We study local and global optimality of geodesics in the left invariant sub-Riemannian problem on the Lie group SH(2). We obtain the complete description of the Maxwell points corresponding to the discrete symmetries of the vertical subsystem of the Hamiltonian system. An effective upper bound on the cut time is obtained in terms of the first Maxwell times. We study the local optimality of extremal trajectories and prove the lower and upper bounds on the first conjugate times. We also obtain the generic time interval for the n-th conjugate time which is important in the study of sub-Riemannian wavefront.
Title: Maxwell strata and conjugate points in the sub-Riemannian problem on the Lie group SH(2)
Authors: Ya. A. Butt, Yu. L. Sachkov, A.I. Bhatti
Journal title: Journal of Dynamical and Control Systems