Abstract
This paper studies the stabilization problem of uniform Euler-Bernoulli beam with a nonlinear locally distributed feedback control. By virtue of nonlinear semigroup theory, energy-perturbed approach and polynomial multiplier skill, the authors show that, corresponding to the different values of the parameters involved in the nonlinear locally distributed feedback control, the energy of the beam under the proposed feedback decays exponentially or in negative power of time t as t → ∞.
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References
G. Chen, et al., Modelling, stabilization and control of serially connected beam, SIAM. J. Control Optim., 1987, 25: 526–546.
F. L. Huang, Characteristic conditions for exponential stability of linear dynamical system in Hilbert space, Ann. Diff. Eqs., 1985, 1(1): 43–53.
G. Chen, et al., The Euler-Bernoulli Equation with Boundary Energy Dissipation, tin Operator Methods for Optimal Control Problems, S. J. Lee ed., Marcel Dekker, New York, 1988.
D. X. Feng and W. T. Zhang, Nonlinear feedback control of timoshenko beam, Science in China, Series A, 1995, 38: 918–927.
Q. X. Yan, Bounary stabilization of Timoshenko beam, Systems Science and Mathematical Sciences, 2000, 13(4): 376–384.
Q. X. Yan, Y. Xie, and D. X. Feng, Nonlinear dissiptive boundary feedback stabilization of Euler-Bernoulli beam, Acta Automatica Sinica, 2003, 29(1): 1–7.
Q. X. Yan, H. C. Zou, and D. X. Feng, Nonlinear boundary stabilization of nonuniform Timoshenko beam, Acta Mathematicae Applicatae Sinica, English Series, 2003, 19(2): 239–246.
S. H. Hou and Q. X. Yan, Nonlinear locally distributed feedback stabilization, Journal of Industrial and Management Optimization, 2008, 4(1): 376–384.
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Nordhoff, International Publishing, Bucuresti, România, 1976.
D. H. Shi and D. X. Feng, Exponential decay of Timoshenko beam with locally distributed feedback, Proc. of the 1999 IFAC World Congress, Beijing, Vol. F, 158–162.
D. H. Shi and D. X. Feng, Optimal energy decay rate of Timoshenko beam with distributed damping, to appear in Australian and Newzealand Industrial and Applied Mathematics.
Abdelaziz Soufyane, Stabilisation de la Poutre de Timoshenko, C. R. Acad. Sci. Paris, 1999, 328(1): 731–734.
K. S. Liu and Z. Y. Liu, Exponential decay of energy of the Euler-Bernoulli beam with locally distributed kelvin-voigt damping, SIAM J. Control and Optimization, 1998, 36: 1086–1098.
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This research is supported by the National Science Foundation of China under Grant Nos. 10671166 and 60673101.
This paper was recommended for publication by Editor Dexing FENG.
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Yan, Q., Hou, S. & Zhang, L. Stabilization of Euler-Bernoulli beam with a nonlinear locally distributed feedback control. J Syst Sci Complex 24, 1100–1109 (2011). https://doi.org/10.1007/s11424-011-8360-5
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DOI: https://doi.org/10.1007/s11424-011-8360-5