December  2013, 2(4): 695-710. doi: 10.3934/eect.2013.2.695

Uniform stabilization of a multilayer Rao-Nakra sandwich beam

1. 

Department of Applied Mathematics, University of Waterloo, Waterloo, ON N2L3G1, Canada

2. 

Department of Mathematics, Iowa State University, Ames, IA 50011, United States

Received  October 2012 Revised  April 2013 Published  November 2013

We consider the problem of boundary feedback stabilization of a multilayer Rao-Nakra sandwich beam. We show that the eigenfunctions of the decoupled system form a Riesz basis. This allows us to deduce that the decoupled system is exponentially stable. Since the coupling terms are compact, the exponential stability of the coupled system follows from the strong stability of the coupled system, which is proved using a unique continuation result for the overdetermined homogenous system in the case of zero feedback.
Citation: A. Özkan Özer, Scott W. Hansen. Uniform stabilization of a multilayer Rao-Nakra sandwich beam. Evolution Equations & Control Theory, 2013, 2 (4) : 695-710. doi: 10.3934/eect.2013.2.695
References:
[1]

A. A. Allen, Stability Results for Damped Multilayer Composite Beams and Plates,, Ph.D. thesis, (2009). 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 (4), 14 (2010), 1279. doi: 10.3934/dcdsb.2010.14.1279. Google Scholar

[3]

A. A. Allen and S. W. Hansen, Analyticity of a multilayer Mead-Markus plate,, Nonlinear Analysis (12), 71 (2009). doi: 10.1016/j.na.2009.02.063. Google Scholar

[4]

G. Chen, A note on the boundary stabilization of the wave equation,, SIAM J. Control Optim., 19 (1981), 106. doi: 10.1137/0319008. Google Scholar

[5]

R. H. Fabiano and S. W. Hansen, Modeling and analysis of a three-layer damped sandwich beam,, Discrete Contin. Dyn. Syst., (2001), 143. Google Scholar

[6]

B. Z. Guo, Basis property of a Rayleigh beam with boundary stabilization,, J. Optim. Theory Appl., 112 (2002), 529. doi: 10.1023/A:1017912031840. Google Scholar

[7]

S. W. Hansen, Several related models for multilayer sandwich plates,, Math. Models Methods Appl. Sci., 14 (2004), 1103. doi: 10.1142/S0218202504003568. Google Scholar

[8]

S. W. Hansen and I. Lasiecka, Analyticity, hyperbolicity and uniform stability of semigroups arising in models of composite beams,, Math. Models Meth. Appl. Sci., 10 (2000), 555. doi: 10.1142/S0218202500000306. Google Scholar

[9]

S. W. Hansen and O. Y. Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions,, Math. Control Relat. Fields, 1 (2011), 189. doi: 10.3934/mcrf.2011.1.189. Google Scholar

[10]

S. W. Hansen and O. Y. Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions,, ESAIM Control Optim. Calc. Var., 17 (2011), 1101. doi: 10.1051/cocv/2010040. Google Scholar

[11]

S. W. Hansen and R. Rajaram, Simultaneous boundary control of a Rao-Nakra sandwich beam,, Proc. 44th IEEE Conference on Decision and Control and the European Control Conference, (2005), 3146. doi: 10.1109/CDC.2005.1582645. Google Scholar

[12]

S. W. Hansen and R. Rajaram, Riesz basis property and related results for a Rao-Nakra sandwich beam,, Discrete Contin. Dyn. Syst., (2005), 365. Google Scholar

[13]

S. W. Hansen and R. D. Spies, Structural damping in laminated beams due to interfacial slip,, Journal of Sound and Vibration, 204 (1997), 183. doi: 10.1006/jsvi.1996.0913. Google Scholar

[14]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl., 69 (1990), 33. Google Scholar

[15]

J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163. doi: 10.1016/0022-0396(83)90073-6. Google Scholar

[16]

I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Kirchhoff plates with boundary controls only in $ \Delta w |_\Sigma$,, J. Differential Equations, 93 (1991), 62. doi: 10.1016/0022-0396(91)90022-2. Google Scholar

[17]

I. Lasiecka and R. Triggiani, Uniform exponential decay of wave equations in a bounded region with $L_2(0,\infty; L_2(\Gamma))$-feedback control in the Dirichlet boundary conditions,, J. Differential Equations, 66 (1987), 340. doi: 10.1016/0022-0396(87)90025-8. Google Scholar

