Maxwell Strata and Conjugate Points in the Sub-Riemannian Problem on the Lie Group SH(2)

We study local and global optimality of geodesics in the left invariant sub-Riemannian problem on the Lie group $\mathrm{SH}(2)$. 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. Based on our results of $n$-th conjugate time and $n$-th Maxwell time, we prove a generalization of Rolle's theorem that between any two consecutive Maxwell points, there is exactly one conjugate point along any geodesic.