Journal of Dynamical and Control Systems

, Volume 9, Issue 1, pp 103–129 | Cite as

Singular Trajectories in Multi-Input Time-Optimal Problems: Application to Controlled Mechanical Systems

  • M. Chyba
  • N.E. Leonard
  • E.D. Sontag


This paper deals with the time-optimal control problem for a class of control systems which includes controlled mechanical systems with possible dissipation terms. The Lie algebras associated with such mechanical systems have certain special properties. These properties are explored and used in conjunction with the Pontryagin maximum principle to determine the structure of singular extremals and, in particular, time-optimal trajectories. The theory is illustrated by an application to a time-optimal problem for a class of underwater vehicles.

Controlled mechanical systems time-optimal problem maximum principle singular extremals 


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  1. 1.
    B. Bonnard and I. Kupka, Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singuli'eres dans le probléme du temps minimal. Forum Math. (1993), 111-159.Google Scholar
  2. 2.
    M. Chyba, N. E. Leonard, and E. D. Sontag, Time-optimal control for underwater vehicles. In: Lagrangian and Hamiltonian Methods for Nonlinear Control (N. E. Leonard and R. Ortega (Eds.)), 2000, 117-122.Google Scholar
  3. 3.
    M. Chyba, N. E. Leonard, and E. D. Sontag, Optimality for underwater vehicles. Proc. IEEE Conf. Decision and Control (2001) 4204-4209.Google Scholar
  4. 4.
    E. B. Lee and L. Markus, Foundations of optimal control theory. Wiley, 1967.Google Scholar
  5. 5.
    N. E. Leonard, Stability of a bottom-heavy underwater vehicle. Automatica 33 (1997), 331-346.Google Scholar
  6. 6.
    A. D. Lewis, The geometry of the maximum principle for affine connection control systems. Preprint, 2000.Google Scholar
  7. 7.
    L. S. Pontryagin, B. Boltyanski, R. Gamkrelidze, and E. Mishchenko, The mathematical theory of optimal processes. Interscience, New York, 1962.Google Scholar
  8. 8.
    H. Schaettler, The local structure of time-optimal trajectories in dimension 3 under generic conditions. SIAM J. Control Optim. 26 (1988), No. 4, 899-918Google Scholar
  9. 9.
    E. D. Sontag, Remarks on the time-optimal control of a class of Hamiltonian systems. In: Proc. IEEE Conf. Decision and Control (1989), 317-221.Google Scholar
  10. 10.
    E. D. Sontag, Mathematical control theory: Deterministic finite dimensional systems. (Second edition.) Springer-Verlag, New York, 1998.Google Scholar
  11. 11.
    E. D. Sontag and H.J. Sussmann, Time-optimal control of manipulators. In: Proc. IEEE Int. Conf. Robotics and Automat. (1986), 1692-1697.Google Scholar
  12. 12.
    H. J. Sussmann, The Markov-Dubins problem with angular acceleration control. In: Proc. IEEE Conf. Decision and Control (1997), 2639-2643.Google Scholar
  13. 13.
    H. J. Sussmann, The maximum principle of optimal control theory. In: Mathematical Control Theory. (J. Baillieul and J.C. Willems (Eds.)), Springer-Verlag, 140-198, 1998.Google Scholar
  14. 14.
    H. J. Sussmann and G. Q. Tang, Shortest paths for the Reeds-Shepp car: a worked out example of the use of geometric techniques in nonlinear optimal control. (to appear in SIAM J. Control Optim.).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • M. Chyba
    • 1
  • N.E. Leonard
    • 2
  • E.D. Sontag
    • 3
  1. 1.Dept. of Mathematics, 379 Applied Sciences BuildingUniversity of Santa Cruz
  2. 2.Dept. of Mechanical and Aerospace Eng.Princeton UniversityPrinceton
  3. 3.Dept. of Mathematics, Hill CenterRutgers UniversityPiscataway

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