Journal of Mathematical Sciences

, Volume 135, Issue 4, pp 3109–3124 | Cite as

Classification of stable time-optimal controls on 2-manifolds

  • U. Boscain
  • I. Nikolaev
  • B. Piccoli


In this paper, we provide a topological classification via graphs of time-optimal flows for generic control systems of the form \(\dot x = F(x) + uG(x)\), xM, |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.


Manifold Control System 19th Century Structural Stability Generic Control 
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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • U. Boscain
    • 1
  • I. Nikolaev
    • 2
  • B. Piccoli
    • 3
  1. 1.SISSA-ISASTriesteItaly
  2. 2.CRM, Université de MontréalMontréalCanada
  3. 3.I.A.C.—C.N.R.RomeItaly

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