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

Abstract

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 nth 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.

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

“…Similarly we proved that the first conjugate time t conj 1 (λ) is bounded as (Theorems 4.1-4.3) [8]:…”

“…We now briefly review the results from [1] and [8] as a ready reference in this paper. System (2.1) satisfies the bracket generating condition and is hence globally controllable [12], [13].…”

“…In [8] Proposition 3.7 we proved that the first Maxwell points corresponding to the reflection symmetries of the vertical subsystem lie on the plane z = 0 and the corresponding Maxwell time t…”

“…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.…”