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Existence of the longest arcs for left-invariant three-dimensional contact sub-Lorentzian structures

A. V. Podobryaev

Abstract

The problem of finding optimal curves (the longest arcs) for sub-Lorentzian structures is an optimal control problem with an unbounded control set and a concave cost functional. The question of existence of an optimal solution is nontrivial for such problems. We solve here this question for some left-invariant three-dimensional contact sub-Lorentzian structures, whose classification is known. We propose sufficient conditions for the existence of the longest arcs for left-invariant (sub-)Lorentzian structures on solvable Lie groups and on the universal cover of the Lie group SL(2, R).

Existence of the longest arcs for left-invariant three-dimensional contact sub-Lorentzian structures

Abstract

The problem of finding optimal curves (the longest arcs) for sub-Lorentzian structures is an optimal control problem with an unbounded control set and a concave cost functional. The question of existence of an optimal solution is nontrivial for such problems. We solve here this question for some left-invariant three-dimensional contact sub-Lorentzian structures, whose classification is known. We propose sufficient conditions for the existence of the longest arcs for left-invariant (sub-)Lorentzian structures on solvable Lie groups and on the universal cover of the Lie group SL(2, R).
Paper Structure (4 sections, 9 theorems, 40 equations, 2 tables)

This paper contains 4 sections, 9 theorems, 40 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Sub-Lorentzian structures on three-dimensional solvable Lie groups
  4. Sub-Lorentzian structures on the universal cover of the Lie group $\mathrm{SL}_2(\mathbb{R})$

Key Result

Theorem 1

(1) For left-invariant sub-Lorentzian structures 1, 11--12(for $\chi = \kappa$) and 13--15(see Table tb-classification) on simply connected solvable Lie groups, for any two points $x_0$ and $x_1$ there exists the longest arc from $x_0$ to $x_1$ if and only if point $x_1$ is attainable from point $x_

Theorems & Definitions (25)

  • Definition 1
  • Theorem 1
  • Theorem 2
  • Remark 1
  • Definition 2
  • Definition 3
  • Theorem 3: lokutsievskiy-podobryaev, Th. 1
  • Remark 2
  • Corollary 1
  • proof
  • ...and 15 more