Estimation of the Reachable Set for the Problem of Vibrating Kirchhoff Plate
- 43 Downloads
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.
KeywordsAlgebraic Number Ukrainian National Academy Approximate Controllability Liouville Theorem Kirchhoff Plate
Unable to display preview. Download preview PDF.
- 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).Google Scholar
- 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.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
- 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.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
- 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.Google Scholar