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

Home/Archive/arXiv

arXiv · 2602.17782

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

View original source ↗Read source document ↗Export metadata ↓

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.

Explore related subjects

math.OC math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Adriano Da Silva, Lino Grama, Douglas Duarte Novaes, Margarita Quispe Tusco. 2026-02-19. Maxwell Strata in the sub-Riemannian problem on solvable, nonnilpotent regular three-dimensional Lie groups. https://arxiv.org/abs/2602.17782

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Strategic Inference in Stackelberg Games: Optimal Control for Revealing Adversary Intent

We study a continuous-time stochastic Stackelberg game in which a leader seeks to accomplish a primary objective while inferring a hidden parameter of a rational follower. The follower solves an entropy-regularized linear-quadratic tracking problem and responds to the leader's trajectory with a randomized policy. Anticipating this response, the leader designs informative controls to maximize the estimation efficiency for the follower's latent intent, through maximum likelihood estimation. Unlike prior work on discrete-time or finite-candidate inverse learning, our framework enables continuous parameter inference without prior assumptions and endogenizes the information source through the follower's strategic feedback. We derive semi-explicit solutions, prove well-posedness, and develop recurrent neural network algorithms to approximate the leader's path-dependent control. Numerical experiments demonstrate how the leader balances task performance and information gain, highlighting the practical value of our approach for adversarial strategic inference.

math.OC↗

arXiv2026

Stratification for Nonlinear Semidefinite Programming

This paper introduces a stratification framework for nonlinear semidefinite programming (NLSDP) that reveals and utilizes the geometry behind the nonsmooth KKT system. Based on the index stratification of $\mathbb{S}^n$ and its lift to the primal-dual space, a stratified variational analysis is developed. Specifically, we define the stratum-restricted regularity property, characterize it by the verifiable weak second order condition (W-SOC) and weak strict Robinson constraint qualification (W-SRCQ), and interpret the W-SRCQ geometrically via transversality, with stability along strata. The interactions of these properties across neighboring strata are further examined, leading to the conclusion that classical strong-form regularity conditions correspond to the local uniform validity of stratum-restricted counterparts. On the algorithmic side, a stratified Gauss--Newton method with normal steps and a correction mechanism is proposed for globally solving the KKT equation through a least-squares merit function. We demonstrate that the algorithm converges globally to directional stationary points. Moreover, under the second order sufficient condition (SOSC) and the strict Robinson constraint qualification (SRCQ) at an accumulation point, with a suitable correction threshold, the whole sequence converges superlinearly to this point, which is a KKT pair, and eventually identifies the active stratum. The rate is quadratic if the problem data are additionally of class $LC^2$ near the solution.

math.OC↗

arXiv2026

Convergence Rate Analysis of SOAP with Arbitrary Orthogonal Projection Matrices

In this short note, we establish, for the first time, the convergence rate of SOAP, an efficient and popular matrix-based optimizer for training deep neural networks. Our analysis extends to a more general variant of SOAP that admits arbitrary orthogonal projection matrices and requires only that these matrices be conditionally independent of the current stochastic gradient at each iteration. For example, they may be constructed from information available up to the preceding step.

math.OC↗