Abstract
In this paper, we provide a topological classification via graphs of time-optimal flows for generic control systems of the form \(\dot x = F(x) + uG(x)\), 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.
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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004.
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Boscain, U., Nikolaev, I. & Piccoli, B. Classification of stable time-optimal controls on 2-manifolds. J Math Sci 135, 3109–3124 (2006). https://doi.org/10.1007/s10958-006-0148-0
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DOI: https://doi.org/10.1007/s10958-006-0148-0