Parametrization of Extremal Trajectories in Sub-Riemannian Problem on Group of Motions of Pseudo Euclidean Plane

We consider the sub-Riemannian length optimization problem on the group of motions of hyperbolic plane i.e. 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 prove the global controllability of control distribution and use Pontryagin 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 transforms 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.