Advertisement

Automation and Remote Control

, Volume 74, Issue 3, pp 385–400 | Cite as

Petrov invariants of Hamiltonian systems with a control parameter

  • A. G. Kushner
  • V. V. Lychagin
Topical Issue

Abstract

The problem is considered of the classification of Hamiltonian systems with a scalar control parameter relative to feedback transformations. Differential invariants of these systems, which are called Petrov invariants, are set up. The dimensions of algebras of these invariants are found. The conditions of global equivalence of regular Hamiltonian systems with the control parameter are found in terms of Petrov invariants.

Keywords

Control Parameter Remote Control Hamiltonian System Symplectic Structure Determine Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agrachev, A.A. and Sachkov, Yu. L., Geometricheskaya teoriya upravleniya (Geometric Control Theory), Moscow: Fizmatlit, 2006.Google Scholar
  2. 2.
    Kim, D.P., Theoriya avtomaticheskogo upravleniya, tom 2: Mnogomernye nelineinye, optimal’nye i adaptivnye sistemy (Automatic Control Theory, vol. 2: Multivariate, Nonlinear, Optimal and Adaptive Systems), Moscow: Fizmatlit, 2004.Google Scholar
  3. 3.
    Gardner, R.B. and Shadwick, W.F., Feedback Equivalence for General Control Systems, Syst. Control Lett., 1990, vol. 15, pp. 15–23.MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Respondek, W., Feedback Classification of Nonlinear Control Systems in R 2 and R 3, in Geometry of Feedback and Optimal Control, Jakubczyk, B. and Respondek, W, Eds., New York: Marcel Dekker, 1997, pp. 347–382.Google Scholar
  5. 5.
    Kushner, A.G., Lychagin, V.V., and Rubtsov, V.N., Contact Geometry and Nonlinear Differential Equations, in Encyclopedia Math. Its Appl., Cambridge: Cambridge Univ. Press, 2007, vol. 101.Google Scholar
  6. 6.
    Agrachev, A. and Zelenko, I., On Feedback Classification of Control-Affine Systems with One- and Two-Dimensional Inputs, SIAM J. Control Optim., 2007, vol. 46, no. 4, pp. 1431–1460.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Jakubczyk, B., Equivalence and Invariants of Nonlinear Control Systems, in Nonlinear Controllability and Optimal Control, Sussmann, H.J., Ed., New York: Marcel Dekker, 1990.Google Scholar
  8. 8.
    Petrov, B.N., Izbrannye trudy, tom 1: Teoriya avtomaticheskogo upravleniya (Selected Works, vol. 1: Automatic Control Theory), Moscow: Nauka, 1983.Google Scholar
  9. 9.
    Kushner, A.G. and Lychagin, V.V., Petrov Invariants for 1-D Control Hamiltonian Systems, Global Stochast. Anal., 2012, vol. 2, no. 1, pp. 89–100.Google Scholar
  10. 10.
    Kushner, A.G. and Lychagin, V.V., Petrov Invariance and Identification of Hamiltonian Systems with a Control Parameter, Proc. IX Int. Conf. “Syst. Identificat. Control Prob.,” SICPROT’12, Moscow, 2012, pp. 75–81.Google Scholar
  11. 11.
    Lychagin, V.V., Feedback Differential Invariants, Acta Appl. Math., 2010, vol. 109, no. 1, pp. 211–222.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Vinogradov, A.M., Krasil’shchik, I.S., and Lychagin, V.V., Vvedenie v geometriyu nelineinykh differentsial’nykh uravnenii (Introduction to Geometry of Nonlinear Differential Equations), Moscow: Nauka, 1986.Google Scholar
  13. 13.
    Cartan, E., Les sous-groupes continus de transformations, Ann. Ecole Normale, 1908, vol. 25, pp. 719–856.MathSciNetGoogle Scholar
  14. 14.
    Krasil’shchik, I.S., Lychagin, V.V., and Vinogradov, A.M., Geometry of Jet Spaces and Nonlinear Partial Differential Equations (Advanced Studies in Contemporary Math., vol. 1), New York: Gordon and Breach, 1986.Google Scholar
  15. 15.
    Kruglikov, B.S. and Lychagin, V.V., Global Lie-Tresse Theorem, arXiv:1111.5480v1 [math.DG], 23 Nov. 2011.Google Scholar
  16. 16.
    Lie, S., Ueber Einige Partielle Differential-Gleichungen Zweiter Ordnung, Math. Ann., 1872, vol. 5, pp. 209–256.MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lie, S., Begrundung Einer Invarianten-Theorie der Beruhrungs-Transformationen, Math. Ann., 1874, vol. 8, pp. 215–303.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lychagin, V.V., Feedback Equivalence of 1-Dimensional Control Systems of the 1st Order, Geometry, Topology and Their Applications, Proc. Inst. Math. NAS of Ukraine, 2009, vol. 6, no. 2, pp. 288–302.MathSciNetzbMATHGoogle Scholar
  19. 19.
    Chernous’ko, F.L., Akulenko, L.D., and Sokolov, B.N., Upravlenie kolebaniyami (Vibration Control), Moscow: Nauka, 1980.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  • A. G. Kushner
    • 1
    • 2
  • V. V. Lychagin
    • 1
    • 3
  1. 1.Trapeznikov Institute of Control SciencesRussian Academy of SciencesMoscowRussia
  2. 2.Moscow State UniversityMoscowRussia
  3. 3.University of TromsøTromsøNorway

Personalised recommendations