# American Institute of Mathematical Sciences

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 and 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, Iowa State University, 2009. [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), 1279-1292. doi: 10.3934/dcdsb.2010.14.1279. [3] A. A. Allen and S. W. Hansen, Analyticity of a multilayer Mead-Markus plate, Nonlinear Analysis (12), 71 (2009), e1835-e1842. doi: 10.1016/j.na.2009.02.063. [4] G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113. doi: 10.1137/0319008. [5] R. H. Fabiano and S. W. Hansen, Modeling and analysis of a three-layer damped sandwich beam, Discrete Contin. Dyn. Syst., (2001), Added Volume, 143-155. [6] B. Z. Guo, Basis property of a Rayleigh beam with boundary stabilization, J. Optim. Theory Appl., 112 (2002), 529-547. doi: 10.1023/A:1017912031840. [7] S. W. Hansen, Several related models for multilayer sandwich plates, Math. Models Methods Appl. Sci., 14 (2004), 1103-1132. doi: 10.1142/S0218202504003568. [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), 555-580. doi: 10.1142/S0218202500000306. [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), 189-230. doi: 10.3934/mcrf.2011.1.189. [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), 1101-1132. doi: 10.1051/cocv/2010040. [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), 3146-3151. doi: 10.1109/CDC.2005.1582645. [12] S. W. Hansen and R. Rajaram, Riesz basis property and related results for a Rao-Nakra sandwich beam, Discrete Contin. Dyn. Syst., (2005), suppl., 365-375. [13] S. W. Hansen and R. D. Spies, Structural damping in laminated beams due to interfacial slip, Journal of Sound and Vibration, 204 (1997), 183-202. doi: 10.1006/jsvi.1996.0913. [14] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54. [15] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182. doi: 10.1016/0022-0396(83)90073-6. [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), 62-101. doi: 10.1016/0022-0396(91)90022-2. [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), 340-390. doi: 10.1016/0022-0396(87)90025-8. [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), 163-175. doi: 10.1016/0022-460X(69)90193-X. [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), 199-222. doi: 10.1007/s00498-011-0069-4. [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. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [22] R. Rajaram, Exact boundary controllability result for a Rao-Nakra sandwich beam, Systems Control Lett., 56 (2007), 558-567. doi: 10.1016/j.sysconle.2007.03.007. [23] B. Rao, A compact perturbation method for the boundary stabilization of the Ragleigh beam equation, Appl. Math. Optim., 33 (1996), 253-264. doi: 10.1007/BF01204704. [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), 309-326. [25] R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl., 52 (1975), 383-403. doi: 10.1016/0022-247X(75)90067-0. [26] R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383. doi: 10.1090/S0002-9939-1989-0953013-0. [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), 675-689. doi: 10.1007/s10957-007-9341-7. [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), 1575-1597. doi: 10.1137/040610003. [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), 12-34. doi: 10.1051/cocv:2005030. [30] M. J. Yan and E. H. Dowell, Governing equations for vibratory constrained-layer damping sandwich plates and beams, J. Appl. Mech., 39 (1972), 1041-1046. [31] R. Young, An Introduction to Nonharmonic Fourier Series, Revised first edition. Academic Press, Inc., San Diego, CA, 2001.

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##### References:
 [1] A. A. Allen, Stability Results for Damped Multilayer Composite Beams and Plates, Ph.D. thesis, Iowa State University, 2009. [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), 1279-1292. doi: 10.3934/dcdsb.2010.14.1279. [3] A. A. Allen and S. W. Hansen, Analyticity of a multilayer Mead-Markus plate, Nonlinear Analysis (12), 71 (2009), e1835-e1842. doi: 10.1016/j.na.2009.02.063. [4] G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim., 19 (1981), 106-113. doi: 10.1137/0319008. [5] R. H. Fabiano and S. W. Hansen, Modeling and analysis of a three-layer damped sandwich beam, Discrete Contin. Dyn. Syst., (2001), Added Volume, 143-155. [6] B. Z. Guo, Basis property of a Rayleigh beam with boundary stabilization, J. Optim. Theory Appl., 112 (2002), 529-547. doi: 10.1023/A:1017912031840. [7] S. W. Hansen, Several related models for multilayer sandwich plates, Math. Models Methods Appl. Sci., 14 (2004), 1103-1132. doi: 10.1142/S0218202504003568. [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), 555-580. doi: 10.1142/S0218202500000306. [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), 189-230. doi: 10.3934/mcrf.2011.1.189. [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), 1101-1132. doi: 10.1051/cocv/2010040. [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), 3146-3151. doi: 10.1109/CDC.2005.1582645. [12] S. W. Hansen and R. Rajaram, Riesz basis property and related results for a Rao-Nakra sandwich beam, Discrete Contin. Dyn. Syst., (2005), suppl., 365-375. [13] S. W. Hansen and R. D. Spies, Structural damping in laminated beams due to interfacial slip, Journal of Sound and Vibration, 204 (1997), 183-202. doi: 10.1006/jsvi.1996.0913. [14] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54. [15] J. Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182. doi: 10.1016/0022-0396(83)90073-6. [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), 62-101. doi: 10.1016/0022-0396(91)90022-2. [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), 340-390. doi: 10.1016/0022-0396(87)90025-8. [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), 163-175. doi: 10.1016/0022-460X(69)90193-X. [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), 199-222. doi: 10.1007/s00498-011-0069-4. [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. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [22] R. Rajaram, Exact boundary controllability result for a Rao-Nakra sandwich beam, Systems Control Lett., 56 (2007), 558-567. doi: 10.1016/j.sysconle.2007.03.007. [23] B. Rao, A compact perturbation method for the boundary stabilization of the Ragleigh beam equation, Appl. Math. Optim., 33 (1996), 253-264. doi: 10.1007/BF01204704. [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), 309-326. [25] R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl., 52 (1975), 383-403. doi: 10.1016/0022-247X(75)90067-0. [26] R. Triggiani, Lack of uniform stabilization for noncontractive semigroups under compact perturbation, Proc. Amer. Math. Soc., 105 (1989), 375-383. doi: 10.1090/S0002-9939-1989-0953013-0. [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), 675-689. doi: 10.1007/s10957-007-9341-7. [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), 1575-1597. doi: 10.1137/040610003. [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), 12-34. doi: 10.1051/cocv:2005030. [30] M. J. Yan and E. H. Dowell, Governing equations for vibratory constrained-layer damping sandwich plates and beams, J. Appl. Mech., 39 (1972), 1041-1046. [31] R. Young, An Introduction to Nonharmonic Fourier Series, Revised first edition. Academic Press, Inc., San Diego, CA, 2001.
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