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Optimal Synthesis in a Radially Symmetric Grushin Space

Michael Albert

Abstract

We study the geometry of $\mathbb{R}^3$ equipped with a rotationally invariant Carnot-Carthéodory metric obtained by weighting motion in the $z$-direction by a function $f(r)$ of the cylindrical radius. When $f$ vanishes only at $r=0$, the space exhibits a Grushin--type singularity along the vertical axis. We provide sufficient conditions on $f$ ensuring a Grushin--like structure and describe the full optimal synthesis at singular points. For Riemannian points, we propose a candidate cut time determined by a discrete symmetry of the Hamiltonian flow. In the integrable case $f(r)=r$, we prove that this candidate coincides with the true cut time and give an explicit description of the cut locus.

Optimal Synthesis in a Radially Symmetric Grushin Space

Abstract

We study the geometry of equipped with a rotationally invariant Carnot-Carthéodory metric obtained by weighting motion in the -direction by a function of the cylindrical radius. When vanishes only at , the space exhibits a Grushin--type singularity along the vertical axis. We provide sufficient conditions on ensuring a Grushin--like structure and describe the full optimal synthesis at singular points. For Riemannian points, we propose a candidate cut time determined by a discrete symmetry of the Hamiltonian flow. In the integrable case , we prove that this candidate coincides with the true cut time and give an explicit description of the cut locus.

Paper Structure

This paper contains 16 sections, 16 theorems, 146 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Background and Definitions
  3. Hamiltonian Theory
  4. Control and Symplectic Theory
  5. Hamiltonian Equations
  6. Initial Riemannian Points
  7. Initial Singular Points
  8. Optimal Synthesis
  9. Optimal Synthesis at Singular Points
  10. Conjectured Cut Time for Riemannian Initial Points
  11. Conjugacy Via Jacobian Determinants
  12. Case of pdf
  13. Conjugate Time Analysis for Covector Edge Cases
  14. Extended Hadamard Argument
  15. Conclusion and Future Work
  16. ...and 1 more sections

Key Result

Theorem 1.4

The Carnot-Carthéodory metric is well defined and induces the Euclidean topology on $\mathbb{R}^3.$ Furthermore, $(\mathbb{R}^3,d_{CC})$ is a complete metric space. $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Three regimes of geodesics: $\gamma_1$ is a geodesic starting from a singular point. $\gamma_2$ is a geodesic starting from a Riemannian point but with $K=0$, which remains in the plane $\Pi_{\theta}$. $\gamma_3$ is a geodesic starting from a Riemannian point with $K\neq 0$, which leaves the plane $\Pi_\theta$ and then returns on the opposite side of $\Sigma$ at the time $T$ given in Theorem \ref{['Geodesic and Cut time Theorem']}.
  • Figure 2: Unit Ball $B_{CC}(q_0,1)$ for the singular point $q_0=(0,0,0)$ in the radial Grushin structure with $f(r)=\log(r+1)^\beta r^\alpha$, and $\alpha=1,\beta=2$.
  • Figure 3: Portion of the cut locus $\operatorname{Cut}(q_0)=\{(-x_0,-y_0,z): \lvert z-z_0\rvert\geq \frac{\pi r_0^2}{2}\}$ for the Riemannian point $q_0=(1,0,0)$ in the Radial Grushin space with $f(r)=r$ and unit speed geodesic trajectories corresponding to $w_0=1/2$, all intersecting at $T=t_{\operatorname{cut}}=\frac{\pi}{\lvert w_0\rvert}$.

Theorems & Definitions (33)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4
  • Theorem 2.1: Pontryagin Maximum Principle
  • proof
  • Theorem 2.2
  • proof
  • Definition 3.1
  • Theorem 3.2
  • ...and 23 more