Advertisement

Journal of Mathematical Sciences

, Volume 160, Issue 2, pp 139–196 | Cite as

Extremals with accumulation of switchings in an infinite-dimensional space

  • B. F. Borisov
  • M. I. Zelikin
  • L. A. Manita
Article

Abstract

This paper studies a class of optimal control problems and the Hamiltonian systems generated by them in the space l 2. The authors prove the existence of extremals with countably many switchings on a finite interval of time that compose a bundle with piecewise fibers over the base of singular extremals of condimension 4.

Keywords

Hamiltonian System Optimal Control Problem Invariant Manifold Timoshenko Beam Optimal Trajectory 
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.
    N. A. Alfutov, Foundations of Stability Calculation of Elastic Systems [in Russian], Mashinostroenie, Moscow (1991).Google Scholar
  2. 2.
    V. I. Bogachev, Foundations of Measure Theory [in Russian], NITs “Regulyarnaya i Khaoticheskaya Dinamika,” Moscow–Izhevsk (2003).Google Scholar
  3. 3.
    A. V. Fursikov, Optimal Control of Distributed Parameter Systems. Theory and Applications [in Russian], Nauchnaya Kniga, Novesibirsk (1999).Google Scholar
  4. 4.
    M. Gugat, “Controllability of a slowly rotating Timoshenko beam,” ESAIM: Control, Optimiztion and Calculus of Variations, 6, 333–360 (2001).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    M. W. Hirsh, C. C. Pugh, and M. Shub, Invariant manifolds, Springer–Verlag, Berlin, Heidelberg, and New Yok (1977).Google Scholar
  6. 6.
    T. Kato, Perturbation Theory Of Linear Operators [Russian translation], Mir, Moscow (1972).Google Scholar
  7. 7.
    W. Krabs and G. M. Sklyar, “On the controllability of a slowly rotating Timoshenko beam,” Z. Anal. Anwend., 18, No. 2, 437–448 (1999).MATHMathSciNetGoogle Scholar
  8. 8.
    W. Krabs and G. M. Sklyar, “On the stabilizability of a slowly rotating Timoshenko beam,” Z. Anal. Anwend., 19, No. 1, 131–145 (2000).MATHMathSciNetGoogle Scholar
  9. 9.
    S. G. Krein, Linear Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1967).Google Scholar
  10. 10.
    V. P. Mikhailov, Partial Diffrential Equations [in Russian], Nauka, Moscow (1976).Google Scholar
  11. 11.
    L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, Mathematical Theory of Optimal Processes [in Russian], Nauka, Moscow (1976).Google Scholar
  12. 12.
    S. Taylor and S. Yau, “Boundary control of a rotating Timoshenko beam,” ANZIAM J., Ser. E., 44, No. 1, 143–184 (2003).Google Scholar
  13. 13.
    S. P. Timoshenko, A Course of Elasticity Theory [in Russian], Naukova Dumka, Kiev (1972).Google Scholar
  14. 14.
    S. P. Timoshenko, D. H. Young, and W. Weaver, Vibration Problems in Engineering, Wiley, New York (1990).Google Scholar
  15. 15.
    H.Weyl, “On uniform distribution of numbers modulo one,” In: H.Weyl, Selected Works [Russian translation], Nauka, Moscow (1984).Google Scholar
  16. 16.
    M. I. Zelikin and V. F. Borisov, “More and more frequent switching regimes in optimal control problems,” Tr. Mat. Inst. USSR Academy of Sciences, 197, 85–167 (1991).MATHGoogle Scholar
  17. 17.
    M. I. Zelikin and V. F. Borisov, Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering, Birkhäuser, Boston, New tork (1994).MATHGoogle Scholar
  18. 18.
    M. I. Zelikin and L. A. Manita, “Accumulation of switchings in distributed parameters problems,” Sovremennaya Matematika, Fundamental’nye Napravleniya, 19, 78–113 (2006).Google Scholar
  19. 19.
    M. I. Zelikin and L. A. Manita, “Optimal regimes with more and more frequent switchings in the Timoshenko beam control problem,” Prikl. Mat. Mekh., 70, No. 2, 295–304 (2006).MATHMathSciNetGoogle Scholar
  20. 20.
    M. I. Zelikin and L. A. Manita, “Accumulation of switchings in distributed parameters problems,” Sovremennaya Matematika. Fundamental’nye Napravleniya, 19, 78–113 (2006).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Moscow State University of Tourism and ServicePushkinoRussia

Personalised recommendations