Maxwell Strata in the sub-Riemannian problem on solvable, nonnilpotent regular three-dimensional Lie groups

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.

💡 Research Summary

This paper investigates the sub‑Riemannian optimal control problem on connected, simply‑connected, solvable, non‑nilpotent, regular three‑dimensional Lie groups. Such groups can be written as a semidirect product (G(\theta)=\mathbb R\times_{\rho}\mathbb R^{2}) where (\rho_{t}=e^{t\theta}) and the (2\times2) matrix (\theta) satisfies (\det\theta\cdot\operatorname{tr}\theta\neq0) (the regularity condition). A left‑invariant 2‑dimensional distribution (\Delta_{\eta}) is chosen, generated by ((1,0)) and ((0,\eta)) with (\omega(\eta,\theta\eta)\neq0); this distribution together with the Euclidean inner product defines a sub‑Riemannian (contact) structure.

The optimal control problem consists of steering a point (g=(z,w)\in G(\theta)) from the identity to a prescribed final point while minimizing the sub‑Riemannian length (equivalently, the energy). Applying the Pontryagin Maximum Principle (PMP) yields a Hamiltonian system on the cotangent bundle. After normalizing the Hamiltonian to the level set (H=1/2) (arc‑length parametrization) the vertical dynamics can be expressed in polar coordinates ((\phi,r)) as \