Ukrainian Mathematical Journal

, Volume 60, Issue 2, pp 199–210 | Cite as

Localization of the limit set of trajectories of the Euler-Bernoulli equation with control

  • A. L. Zuev


We investigate a differential equation in a Hilbert space that describes vibrations of the Euler-Bernoulli elastic beam with feedback control. The relative compactness of positive semitrajectories of the considered equation is proved. Constructing a Lyapunov functional in explicit form and using the invariance principle, we obtain representations of limit sets.


Feedback Control Invariance Principle Elastic Beam BERNOULLI Equation Abstract Cauchy Problem 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. Z. Zgurovskii and V. S. Mel'nik, Nonlinear Analysis and Control by Finite-Dimensional Systems [in Russian], Naukova Dumka, Kiev (1999).Google Scholar
  2. 2.
    V. I. Korobov and G. M. Sklyar, “On the problem of strong stabilization of contracting systems in Hilbert spaces,” Differents. Uravn., 20, No. 11, 1862–1869 (1984).MathSciNetGoogle Scholar
  3. 3.
    V. Barbu, Analysis and Control of Nonlinear Infinite-Dimensional Systems, Academic Press, San Diego (1992).Google Scholar
  4. 4.
    R. F. Curtain and H. Zwart, An Introduction to Infinite-Dimensional Linear Systems Theory, Springer, New York (1995).zbMATHGoogle Scholar
  5. 5.
    Z.-H. Luo, B.-Z. Guo, and O. Morgul, Stability and Stabilization of Infinite-Dimensional Systems with Applications, Springer, London (1999).zbMATHGoogle Scholar
  6. 6.
    J. Oostveen, Strongly Stabilizable Distributed Systems, SIAM, Philadelphia (2000).zbMATHGoogle Scholar
  7. 7.
    A. A. Shestakov, Generalized Direct Lyapunov Method for Systems with Distributed Parameters [in Russian], Nauka, Moscow (1990).zbMATHGoogle Scholar
  8. 8.
    A. L. Zuyev, “Partial asymptotic stabilization of nonlinear distributed parameter systems,” Automatica, 41, No. 1, 1–10 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    J. E. Lagnese and G. Leugering, “Controllability of thin elastic beams and plates,” in: W. S. Levine (editor), Control Handbook, CRS Press, Boca Raton (1996), pp. 1139–1156.Google Scholar
  10. 10.
    A. L. Zuev, “Simulation of a space elastic manipulator with telescopic motion of chains,” Tr. Inst. Prikl. Mat. Mekh. Nats. Akad. Nauk Ukr., 10, 51–58 (2005).MathSciNetGoogle Scholar
  11. 11.
    A. L. Zuev, “Stabilization of the spatial oscillations of an elastic system model,” Visn. Kyiv Univ., Ser. Fiz.-Mat. Nauk., No. 3, 33–38 (2004) (arXiv: 0707.2209v1).Google Scholar
  12. 12.
    J. P. LaSalle, “Stability theory and invariance principles,” in: Proceedings of the International Symposium on Dynamical Systems (Providence, 1974), Vol. 1, Academic Press, New York (1976), pp. 211–222.Google Scholar
  13. 13.
    A. L. Zuev, “Partial asymptotic stability of abstract differential equations,” Ukr. Mat. Zh., 58, No. 5, 629–637 (2006).zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    A. L. Zuev, “On relative compactness of trajectories of differential equations in a Banach space,” Dopov. Akad. Nauk Ukr., No. 2, 7–12 (2007).Google Scholar
  15. 15.
    S. G. Krein (editor), Functional Analysis [in Russian], Nauka, Moscow (1972).Google Scholar
  16. 16.
    C. M. Dafermos and M. Slemrod, “Asymptotic behavior of nonlinear contraction semigroups,” J. Funct. Anal., 13, 97–106 (1973).zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  • A. L. Zuev
    • 1
  1. 1.Institute of Applied Mathematics and MechanicsUkrainian National Academy of SciencesDonetskUkraine

Personalised recommendations