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Feedback stabilization with one simultaneous control for systems of parabolic equations
Weak stability of a laminated beam
1. | College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China |
2. | Department of Mathematics and Statistics, University of Minnesota, Duluth, MN 55812, USA |
3. | College of Information Science and Technology, Donghua University, School of Mathematics, Physics and Statistics, Shanghai University of Engineering Science, Shanghai 201620, China |
In this paper, we consider the stability of a laminated beam equation, derived by Liu, Trogdon, and Yong [
References:
[1] |
A. A. Allen and S. W. Hansen,
Analyticity of a multilayer Mead-Markus plate, Nonliear Anal., 71 (2009), e1835-e1842.
doi: 10.1016/j.na.2009.02.063. |
[2] |
A. A. Allen and S. W. Hansen,
Analyticity and optimal damping for a multilayer Mead-Markus sandwich beam, Discrete Contin. Dyn. Syst. Ser. B., 14 (2010), 1279-1292.
doi: 10.3934/dcdsb.2010.14.1279. |
[3] |
A. Borichev and Y. Tomilov,
Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.
doi: 10.1007/s00208-009-0439-0. |
[4] |
S. W. Hansen and Z. Liu, Analyticity of semigroup associated with a laminated composite
beam, Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998), Kluwer
Acad. Publ., Boston, MA, 1999, 47–54. |
[5] |
S. W. Hansen and R. Spies,
Structural damping in a laminated beam due to interfacial slip, J. Sound and Vibration, 204 (1997), 183-202.
doi: 10.1006/jsvi.1996.0913. |
[6] |
Z. Liu, S. A. Trogdon and J. Yong,
Modeling and analysis of a laminated beam, Math. Comput. Modeling, 30 (1999), 149-167.
doi: 10.1016/S0895-7177(99)00122-3. |
[7] |
Z. Liu and S. Zheng,
Semigroup Associated with Dissipative System, Res. Notes Math., Vol 394, Chapman & Hall/CRC, Boca Raton, 1999. |
[8] |
D. J. Mead and S. Markus,
The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions, J. Sound Vibr.(2), 10 (1969), 163-175.
doi: 10.1016/0022-460X(69)90193-X. |
[9] |
A. Özkan Özer and S. W. Hansen,
Uniform stabilization of a multilayer Rao-Nakra sandwich beam, Evol. Equ. Control Theorey, 2 (2013), 695-710.
doi: 10.3934/eect.2013.2.695. |
[10] |
Y. V. K. S Rao and B. C. Nakra,
Vibrations of unsymmetrical sanwich beams and plates with viscoelastic cores, J. Sound Vibr.(3), 34 (1974), 309-326.
|
[11] |
C. A. Raposo,
Exponential stability of a structure with interfacial slip and frictional damping, Applied Math. Letter, 53 (2016), 85-91.
doi: 10.1016/j.aml.2015.10.005. |
[12] |
J. M. Wang, G. Q. Xu and S. P. Yung,
Stabilization of laminated beams with structural damping by boundary feedback controls, SIAM Control Optim., 44 (2005), 1575-1597.
doi: 10.1137/040610003. |
[13] |
M. J. Yan and E. H. Dowell,
Governing equations for vibratory constrained-layer damping sandwich plates and beams, J. Appl. Mech.(4), 39 (1972), 1041-1046.
|
show all references
References:
[1] |
A. A. Allen and S. W. Hansen,
Analyticity of a multilayer Mead-Markus plate, Nonliear Anal., 71 (2009), e1835-e1842.
doi: 10.1016/j.na.2009.02.063. |
[2] |
A. A. Allen and S. W. Hansen,
Analyticity and optimal damping for a multilayer Mead-Markus sandwich beam, Discrete Contin. Dyn. Syst. Ser. B., 14 (2010), 1279-1292.
doi: 10.3934/dcdsb.2010.14.1279. |
[3] |
A. Borichev and Y. Tomilov,
Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.
doi: 10.1007/s00208-009-0439-0. |
[4] |
S. W. Hansen and Z. Liu, Analyticity of semigroup associated with a laminated composite
beam, Control of Distributed Parameter and Stochastic Systems (Hangzhou, 1998), Kluwer
Acad. Publ., Boston, MA, 1999, 47–54. |
[5] |
S. W. Hansen and R. Spies,
Structural damping in a laminated beam due to interfacial slip, J. Sound and Vibration, 204 (1997), 183-202.
doi: 10.1006/jsvi.1996.0913. |
[6] |
Z. Liu, S. A. Trogdon and J. Yong,
Modeling and analysis of a laminated beam, Math. Comput. Modeling, 30 (1999), 149-167.
doi: 10.1016/S0895-7177(99)00122-3. |
[7] |
Z. Liu and S. Zheng,
Semigroup Associated with Dissipative System, Res. Notes Math., Vol 394, Chapman & Hall/CRC, Boca Raton, 1999. |
[8] |
D. J. Mead and S. Markus,
The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions, J. Sound Vibr.(2), 10 (1969), 163-175.
doi: 10.1016/0022-460X(69)90193-X. |
[9] |
A. Özkan Özer and S. W. Hansen,
Uniform stabilization of a multilayer Rao-Nakra sandwich beam, Evol. Equ. Control Theorey, 2 (2013), 695-710.
doi: 10.3934/eect.2013.2.695. |
[10] |
Y. V. K. S Rao and B. C. Nakra,
Vibrations of unsymmetrical sanwich beams and plates with viscoelastic cores, J. Sound Vibr.(3), 34 (1974), 309-326.
|
[11] |
C. A. Raposo,
Exponential stability of a structure with interfacial slip and frictional damping, Applied Math. Letter, 53 (2016), 85-91.
doi: 10.1016/j.aml.2015.10.005. |
[12] |
J. M. Wang, G. Q. Xu and S. P. Yung,
Stabilization of laminated beams with structural damping by boundary feedback controls, SIAM Control Optim., 44 (2005), 1575-1597.
doi: 10.1137/040610003. |
[13] |
M. J. Yan and E. H. Dowell,
Governing equations for vibratory constrained-layer damping sandwich plates and beams, J. Appl. Mech.(4), 39 (1972), 1041-1046.
|
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