On geometry of affine control systems with one input Boris Doubrov and Igor Zelenko Abstract We demonstrate how the novel approach to the local geometry of struc- tures of nonholonomic nature, originated by Andrei Agrachev, works for rank 2 dis- tributions of maximal class in R n with additional structures such as affine control systems with one input spanning these distributions,sub-(pseudo)Riemannian struc- tures etc. In contrast to the case of an arbitrary rank 2 distributionwithout additional structures, in the considered cases each abnormal extremal (of the underlying rank 2 distribution) possesses a distinguished parametrization. This fact allows one to con- struct the canonical frame on a .2n3/-dimensional for arbitrary n  5 . The moduli spaces of the most symmetric models are described as well. 1 Introduction About seventeen years ago Andrei Agrachev proposed the idea to study the local geometry of control systems and geometric structures on manifolds by studying the flow of extremals of optimal control problems naturally associated with these ob- jects [1–3]. Originally he considered situations when one can assign a curve of La- grangian subspaces of a linear symplectic space or, in other words, a curve in a Lagrangian Grassmannian to an extremal of these optimal control problems. This curve was called the Jacobi curve of this extremal, because it contains all informa- tion about the solutions of the Jacobi equations along it. Agrachev’s constructions of Jacobi curves worked in particular for normal extremals of sub-Riemannian struc- tures and abnormal extremals of rank 2 distributions. Similar idea can be used for B. Doubrov Belarussian State University, Nezavisimosti Ave. 4, Minsk 220030, Belarus e-mail: doubrov@islc.org I. Zelenko ( ) Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA e-mail: zelenko@math.tamu.edu G. Stefani, U. Boscain, J.-P. Gauthier, A. Sarychev, M. Sigalotti (eds.): Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Series 5, DOI 10.1007/978-3-319-02132-4_9, © Springer International Publishing Switzerland 2014