Journal of Dynamical and Control Systems

, Volume 25, Issue 2, pp 289–307 | Cite as

Optimality of Broken Extremals

  • Andrei A. Agrachev
  • Carolina BioloEmail author


In this paper, we analyse the optimality of broken Pontryagin extremal for an n-dimensional affine control system with a control parameter, taking values in a k-dimensional closed ball. We prove the optimality of broken normal extremals in many cases that include (but not are exhausted by) the cases of the involutive driftless part of the system for any n > k and of the contact driftless part for n = 3 and k = 2.


Optimal control Lie brackets Switching 



The work of A.A. Agrachev was supported by the Russian Science Foundation under grant No 17-11-01387.


  1. 1.
    Agrachev A. Some open problems arXiv:13042590. 2013.
  2. 2.
    Agrachev A, Barilari D, Boscain U. Introduction to Riemannian and Sub-Riemannian geometry, Book in preparation, p. 457
  3. 3.
    Agrachev A, Biolo C. 2016. Switching in time-optimal problem: the 3-D case with 2-D control, J Dyn Control Syst.
  4. 4.
    Agrachev AA, Biolo C. 2016. Switching in time-optimal problem with control in a ball, to appear in SIAM J Control Optim arXiv:1610.06755.
  5. 5.
    Agrachev A, Sachkov Yu L. 2004. Control theory from the geometric viewpoint. Springer.Google Scholar
  6. 6.
    Agrachev A, Sigalotti M. On the local structure of optimal trajectories in R3. SIAM J Control Optim 2003;42:513–31.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Biolo C. Switching in time-optimal problem, PhD thesis at SISSA under the supervision of A. Agrachev, academic year 2016-2017.Google Scholar
  8. 8.
    Brin M, Pesin Ya. Partially hyperbolic dynamical systems. Math USSR-Izv 1974; 8:177–218.CrossRefzbMATHGoogle Scholar
  9. 9.
    Boscain U, Piccoli B. Optimal syntheses for control systems on 2-D manifolds. Berlin: Springer; 2004, p. xiv+ 261.zbMATHGoogle Scholar
  10. 10.
    Fuller AT. Study of an optimum nonlinear system. J Electronics Control 1963; 15:63–71.MathSciNetCrossRefGoogle Scholar
  11. 11.
    Gamkrelidze RV. Principles of optimal control theory. New York: Plenum Publishing Corporation; 1978.CrossRefzbMATHGoogle Scholar
  12. 12.
    Kupka I. The ubiquity of Fuller’s phenomenon. Nonlinear controllability and optimal control. Marcel Dekker. In: Sussmann H, editors; 1990.Google Scholar
  13. 13.
    Lee JM. Introduction to Smooth Manifolds, 2nd ed. New York: Springer; 2012. ISBN 978-1-4419-99 82-5.CrossRefGoogle Scholar
  14. 14.
    Pontryagin LS, Boltyanskij VG, Gamkrelidze RV, Miskchenko EF. The mathematica theory of optimal processes. Oxford: Pergamon Press; 1964.Google Scholar
  15. 15.
    Schättler H. Regularity properties of optimal trajectories: Recently developed techniques Sussmann H, editor.Google Scholar
  16. 16.
    Schättler H, Sussmann H. On the regularity of optimal controls. Zeitr Angew Math Phys 1987;38 (2):292–301.MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Shoshitaishvili AN. 1975. Bifurcations of the topological type of a vector field near a singular point. Trudy Seminarov I.G.Petrovskogo 1, pp. 279–309 (in Russian). English translation in American Math. Soc. Translations 118(2) (1982).Google Scholar
  18. 18.
    Sigalotti M. Local regularity of optimal trajectories for control problems with general boundary conditions. J Dynam Control Systems 2005;11:91–123.MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Sigalotti M. Regularity properties of optimal trajectories of single-input control systems in dimension three. J Math Sci 2005;126:1561–73.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sussmann H. Time-optimal control in the plane. Feedback control of linear and nonlinear systems, Lecture notes in control and information scienced. Berlin: Springer; 1985. p. 244–60.Google Scholar
  21. 21.
    Sussmann H. Envelopes, conjugate points and optimal bang-bang extremals. In: Fliess M and Hazewinkel M, editors. Proceedings of the 1985 Paris Conference on Nonlinear Systems. Dordrecht: D. Reidel; 1986.Google Scholar
  22. 22.
    Sussmann HJ. The Markov-Dubins problem with angular acceleration control. Proceedings of the 36th Conference on Decision and Control, San Diego; 1997.Google Scholar
  23. 23.
    Zelikin MI, Borisov VF. Theory of chattering control with applications to astronautics, robotics, economics and engineering. Systems and control Foundations and applications. Boston: Birkhäuser; 1994.Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.SISSATriesteItaly
  2. 2.PSI RASPereslavl-ZalesskyRussia

Personalised recommendations