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.

show all references

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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