Maxwell Strata in the sub-Riemannian problem on solvable, nonnilpotent regular three-dimensional Lie groups
Adriano Da Silva, Lino Grama, Douglas Duarte Novaes, Margarita Quispe Tusco
TL;DR
This work analyzes the sub-Riemannian optimal control problem on regular three-dimensional solvable Lie groups, where the vertical dynamics reduce to a perturbed pendulum on $G(\theta)$. Using qualitative phase-space analysis and the discrete Klein four symmetry group, the authors fully characterize the Maxwell set and prove that the first Maxwell time equals the pendulum period $\tau(\lambda)$ for almost all geodesics, yielding an explicit upper bound for the cut time. The approach hinges on the left-invariant sub-Riemannian structure $\Delta_{\eta}$, the geometry of the exponential map, and symmetry-induced relations among extremals. This period-based bound extends known Maxwell-based optimality results for related groups (e.g., SE(2), SH(2)) and provides a concrete tool for assessing the loss of optimality in this class of sub-Riemannian problems.
Abstract
In this paper, we study the sub-Riemannian problem associated with contact structures on connected, simply connected, solvable, non-nilpotent, regular three-dimensional Lie groups. For these groups, the vertical component of the Hamiltonian system takes the form of a perturbed pendulum. A qualitative phase-space analysis allows us to prove that this vertical component exhibits nontrivial symmetries. In particular, we are able to fully characterize the Maxwell set corresponding to these symmetries, and show that its first Maxwell time coincides with the period of the pendulum for almost all geodesics. This result yields an explicit upper bound for the cut time in terms of the period of the pendulum.