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

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Abstract

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.

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Authors and Affiliations

  1. Department of Electronic Engineering, Muhammad Ali Jinnah University, Islamabad, Pakistan

    Yasir Awais Butt & Aamer Iqbal Bhatti

  2. Program Systems Institute, Pereslavl-Zalessky, Russia

    Yuri L. Sachkov

Authors
  1. Yasir Awais Butt
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  2. Yuri L. Sachkov
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  3. Aamer Iqbal Bhatti
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Correspondence to Yasir Awais Butt.

Additional information

Work of the second author is supported by Grant of the Russian Federation for the State Support of Researches (Agreement No 14.B25.31.0029).

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Butt, Y.A., Sachkov, Y.L. & Bhatti, A.I. Extremal Trajectories and Maxwell Strata in Sub-Riemannian Problem on Group of Motions of Pseudo-Euclidean Plane. J Dyn Control Syst 20, 341–364 (2014). https://doi.org/10.1007/s10883-014-9239-2

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  • DOI: https://doi.org/10.1007/s10883-014-9239-2

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