[18]

D. J. Mead and S. Markus, The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions,, J. Sound Vibr., 10 (1969), 163. doi: 10.1016/0022-460X(69)90193-X. Google Scholar

[19]

A. Ö. Özer and S. W. Hansen, Exact controllability of a Rayleigh beam with a single boundary control,, Math. Control Signals Systems, 23 (2011), 199. doi: 10.1007/s00498-011-0069-4. Google Scholar

[20]

A. Ö. Özer and S. W. Hansen, Exact boundary controllability results for a multilayer Rao-Nakra sandwich beam,, to appear in SIAM J. Cont. Optim., (). Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[22]

R. Rajaram, Exact boundary controllability result for a Rao-Nakra sandwich beam,, Systems Control Lett., 56 (2007), 558. doi: 10.1016/j.sysconle.2007.03.007. Google Scholar

[23]

B. Rao, A compact perturbation method for the boundary stabilization of the Ragleigh beam equation,, Appl. Math. Optim., 33 (1996), 253. doi: 10.1007/BF01204704. Google Scholar

[24]

Y. V. K. S Rao and B. C. Nakra, Vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores,, J. Sound Vibr., 34 (1974), 309. Google Scholar

[25]

R. Triggiani, On the stabilizability problem in Banach space,, J. Math. Anal. Appl., 52 (1975), 383. doi: 10.1016/0022-247X(75)90067-0. Google Scholar

[26]

R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation,, Proc. Amer. Math. Soc., 105 (1989), 375. doi: 10.1090/S0002-9939-1989-0953013-0. Google Scholar

[27]

J. M. Wang and B. Z. Guo, Analyticity and dynamic behavior of a damped three-layer sandwich beam,, J. Optim. Theory Appl., 137 (2008), 675. doi: 10.1007/s10957-007-9341-7. Google Scholar

[28]

J. M. Wang, G. Q. Xu and S. P. Yung, Exponential stabilization of laminated beams with structural damping and boundary feedback controls,, SIAM J. Cont. Optim., 44 (2005), 1575. doi: 10.1137/040610003. Google Scholar

[29]

J. M. Wang, B. Z. Guo and B. Chentouf, Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach,, ESAIM Control Optim. Calc. Var., 12 (2006), 12. doi: 10.1051/cocv:2005030. Google Scholar

[30]

M. J. Yan and E. H. Dowell, Governing equations for vibratory constrained-layer damping sandwich plates and beams,, J. Appl. Mech., 39 (1972), 1041. Google Scholar

[31]

R. Young, An Introduction to Nonharmonic Fourier Series,, Revised first edition. Academic Press, (2001). Google Scholar

show all references

References:
[1]

A. A. Allen, Stability Results for Damped Multilayer Composite Beams and Plates,, Ph.D. thesis, (2009). 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 (4), 14 (2010), 1279. doi: 10.3934/dcdsb.2010.14.1279. Google Scholar

[3]

A. A. Allen and S. W. Hansen, Analyticity of a multilayer Mead-Markus plate,, Nonlinear Analysis (12), 71 (2009). doi: 10.1016/j.na.2009.02.063. Google Scholar

[4]

G. Chen, A note on the boundary stabilization of the wave equation,, SIAM J. Control Optim., 19 (1981), 106. doi: 10.1137/0319008. Google Scholar

[5]

R. H. Fabiano and S. W. Hansen, Modeling and analysis of a three-layer damped sandwich beam,, Discrete Contin. Dyn. Syst., (2001), 143. Google Scholar

[6]

B. Z. Guo, Basis property of a Rayleigh beam with boundary stabilization,, J. Optim. Theory Appl., 112 (2002), 529. doi: 10.1023/A:1017912031840. Google Scholar

[7]

S. W. Hansen, Several related models for multilayer sandwich plates,, Math. Models Methods Appl. Sci., 14 (2004), 1103. doi: 10.1142/S0218202504003568. Google Scholar

[8]

S. W. Hansen and I. Lasiecka, Analyticity, hyperbolicity and uniform stability of semigroups arising in models of composite beams,, Math. Models Meth. Appl. Sci., 10 (2000), 555. doi: 10.1142/S0218202500000306. Google Scholar

[9]

S. W. Hansen and O. Y. Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions,, Math. Control Relat. Fields, 1 (2011), 189. doi: 10.3934/mcrf.2011.1.189. Google Scholar

