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

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Ukrainian Mathematical Journal Aims and scope

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.

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References

  1. M. K. Nabiullin, Stationary Motions and Stability of Elastic Satellites [in Russian], Nauka, Sibirskoe Otdelenie, Novosibirsk (1990).

    Google Scholar 

  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. 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).

  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. 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.

  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).

    MathSciNet  Google Scholar 

  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. 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. B. Jacob and J. R. Partington, “On controllability of diagonal systems with one-dimensional input space,” Syst. Contr. Lett., 55, 321–328 (2006).

    Article  MathSciNet  Google Scholar 

  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).

  11. A. L. Zuev and Yu. V. Novikova, “Optimal control over the model of Kirchhoff plate,” Mekh. Tverd. Tela, Issue 42, 163–176 (2012).

  12. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York (1983).

    Book  Google Scholar 

  13. H. O. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge University Press, Cambridge (1999).

    Book  Google Scholar 

  14. N. Levan and L. Rigby, “Strong stabilizability of linear contractive control systems on Hilbert space,” SIAM J. Contr. Optimiz., 17, 23–35 (1979).

    Article  MathSciNet  Google Scholar 

  15. A. A. Bukhshtab, Number Theory [in Russian], Prosveshchenie, Moscow (1966).

    Google Scholar 

  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.

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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 11, pp. 1463–1476, November, 2014.

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Zuev, A.L., Novikova, Y.V. Estimation of the Reachable Set for the Problem of Vibrating Kirchhoff Plate. Ukr Math J 66, 1639–1653 (2015). https://doi.org/10.1007/s11253-015-1041-0

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  • DOI: https://doi.org/10.1007/s11253-015-1041-0

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