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

The problem of finding optimal (longest) curves 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. In this paper, we address 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 covering of the Lie group $\mathrm{SL}_2(\mathbb{R})$.