[10]

S. W. Hansen and O. Y. Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions,, ESAIM Control Optim. Calc. Var., 17 (2011), 1101. doi: 10.1051/cocv/2010040. Google Scholar

[11]

S. W. Hansen and R. Rajaram, Simultaneous boundary control of a Rao-Nakra sandwich beam,, Proc. 44th IEEE Conference on Decision and Control and the European Control Conference, (2005), 3146. doi: 10.1109/CDC.2005.1582645. Google Scholar

[12]

S. W. Hansen and R. Rajaram, Riesz basis property and related results for a Rao-Nakra sandwich beam,, Discrete Contin. Dyn. Syst., (2005), 365. Google Scholar

[13]

S. W. Hansen and R. D. Spies, Structural damping in laminated beams due to interfacial slip,, Journal of Sound and Vibration, 204 (1997), 183. doi: 10.1006/jsvi.1996.0913. Google Scholar

[14]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl., 69 (1990), 33. Google Scholar

[15]

J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163. doi: 10.1016/0022-0396(83)90073-6. Google Scholar

[16]

I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Kirchhoff plates with boundary controls only in $ \Delta w |_\Sigma$,, J. Differential Equations, 93 (1991), 62. doi: 10.1016/0022-0396(91)90022-2. Google Scholar

[17]

I. Lasiecka and R. Triggiani, Uniform exponential decay of wave equations in a bounded region with $L_2(0,\infty; L_2(\Gamma))$-feedback control in the Dirichlet boundary conditions,, J. Differential Equations, 66 (1987), 340. doi: 10.1016/0022-0396(87)90025-8. Google Scholar

[18]

D. J. Mead and S. Markus, The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions,, J. Sound Vibr., 10 (1969), 163. doi: 10.1016/0022-460X(69)90193-X. Google Scholar

[19]

A. Ö. Özer and S. W. Hansen, Exact controllability of a Rayleigh beam with a single boundary control,, Math. Control Signals Systems, 23 (2011), 199. doi: 10.1007/s00498-011-0069-4. Google Scholar

[20]

A. Ö. Özer and S. W. Hansen, Exact boundary controllability results for a multilayer Rao-Nakra sandwich beam,, to appear in SIAM J. Cont. Optim., (). Google Scholar

[21]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[22]

R. Rajaram, Exact boundary controllability result for a Rao-Nakra sandwich beam,, Systems Control Lett., 56 (2007), 558. doi: 10.1016/j.sysconle.2007.03.007. Google Scholar

[23]

B. Rao, A compact perturbation method for the boundary stabilization of the Ragleigh beam equation,, Appl. Math. Optim., 33 (1996), 253. doi: 10.1007/BF01204704. Google Scholar

[24]

Y. V. K. S Rao and B. C. Nakra, Vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores,, J. Sound Vibr., 34 (1974), 309. Google Scholar

[25]

R. Triggiani, On the stabilizability problem in Banach space,, J. Math. Anal. Appl., 52 (1975), 383. doi: 10.1016/0022-247X(75)90067-0. Google Scholar

[26]

R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation,, Proc. Amer. Math. Soc., 105 (1989), 375. doi: 10.1090/S0002-9939-1989-0953013-0. Google Scholar

[27]

J. M. Wang and B. Z. Guo, Analyticity and dynamic behavior of a damped three-layer sandwich beam,, J. Optim. Theory Appl., 137 (2008), 675. doi: 10.1007/s10957-007-9341-7. Google Scholar

[28]

J. M. Wang, G. Q. Xu and S. P. Yung, Exponential stabilization of laminated beams with structural damping and boundary feedback controls,, SIAM J. Cont. Optim., 44 (2005), 1575. doi: 10.1137/040610003. Google Scholar

[29]

J. M. Wang, B. Z. Guo and B. Chentouf, Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach,, ESAIM Control Optim. Calc. Var., 12 (2006), 12. doi: 10.1051/cocv:2005030. Google Scholar

[30]

M. J. Yan and E. H. Dowell, Governing equations for vibratory constrained-layer damping sandwich plates and beams,, J. Appl. Mech., 39 (1972), 1041. Google Scholar

[31]

R. Young, An Introduction to Nonharmonic Fourier Series,, Revised first edition. Academic Press, (2001). Google Scholar

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