Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions
ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, p. 1101-1132

Exact controllability results for a multilayer plate system are obtained from the method of Carleman estimates. The multilayer plate system is a natural multilayer generalization of a classical three-layer “sandwich plate” system due to Rao and Nakra. The multilayer version involves a number of Lamé systems for plane elasticity coupled with a scalar Kirchhoff plate equation. The plate is assumed to be either clamped or hinged and controls are assumed to be locally distributed in a neighborhood of a portion of the boundary. The Carleman estimates developed for the coupled system are based on some new Carleman estimates for the Kirchhoff plate as well as some known Carleman estimates due to Imanuvilov and Yamamoto for the Lamé system.

DOI : https://doi.org/10.1051/cocv/2010040
Classification:  93B05,  93C20,  74K20
Keywords: Carleman estimates, exact controllability, multilayer plate, lamé system, Kirchhoff plate
@article{COCV_2011__17_4_1101_0,
     author = {Hansen, Scott W. and Imanuvilov, Oleg},
     title = {Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {4},
     year = {2011},
     pages = {1101-1132},
     doi = {10.1051/cocv/2010040},
     zbl = {1238.93012},
     mrnumber = {2859867},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_4_1101_0}
}
Hansen, Scott W.; Imanuvilov, Oleg. Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 4, pp. 1101-1132. doi : 10.1051/cocv/2010040. http://www.numdam.org/item/COCV_2011__17_4_1101_0/

[1] R. Dautray and J.-L. Lions (with collaboration of M. Artola, M. Cessenat, H. Lanchon), Mathematical Analysis and Numerical Methods for Science and Technology, Volume 5: Evolution Problems I. Springer-Verlag (1992). | MR 1156075 | Zbl 0755.35001

[2] R.A. Ditaranto, Theory of vibratory bending for elastic and viscoelastic layered finite-length beams. J. Appl. Mech. 32 (1965) 881-886.

[3] S.W. Hansen, Several related models for multilayer sandwich plates. Math. Models Methods Appl. Sci. 14 (2004) 1103-1132. | MR 2078434 | Zbl 1077.74028

[4] S.W. Hansen, Semigroup well-posedness of a multilayer Mead-Markus plate with shear damping, in Control and Boundary Analysis, Lect. Not. Pure Appl. Math. 240, Chapman & Hall/CRC, Boca Raton (2005) 243-256. | MR 2144364 | Zbl 1099.74038

[5] S.W. Hansen and R. Rajaram, Riesz basis property and related results for a Rao-Nakra sandwich beam. Discrete Contin. Dynam. Syst. Suppl. (2005) 365-375. | MR 2192693 | Zbl 1143.93308

[6] L. Hörmander, Linear Partial Differential Equations. Springer-Verlag, Berlin (1963).

[7] O.Y. Imanuvilov, On Carleman estimates for hyperbolic equations. Asymptotic Anal. 32 (2002) 185-220. | MR 1993649 | Zbl 1050.35046

[8] O.Y. Imanuvilov and J.P. Puel, Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. Int. Math. Res. Not. 16 (2003) 883-913. | MR 1959940 | Zbl 1146.35340

[9] O.Y. 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. | Numdam | MR 2110612 | Zbl 1089.35086

[10] O.Y. Imanuvilov and M. Yamamoto, Carleman estimates for the three dimensional Lamé system and applications to an inverse problem, in Control Theory of Partial Differential Equations, Lect. Notes Pure. Appl. Math. 242 (2005) 337-374. | MR 2149172 | Zbl 1084.35122

[11] O.Y. Imanuvilov and M. Yamamoto, Carleman estimates for the Lamé system with stress boundary conditions and the application to an inverse problem. Publications of the Research Institute for Mathematical Sciences Kyoto University 43 (2007) 1023-1093. | MR 2389792 | Zbl 1180.35580

[12] V. Komornik, A new method of exact controllability in short time and applications. Ann. Fac. Sci. Toulouse Math. 10 (1989) 415-464. | Numdam | MR 1425495 | Zbl 0702.93010

[13] J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics 10. Society for Industrial and Applied Mathematics (1989). | MR 1061153 | Zbl 0696.73034

[14] J.E. Lagnese and J.-L Lions, Modelling, Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées RMA 6. Springer-Verlag (1989). | MR 953313 | Zbl 0662.73039

[15] I. Lasiecka and R. