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

Cut Locus and Optimal Synthesis in Sub-Riemannian Problem on the Lie Group SH(2)

“…The sub-Riemannian problem on the Lie group SH(2) reads as follows [1]: By Cauchy-Schwarz inequality, the sub-Riemannian length functional l minimization problem (2.4) is equivalent to the problem of minimizing the following action functional with fixed t 1 [11]:…”

“…The work was initiated in [1] where we defined the sub-Riemannian problem. The control system comprises two 3-dimensional left invariant vector fields and a 2-dimensional linear control vector.…”

“…In [2] we proved the Liouville integrability of the Hamiltonian system. We calculated the Hamiltonian flow such that the extremal trajectories were parametrized in terms of Jacobi elliptic functions [1]. Since PMP states only the first order optimality conditions, the trajectory resulting from PMP are only potentially optimal called extremal trajectories or geodesics.…”

“…Rigorous techniques for this optimality analysis have evolved over the years from research on related sub-Riemannian problems on various Lie groups, see e.g., [4], [5], [6], [7]. These techniques were employed in [1] and [8] to compute the Maxwell strata and the conjugate locus in the problem under investigation. An effective upper bound on the cut time was also computed.…”

“…In Section 2, we review the results from [1] and [8] as ready reference. Sections 3 and 4 contain the main results of this work.…”