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

TL;DR

This paper investigates the existence of longest optimal control arcs in three-dimensional contact sub-Lorentzian structures, providing conditions for their existence on specific Lie groups, advancing understanding in geometric control theory.

Contribution

It establishes sufficient conditions for the existence of longest arcs in certain left-invariant sub-Lorentzian structures, addressing a nontrivial problem in optimal control theory.

Findings

Existence conditions for longest arcs on solvable Lie groups.

Existence results for structures on the universal cover of SL(2, R).

Advances in understanding optimal control in sub-Lorentzian geometry.

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).