Using infinitesimal symmetries for determining the first Maxwell time of geometric control problem on SH(2)

TL;DR

This work targets determining the first Maxwell time for sub-Riemannian geodesics on SH$(2)$ by exploiting infinitesimal symmetries. The authors develop a systematic method to compute the symmetry algebra via L$_v$(Δ) ⊆ Δ and L$_v$(g) = 0, then apply it to the left-invariant SH$(2)$ control problem, reducing extremal dynamics to a pendulum and identifying a symmetry group isomorphic to SO$(1,2)$. They categorize Maxwell points into PMP-induced strata C$_1$–C$_5$, derive explicit Maxwell times, and show how the SO$(1,2)$ action fixes a common set S whose intersection with geodesics yields the first loss of optimality. The results provide a concrete, symmetry-driven route to quantify optimality horizons for motion planning on SH$(2)$, with implications for geometric control and sub-Riemannian analysis.

Abstract

In this work, we utilize infinitesimal symmetries to compute Maxwell points which play a crucial role in studying sub-Riemannian control problems. By examining the infinitesimal symmetries of the geometric control problem on the SH(2) group, particularly through its Lie algebraic structure, we identify invariant quantities and constraints that streamline the Maxwell point computation.

Figures (10)

• Figure 1: The graph of the function $g$
• Figure 2: Local minimizer, in the case $\lambda=(\varphi, k) \in C_1$
• Figure 3: The trajectory's symmetric
• Figure 4: The trajectory $(x(t), y(t))$ and its symmetry, in the case $\lambda \in C_1$
• Figure 5: The local minimizer for numerical values, where $\lambda=(\varphi, k) \in C_1$.

Theorems & Definitions (12)

• Proposition 2.1
• Definition 3.1
• Proposition 3.2
• Theorem 4.1
• Proposition 4.2
• proof
• Proposition 4.3
• Corollary 4.4
• Proposition 4.5
• proof