American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Optimal control problems for some ordinary differential equations with behavior of blowup or quenching
  • MCRF Home
  • This Issue
  • Next Article
    Feedback stabilization with one simultaneous control for systems of parabolic equations
September & December  2018, 8(3&4): 789-808. doi: 10.3934/mcrf.2018035

Weak stability of a laminated beam

Yanfang Li 1, , Zhuangyi Liu 2, and  Yang Wang 3,,

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

* Corresponding authorr: Yang Wan

Received  October 2017 Revised  May 2018 Published  September 2018

In this paper, we consider the stability of a laminated beam equation, derived by Liu, Trogdon, and Yong [6], subject to viscous or Kelvin-Voigt damping. The model is a coupled system of two wave equations and one Euler-Bernoulli beam equation, which describes the longitudinal motion of the top and bottom layers of the beam and the transverse motion of the beam. We first show that the system is unstable if one damping is only imposed on the beam equation. On the other hand, it is easy to see that the system is exponentially stable if direct damping are imposed on all three equations. Hence, we investigate the system stability when two of the three equations are directly damped. There are a total of seven cases from the combination of damping locations and types. Polynomial stability of different orders and their optimality are proved. Several interesting properties are revealed.

Keywords: Laminated beam, polynomial stability, exponential stability, optimal decay rate, semigroup.
Mathematics Subject Classification: Primary: 35B35, 47D03; Secondary: 93D05.
Citation: Yanfang Li, Zhuangyi Liu, Yang Wang. Weak stability of a laminated beam. Mathematical Control & Related Fields, 2018, 8 (3&4) : 789-808. doi: 10.3934/mcrf.2018035
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Z. Liu and S. Zheng, Semigroup Associated with Dissipative System, Res. Notes Math., Vol 394, Chapman & Hall/CRC, Boca Raton, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Z. Liu and S. Zheng, Semigroup Associated with Dissipative System, Res. Notes Math., Vol 394, Chapman & Hall/CRC, Boca Raton, 1999.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[1]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete & Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721
[2]

Qiong Zhang. Exponential stability of a joint-leg-beam system with memory damping. Mathematical Control & Related Fields, 2015, 5 (2) : 321-333. doi: 10.3934/mcrf.2015.5.321
[3]

Manoel J. Dos Santos, João C. P. Fortes, Marcos L. Cardoso. Exponential stability for a piezoelectric beam with a magnetic effect and past history. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021283
[4]

Maya Bassam, Denis Mercier, Ali Wehbe. Optimal energy decay rate of Rayleigh beam equation with only one boundary control force. Evolution Equations & Control Theory, 2015, 4 (1) : 21-38. doi: 10.3934/eect.2015.4.21
[5]

Farah Abdallah, Denis Mercier, Serge Nicaise. Spectral analysis and exponential or polynomial stability of some indefinite sign damped problems. Evolution Equations & Control Theory, 2013, 2 (1) : 1-33. doi: 10.3934/eect.2013.2.1
[6]

Alaa Hayek, Serge Nicaise, Zaynab Salloum, Ali Wehbe. Exponential and polynomial stability results for networks of elastic and thermo-elastic rods. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021142
[7]

Yi Cheng, Zhihui Dong, Donal O' Regan. Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021230
[8]

Tian Zhang, Chuanhou Gao. Stability with general decay rate of hybrid neutral stochastic pantograph differential equations driven by Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021204
[9]

Kai Liu, Zhi Li. Global attracting set, exponential decay and stability in distribution of neutral SPDEs driven by additive $\alpha$-stable processes. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3551-3573. doi: 10.3934/dcdsb.2016110
[10]

Xun-Yang Wang, Khalid Hattaf, Hai-Feng Huo, Hong Xiang. Stability analysis of a delayed social epidemics model with general contact rate and its optimal control. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1267-1285. doi: 10.3934/jimo.2016.12.1267
[11]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219
[12]

Augusto Visintin. Structural stability of rate-independent nonpotential flows. Discrete & Continuous Dynamical Systems - S, 2013, 6 (1) : 257-275. doi: 10.3934/dcdss.2013.6.257
[13]

Takayuki Niimura. Attractors and their stability with respect to rotational inertia for nonlocal extensible beam equations. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2561-2591. doi: 10.3934/dcds.2020141
[14]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424
[15]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. Stability analysis of inhomogeneous equilibrium for axially and transversely excited nonlinear beam. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1447-1462. doi: 10.3934/cpaa.2011.10.1447
[16]

Leif Arkeryd, Raffaele Esposito, Rossana Marra, Anne Nouri. Exponential stability of the solutions to the Boltzmann equation for the Benard problem. Kinetic & Related Models, 2012, 5 (4) : 673-695. doi: 10.3934/krm.2012.5.673
[17]

Sigurdur Freyr Hafstein. A constructive converse Lyapunov theorem on exponential stability. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 657-678. doi: 10.3934/dcds.2004.10.657
[18]

Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations & Control Theory, 2016, 5 (2) : 235-250. doi: 10.3934/eect.2016003
[19]

Litan Yan, Wenyi Pei, Zhenzhong Zhang. Exponential stability of SDEs driven by fBm with Markovian switching. Discrete & Continuous Dynamical Systems, 2019, 39 (11) : 6467-6483. doi: 10.3934/dcds.2019280
[20]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete & Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773

2020 Impact Factor: 1.284

Tools

Metrics

  • PDF downloads (266)
  • HTML views (373)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy