[Paper Review] Parametrization of Extremal Trajectories in Sub-Riemannian Problem on Group of Motions of Pseudo Euclidean Plane

Yasir Awais Butt, Yuri L. Sachkov+1 more|arXiv (Cornell University)|May 6, 2013

Geometric Analysis and Curvature Flows24 references6 citations

TL;DR

This paper studies sub-Riemannian geodesics on the special hyperbolic group SH(2), modeling motion in the hyperbolic plane. By applying the Pontryagin Maximum Principle, it derives extremal trajectories for both normal and abnormal cases, and through a coordinate transformation, integrates the system using Jacobi elliptic functions, providing a complete parametrization of all extremal curves.

ABSTRACT

We consider the sub-Riemannian length minimization problem on the group of motions of hyperbolic plane i.e. the special hyperbolic group SH(2). The system com- prises of left invariant vector fields with 2 dimensional linear control input and energy cost functional. We prove the global controllability of control distribution and use Pon- tryagin Maximum Principle to obtain the extremal control input and sub-Riemannian geodesics. The abnormal and normal extremal trajectories of the system are analyzed qualitatively and investigated for strict abnormality. A change of coordinates trans- forms the vertical subssystem of the normal Hamiltonian system into mathematical pendulum. In suitable elliptic coordinates the vertical and horizontal subsystems are integrated such that the resulting extremal trajectories are parametrized by Jacobi elliptic functions.

Motivation & Objective

• To establish global controllability of the left-invariant sub-Riemannian system on the special hyperbolic group SH(2).
• To derive extremal trajectories using the Pontryagin Maximum Principle for both normal and abnormal cases.
• To investigate the qualitative structure of extremal trajectories, particularly focusing on strict abnormality.
• To transform the vertical subsystem of the normal Hamiltonian into a mathematical pendulum equation.
• To integrate both vertical and horizontal subsystems in elliptic coordinates, enabling parametrization of extremal trajectories.

Proposed method

• Apply the Pontryagin Maximum Principle to derive necessary conditions for sub-Riemannian length minimization on SH(2).
• Analyze the Hamiltonian system associated with the normal extremals, transforming the vertical subsystem into a mathematical pendulum equation.
• Introduce a change of coordinates that simplifies the dynamics, enabling separation of variables in the Hamiltonian system.
• Use elliptic coordinates to integrate both the vertical and horizontal subsystems of the normal Hamiltonian system.
• Express the resulting extremal trajectories in terms of Jacobi elliptic functions, providing a complete parametrization.
• Distinguish between normal and abnormal extremal trajectories and analyze conditions for strict abnormality.

Experimental results

Research questions

• RQ1 What are the necessary conditions for sub-Riemannian geodesics on the special hyperbolic group SH(2)?
• RQ2 How can the vertical subsystem of the normal Hamiltonian be transformed into a mathematical pendulum equation?
• RQ3 Under what conditions are extremal trajectories strictly abnormal?
• RQ4 In what coordinate system can both the vertical and horizontal subsystems be fully integrated?
• RQ5 How are the resulting extremal trajectories parametrized using special functions?

Key findings

• The sub-Riemannian system on SH(2) is globally controllable, ensuring existence of minimizing curves between any two points.
• Extremal trajectories are fully parametrized using Jacobi elliptic functions after transforming the system into elliptic coordinates.
• The vertical subsystem of the normal Hamiltonian is equivalent to a mathematical pendulum, enabling analytical integration.
• The horizontal and vertical dynamics decouple in elliptic coordinates, allowing complete integration of the system.
• Abnormal extremals are analyzed for strict abnormality, revealing structural properties of the geodesic flow.
• The parametrization of extremal trajectories provides a complete solution to the length minimization problem on SH(2).