Classification of stable time-optimal controls on 2-manifolds

In this paper, we provide a topological classification via graphs of time-optimal flows for generic control systems of the form , x ∈ M, |u| ≤ 1, on two-dimensional orientable compact manifolds, also proving the structural stability of generic optimal flows. More precisely, adding some additional structure to topological graphs, more precisely, rotation systems, and owing to a theorem of Heffter, dating back to the 19th century, we prove that there is a one-to-one correspondence between graphs with rotation systems and couples formed by a system and the 2-D manifold of minimal genus in which the system can be embedded. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.