Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Families of Lorentzian problems on the Heisenberg group

Yu. Sachkov

TL;DR

This work analyzes two Lorentzian optimal control families on the Heisenberg group. The first family, with metric $g=-\omega_1^2+\omega_2^2+\omega_3^2/\varepsilon^2$, yields invariant sets, explicit normal and abnormal extremals, conjugate-point scarcity for $h_3\neq0$, a diffeomorphic exponential map, and complete characterizations of attainable sets and Lorentzian spheres, together with a detailed limit analysis as $\varepsilon\to0$ toward a sub-Lorentzian problem. The second family, with a reversed cone orientation, is globally controllable but admits no length-maximizing optimizers due to a periodic timelike trajectory, and it provides explicit normal and abnormal extremals and conjugate-point structure, with implications for general left-invariant problems on the Heisenberg group. Overall, the paper illuminates how cone geometry and the placement of the commutant influence optimality, controllability, and semiclassical convergence in Lorentzian and sub-Lorentzian Heizberg-type systems.

Abstract

We consider two families of Lorentzian problems on the Heisenberg group and their asymptotic behaviour as the parameter of a family tends to a limit.

Families of Lorentzian problems on the Heisenberg group

TL;DR

This work analyzes two Lorentzian optimal control families on the Heisenberg group. The first family, with metric , yields invariant sets, explicit normal and abnormal extremals, conjugate-point scarcity for , a diffeomorphic exponential map, and complete characterizations of attainable sets and Lorentzian spheres, together with a detailed limit analysis as toward a sub-Lorentzian problem. The second family, with a reversed cone orientation, is globally controllable but admits no length-maximizing optimizers due to a periodic timelike trajectory, and it provides explicit normal and abnormal extremals and conjugate-point structure, with implications for general left-invariant problems on the Heisenberg group. Overall, the paper illuminates how cone geometry and the placement of the commutant influence optimality, controllability, and semiclassical convergence in Lorentzian and sub-Lorentzian Heizberg-type systems.

Abstract

We consider two families of Lorentzian problems on the Heisenberg group and their asymptotic behaviour as the parameter of a family tends to a limit.

Paper Structure

This paper contains 26 sections, 24 theorems, 71 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The first family of Lorentzian problems
  3. Problem statement
  4. Invariant set
  5. Extremal trajectories
  6. Conjugate points
  7. Diffeomorphic property of the exponential mapping
  8. Attainable sets
  9. Existence of optimal trajectories
  10. Optimality of extremal trajectories
  11. Spheres and distance
  12. Convergence of the problem as $\varepsilon \to 0$
  13. Convergence of control system and future cone
  14. Convergence of the attainable set
  15. Convergence of the exponential mapping
  16. ...and 11 more sections

Key Result

Lemma 1

There holds the inclusion thus system $(pr1)$, $(pr2)$ is not globally controllable.

Figures (5)

  • Figure 1: The surface $S$ for $\varepsilon=1$
  • Figure 2: Boundaries of the attainable sets $\mathcal{A}_{0}$ and $\mathcal{A}_{1}$
  • Figure 4: The sphere $S_0(1)$
  • Figure 6: The surface $S$ for $\varepsilon = 1$
  • Figure 7: Periodic trajectory for system $(\ref{['pr31']})$--$(\ref{['pr23']})$

Theorems & Definitions (44)

  • Lemma 1
  • proof
  • Theorem 1: notes
  • Proposition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Corollary 1
  • Proposition 4
  • ...and 34 more