Abstract
In this paper, we consider a vibrating beam with one segment made of viscoelastic material of a Kelvin–Voigt (shorted as K–V) type and other parts made of elastic material by means of the Timoshenko model. We have deduced mathematical equations modelling its vibration and studied the stability of the semigroup associated with the equation system. We obtain the exponential stability under certain hypotheses of the smoothness and structural condition of the coefficients of the system, and obtain the strong asymptotic stability under weaker hypotheses of the coefficients.
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Chen, G., Fulling, S. A., Narcowich, F. J., Sun, S.: Exponential decay of energy of evolution equations with locally distributed damping. SIAM J. Appl. Math., 51(1), 266–301 (1991)
Liu, K.: Locally distributed control and damping for the conservative system. SIAM J. Cont. Optim., 35(5), 1574–1590 (1997)
Zuazua, E.: Exponential decay for the semilinear wave equation with localized damping. Comm. Part. Diff. Eq., 15, 205–235 (1990)
Liu, K., Liu, Z.: Exponential decay of energy of vibrating strings with local viscoelasticity. Z. Angew Math. Phys., 53, 265–280 (2002)
Liu, K., Liu, Z.: Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin– Voigt damping. SIAM J. Control Optim., 36, 1086–1098 (1998)
Bardos, C., Lebeau, G., Rauch, J.: Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary. SIAM J. Cont. Optim., 30(5), 1024–1065 (1992)
Rivera, J. E. M., Oquendo, H. P.: The transmission problem of viscoelastic waves. Acta Appl. Math., 62, 1–21 (2000)
Rivera, J. E. M., Salvatierra, A. P.: Asymptotic behaviour of the energy in partially viscoelastic materials. Quaterly of Appl. Math., LIX(3), 557–578 (2001)
Timoshenko: Vibration Problem in Engineering, Von Nostrand, New York, 1955
Washizu, K.: Variational Method in Elasticity and Plasticity, 2nd ed., Pergamon Press, Elinsford, New York, 1975
Christensen, R. M.: Theory of Viscoelasticity, 2nd ed., Academic Press Inc., New York, 1982
Adams, A. R.: Sobolev Space. Acadamic Press, New York, 1975
Kato, T.: Perturbation theory for linear operators, Springer–Verlag, New York, 1980
Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983
Chen, S., Liu, K., Liu, Z.: Spectrum and stability for elastic systems with global or local Kelvin–Voigt damping. SIAM J. Appl. Math., 59(2), 651–668 (1998)
Hartman, P.: Ordinary differential equations, 2nd ed., Birkh¨auser, Boston, Basel, Stuttgart, 1982
Huang, F.: Strong asymptotic stability of linear dynamical systems in Banach spaces. J. Diff. Eqs., 104, 307–324 (1995)
Huang, F.: Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces. Ann. of Diff. Eqs., 1(1), 43–56 (1985)
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This project is supported partially by the National Natural Science Foundation of China Grants 69874034 and 10271111
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Zhao, H.L., Liu, K.S. & Zhang, C.G. Stability for the Timoshenko Beam System with Local Kelvin–Voigt Damping. Acta Math Sinica 21, 655–666 (2005). https://doi.org/10.1007/s10114-003-0256-4
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DOI: https://doi.org/10.1007/s10114-003-0256-4