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The normal sub-Riemannian geodesic flow on E(2) generated by a left-invariant metric and a right-invariant distribution

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Abstract

We consider a sub-Riemannian problem on the three-dimensional solvable Lie group E(2) endowed with a left-invariant metric and a right-invariant distribution. The problem is based on construction of a Hamilton structure for the given metric by the Pontryagin maximum principle.

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References

  1. Mazhitova A. D., “The geodesic flow of a sub-Riemannian metric on a three-dimensional solvable Lie group,” Siberian Adv. Math., 23, No. 2, 99–105 (2013).

    Article  Google Scholar 

  2. Mazhitova A. D., “Sub-Riemannian geodesics on the three-dimensional solvable non-nilpotent Lie group SOLV−,” J. Dynam. Control Systems, 18, No. 3, 309–322 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  3. Agrachev A. and Barilari D., “Sub-Riemannian structures on 3D Lie groups,” J. Dynam. Control Systems, 18, No. 3, 21–44 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  4. Taimanov I. A., “Integrable geodesic flows of non-holonomic metrics,” J. Dynam. Control Systems, 3, 129–147 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  5. Boscain U. and Rossi F., “Invariant Carnot-Carathéodory metrics on S 3, SO(3), SL(2) and lens spaces,” SIAM J. Control Optim., 47, 1851–1878 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  6. Calin O., Chang D.-Ch., and Markina I., “Sub-Riemannian geometry on the sphere S 3,” Canad. J. Math., 61, 721–739 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  7. Butt Ya. A., Sachkov Yu. L., and Bhatti A. I., “Parametrization of extremal trajectories in sub-Riemannian problem on group of motions of pseudo Euclidean plane,” J. Dynam. Control Systems (to be published).

  8. Agrachev A. A. and Sachkov Yu. L., Control Theory from the Geometric Viewpoint, Springer-Verlag, Berlin (2004).

    Book  MATH  Google Scholar 

  9. Sachkov Yu. L., Control and Symmetry of Invariant Systems on Lie Groups and Homogeneous Spaces [in Russian], Fizmatlit, Moscow (2007).

    Google Scholar 

  10. Bateman H. and Erdélyi A., Higher Transcendental Functions. Vol. 3: Elliptic and Automorphic Functions, Lamé and Mathieu Functions, McGraw-Hill Book Comp. Inc., New York (1955).

    Google Scholar 

  11. Whittaker E. T. and Watson G. N., A Course of Modern Analysis. Part 2: Transcendental Functions, Cambridge Univ. Press, Cambridge (1996).

    Book  Google Scholar 

  12. Gradshteĭn I. S. and Ryzhik I. M., Tables of Integrals, Sums, Series, and Products, Academic Press, Boston (1994).

    Google Scholar 

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Authors and Affiliations

  1. Al-Farabi Kazakh National University, Almaty, Kazakhstan

    A. D. Mazhitova

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  1. A. D. Mazhitova
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Correspondence to A. D. Mazhitova.

Additional information

Original Russian Text Copyright © 2014 Mazhitova A.D.

The author was supported by Grant 1431/GF of the Ministry of Education and Science of the Republic of Kazakhstan and a Grant of the Government of the Russian Federation for the State Maintenance of Scientific Research (Agreement 14.B25.31.0029). Almaty.

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 5, pp. 1160–1166, September–October, 2014.

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Mazhitova, A.D. The normal sub-Riemannian geodesic flow on E(2) generated by a left-invariant metric and a right-invariant distribution. Sib Math J 55, 948–953 (2014). https://doi.org/10.1134/S0037446614050127

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  • DOI: https://doi.org/10.1134/S0037446614050127

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