Journal of Dynamical and Control Systems

, Volume 20, Issue 3, pp 341–364 | Cite as

Extremal Trajectories and Maxwell Strata in Sub-Riemannian Problem on Group of Motions of Pseudo-Euclidean Plane

  • Yasir Awais ButtEmail author
  • Yuri L. Sachkov
  • Aamer Iqbal Bhatti


We consider the sub-Riemannian length minimization problem on the group of motions of pseudo-Euclidean plane that form the special hyperbolic group SH(2). The system comprises of left invariant vector fields with 2-dimensional linear control input and energy cost functional. We apply the Pontryagin maximum principle to obtain the extremal control input and the sub-Riemannian geodesics. A change of coordinates transforms the vertical subsystem of the normal Hamiltonian system into the mathematical pendulum. In suitable elliptic coordinates, the vertical and the horizontal subsystems are integrated such that the resulting extremal trajectories are parametrized by the Jacobi elliptic functions. Qualitative analysis reveals that the projections of normal extremal trajectories on the xy-plane have cusps and inflection points. The vertical subsystem being a generalized pendulum admits reflection symmetries that are used to obtain a characterization of the Maxwell strata.


Sub-Riemannian geometry Special hyperbolic group SH(2) Extremal trajectories Parametrization Elliptic coordinates Jacobi elliptic functions Maxwell strata 

Mathematics Subject Classifications (2010)

49J15 93B27 93C10 53C17 22E30 


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Yasir Awais Butt
    • 1
    Email author
  • Yuri L. Sachkov
    • 2
  • Aamer Iqbal Bhatti
    • 1
  1. 1.Department of Electronic EngineeringMuhammad Ali Jinnah UniversityIslamabadPakistan
  2. 2.Program Systems InstitutePereslavl-ZalesskyRussia

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