# American Institute of Mathematical Sciences

June  2011, 1(2): 189-230. doi: 10.3934/mcrf.2011.1.189

## Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions

 1 Department of Mathematics, Iowa State University, Ames, IA 50011, United States 2 Department of Mathematics, Colorado State University, Ft. Collins, CO 80523, United States

Received  October 2010 Revised  April 2011 Published  June 2011

Exact controllability of a multilayer plate system with free boundary conditions are obtained by the method of Carleman estimates. The multilayer plate system is a natural multilayer generalization of a three-layer "sandwich plate'' system due to Rao and Nakra. In the multilayer version, $m$ shear deformable layers alternate with $m+1$ layers modeled under Kirchoff plate assumptions. The resulting system involves $m+1$ Lamé systems coupled with a scalar Kirchhoff plate equation. The controls are taken to be distributed in a neighborhood of the boundary. This paper is the sequel to [2] in which only clamped and hinged boundary conditions are considered.
Citation: Scott W. Hansen, Oleg Yu Imanuvilov. Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions. Mathematical Control and Related Fields, 2011, 1 (2) : 189-230. doi: 10.3934/mcrf.2011.1.189
##### References:
 [1] S. W. Hansen, Several related models for multilayer sandwich plates, Math. Models Methods Appl. Sci., 14 (2004), 1103-1132. doi: 10.1142/S0218202504003568. [2] S. W. Hansen and O. Yu. Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions,, to appear, (). [3] S. W. Hansen and R. Rajaram, Riesz basis property and related results for a Rao-Nakra sandwich beam,, Discrete Contin. Dynam. Syst., 2005 (): 365. [4] L. Hörmander, "Linear Partial Differential Equations," Springer-Verlag, Berlin, 1963. [5] O. Yu. Imanuvilov and J.-P. Puel, Global carleman estimates for weak solutions of elliptic nonhomogeneous dirichlet problems,, Int. Math. Res. Not., 2003 (): 883.  doi: 10.1155/S107379280321117X. [6] O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations, Asymptotic Analysis, 32 (2002), 185-220. [7] O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates and the non-stationary Lamé system and the application to an inverse problem, ESIAM COCV, 11 (2005), 1-56. doi: 10.1051/cocv:2004030. [8] O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates for the Lame system with the stress boundary condition, Publ. Research Inst. Math Sciences, 43 (2007), 1023-1093. doi: 10.2977/prims/1201012379. [9] V. Komornik, A new method of exact controllability in short time and applications, Ann. Fac. Sci. Toulouse Math. (5), 10 (1989), 415-464. [10] H. Kumano-go, "Pseudodifferential Operators," MIT Press, Cambrige, Mass.-London, 1981. [11] J. E. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Studies in Applied Mathematics, 10 (). [12] J. E. Lagnese and J.-L Lions, "Modelling, Analysis and Control of Thin Plates," Recherches en Mathématiques Appliquées, [Research in Applied Mathematics], 6, Masson, Paris, 1988. [13] I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Kirchoff plates with boundary controls only on $\Delta$|$\Sigma$ and homogeneous boundary displacement, J. Diff. Eqns., 93 (1991), 62-101. doi: 10.1016/0022-0396(91)90022-2. [14] I. Lasiecka and R. Triggiani, Sharp regularity for elastic and thermoelastic Kirchoff equations with free boundary conditions, Rocky Mountain J. Math., 30 (2000), 981-1024. doi: 10.1216/rmjm/1021477256. [15] G. Lebeau and L. Robbiano, Contrôle exact de l'equation de la chaleur, (French) [Exact control of the heat equation], Séminaire sur les Équations aux Dérivées Partielles, École Polytech., Palaiseau, 1995. [16] J.-L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations," Springer-Verlag, New York-Berlin, 1971. [17] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001. [18] 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. doi: 10.1016/S0022-460X(74)80315-9. [19] R. Rajaram, Exact boundary controllability results for a Rao-Nakra sandwich beam, Systems Control Lett., 56 (2007), 558-567. doi: 10.1016/j.sysconle.2007.03.007. [20] M. Taylor, "Pseudodifferential Operators,", Princeton Mathematical Series, 34, Princeton University Press, Princeton, New Jersey, 1981. [21] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures Appl. (9), 75 (1996), 367-408. [22] X. Zhang, Exact controllability of the semilinear plate equations, Asymptot. Anal., 27 (2001), 95-125. [23] X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities, SIAM J. Control and Optimization, 39 (2000), 812-834. doi: 10.1137/S0363012999350298.

