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Geometric Control Theory and Sub-Riemannian Geometry pp 133–152Cite as

On geometry of affine control systems with one input

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 5))

Abstract

We demonstrate how the novel approach to the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works for rank 2 distributions of maximal class in ℝn with additional structures such as affine control systems with one input spanning these distributions, sub-(pseudo)Riemannian structures etc. In contrast to the case of an arbitrary rank 2 distribution without additional structures, in the considered cases each abnormal extremal (of the underlying rank 2 distribution) possesses a distinguished parametrization. This fact allows one to construct the canonical frame on a (2n−3)-dimensional for arbitrary n ≥ 5. The moduli spaces of the most symmetric models are described as well.

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Authors and Affiliations

  1. Belarussian State University, Nezavisimosti Ave. 4, Minsk, 220030, Belarus

    Boris Doubrov

  2. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Igor Zelenko

Authors
  1. Boris Doubrov
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  2. Igor Zelenko
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Corresponding author

Correspondence to Igor Zelenko .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica e Informatica “U.Dini”, Università degli Studi di Firenze, Firenze, Italy

    Gianna Stefani  & Andrey Sarychev  & 

  2. CNRS, CMAP, École Polytechnique, INRIA Saclay, Team GECO, Palaiseau, France

    Ugo Boscain

  3. LSIS, Université de Toulon, La Garde Cedex, France

    Jean-Paul Gauthier

  4. INRIA Saclay, Team GECO, CMAP, École Polytechnique, Palaiseau, France

    Mario Sigalotti

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Doubrov, B., Zelenko, I. (2014). On geometry of affine control systems with one input. In: Stefani, G., Boscain, U., Gauthier, JP., Sarychev, A., Sigalotti, M. (eds) Geometric Control Theory and Sub-Riemannian Geometry. Springer INdAM Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-02132-4_9

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