Ukrainian Mathematical Journal

, Volume 66, Issue 11, pp 1639–1653 | Cite as

Estimation of the Reachable Set for the Problem of Vibrating Kirchhoff Plate

  • A. L. Zuev
  • Yu. V. Novikova
Article

We consider a dynamical system with distributed parameters for the description of controlled vibrations of a Kirchhoff plate without polar moment of inertia. A class of optimal controls corresponding to finite-dimensional approximations is used to study the reachable set. Analytic estimates for the norm of these control functions are obtained depending on the boundary conditions. These estimates are used to study the reachable set for the infinite-dimensional system. For a model with incommensurable frequencies, an estimate of the reachable set is obtained under the condition of power decay of the amplitudes o generalized coordinates.

Keywords

Algebraic Number Ukrainian National Academy Approximate Controllability Liouville Theorem Kirchhoff Plate 
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.

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References

  1. 1.
    M. K. Nabiullin, Stationary Motions and Stability of Elastic Satellites [in Russian], Nauka, Sibirskoe Otdelenie, Novosibirsk (1990).Google Scholar
  2. 2.
    G. L. Degtyarev and T. K. Sirazetdinov, Theoretical Foundations of the Optimal Control of Elastic Spacecrafts [in Russian], Mashinostroenie, Moscow (1986).Google Scholar
  3. 3.
    P. A. Zhilin, “On the Poisson and Kirchhoff theories of plates from the viewpoint of the contemporary theory of plates,” Izv. Ros. Akad. Nauk, Mekh. Tverd. Tela, No. 3, 48–64 (1992).Google Scholar
  4. 4.
    J. E. Lagnese and G. Leugering, “Controllability of thin elastic beams and plates,” in: W. S. Levine (editor), The Control Handbook, CRC Press – IEEE Press, Boca Raton (1996), pp. 1139–1156.Google Scholar
  5. 5.
    A. Zuyev, “Approximate controllability of a rotating Kirchhoff plate model,” in: Proce. of the 49 th IEEE Conference on Decision and Control, Atlanta (USA) (2010), pp. 6944–6948.Google Scholar
  6. 6.
    M. Bradley and S. Lenhart, “Bilinear spatial control of the velocity term in a Kirchhoff plate equation,” Electron. J. Different. Equat., 27, 1–15 (2001).MathSciNetGoogle Scholar
  7. 7.
    L. Chen, J. Pan, and G. Cai, “Active control of a flexible cantilever plate with multiple time delays,” Acta Mech. Solida Sinica, 21, 258–266 (2008).Google Scholar
  8. 8.
    V. V. Novyts’kyi, Decomposition and Control in Linear Systems [in Ukrainian], Proc. of the Institute of Mathematics, Ukrainian National Academy of Sciences, Kyiv (1995).Google Scholar
  9. 9.
    B. Jacob and J. R. Partington, “On controllability of diagonal systems with one-dimensional input space,” Syst. Contr. Lett., 55, 321–328 (2006).MathSciNetCrossRefGoogle Scholar
  10. 10.
    A. L. Zuev and Yu. V. Novikova, “Small vibrations of the Kirchhoff plate with two-dimensional control,” Mekh. Tverd. Tela, Issue 41, 187–198 (2011).Google Scholar
  11. 11.
    A. L. Zuev and Yu. V. Novikova, “Optimal control over the model of Kirchhoff plate,” Mekh. Tverd. Tela, Issue 42, 163–176 (2012).Google Scholar
  12. 12.
    A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983).CrossRefGoogle Scholar
  13. 13.
    H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge University Press, Cambridge (1999).CrossRefGoogle Scholar
  14. 14.
    N. Levan and L. Rigby, “Strong stabilizability of linear contractive control systems on Hilbert space,” SIAM J. Contr. Optimiz., 17, 23–35 (1979).MathSciNetCrossRefGoogle Scholar
  15. 15.
    A. A. Bukhshtab, Number Theory [in Russian], Prosveshchenie, Moscow (1966).Google Scholar
  16. 16.
    A. Zuyev, “Approximate controllability and spillover analysis of a class of distributed parameter systems,” in: Proc. of the 48th IEEE Conference on Decision and Control and the 28th Chinese Control Conference, Shanghai, China (2009), pp. 3270–3275.Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

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

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