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##### References:
 [1] S. W. Hansen, Several related models for multilayer sandwich plates, Math. Models Methods Appl. Sci., 14 (2004), 1103-1132. doi: 10.1142/S0218202504003568. [2] S. W. Hansen and O. Yu. Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions,, to appear, (). [3] S. W. Hansen and R. Rajaram, Riesz basis property and related results for a Rao-Nakra sandwich beam,, Discrete Contin. Dynam. Syst., 2005 (): 365. [4] L. Hörmander, "Linear Partial Differential Equations," Springer-Verlag, Berlin, 1963. [5] O. Yu. Imanuvilov and J.-P. Puel, Global carleman estimates for weak solutions of elliptic nonhomogeneous dirichlet problems,, Int. Math. Res. Not., 2003 (): 883.  doi: 10.1155/S107379280321117X. [6] O. Yu. Imanuvilov, On Carleman estimates for hyperbolic equations, Asymptotic Analysis, 32 (2002), 185-220. [7] O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates and the non-stationary Lamé system and the application to an inverse problem, ESIAM COCV, 11 (2005), 1-56. doi: 10.1051/cocv:2004030. [8] O. Yu. Imanuvilov and M. Yamamoto, Carleman estimates for the Lame system with the stress boundary condition, Publ. Research Inst. Math Sciences, 43 (2007), 1023-1093. doi: 10.2977/prims/1201012379. [9] V. Komornik, A new method of exact controllability in short time and applications, Ann. Fac. Sci. Toulouse Math. (5), 10 (1989), 415-464. [10] H. Kumano-go, "Pseudodifferential Operators," MIT Press, Cambrige, Mass.-London, 1981. [11] J. E. Lagnese, "Boundary Stabilization of Thin Plates,", SIAM Studies in Applied Mathematics, 10 (). [12] J. E. Lagnese and J.-L Lions, "Modelling, Analysis and Control of Thin Plates," Recherches en Mathématiques Appliquées, [Research in Applied Mathematics], 6, Masson, Paris, 1988. [13] I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Kirchoff plates with boundary controls only on $\Delta$|$\Sigma$ and homogeneous boundary displacement, J. Diff. Eqns., 93 (1991), 62-101. doi: 10.1016/0022-0396(91)90022-2. [14] I. Lasiecka and R. Triggiani, Sharp regularity for elastic and thermoelastic Kirchoff equations with free boundary conditions, Rocky Mountain J. Math., 30 (2000), 981-1024. doi: 10.1216/rmjm/1021477256. [15] G. Lebeau and L. Robbiano, Contrôle exact de l'equation de la chaleur, (French) [Exact control of the heat equation], Séminaire sur les Équations aux Dérivées Partielles, École Polytech., Palaiseau, 1995. [16] J.-L. Lions, "Optimal Control of Systems Governed by Partial Differential Equations," Springer-Verlag, New York-Berlin, 1971. [17] J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Review, 30 (1988), 1-68. doi: 10.1137/1030001. [18] 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. doi: 10.1016/S0022-460X(74)80315-9. [19] R. Rajaram, Exact boundary controllability results for a Rao-Nakra sandwich beam, Systems Control Lett., 56 (2007), 558-567. doi: 10.1016/j.sysconle.2007.03.007. [20] M. Taylor, "Pseudodifferential Operators,", Princeton Mathematical Series, 34, Princeton University Press, Princeton, New Jersey, 1981. [21] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems, J. Math. Pures Appl. (9), 75 (1996), 367-408. [22] X. Zhang, Exact controllability of the semilinear plate equations, Asymptot. Anal., 27 (2001), 95-125. [23] X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities, SIAM J. Control and Optimization, 39 (2000), 812-834. doi: 10.1137/S0363012999350298.
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