Sub-Lorentzian extremals defined by an antinorm (Q6570087)

The paper represents a study of an optimal control problem defined through a left invariant sub-Lorentzian structure on a finite-dimensional real Lie group. The sub-Lorentzian structure is induced by a continuous antinorm associated with a closed convex cone (light cone) in the corresponding Lie algebra. The author proves that every extremal trajectory is a solution to a Hamiltonian system, which Hamiltonian function is defined through the antinorm. In the generic case, the tangent vectors of extremal trajectories (normal extremal trajectories) are either time-like or null (isotropic) vectors. In the degenerate case when the antinorm vanishes from the Hamiltonian, the tangent vectors of trajectories (abnormal extremal trajectories) are either null vectors or tangent vectors of sub-Riemannian abnormal trajectories determined by a distribution in the tangent bundle of the Lie group defined through the cone.