talk were obtained in collaboration with A. Agrachev and J.-B. Pomet.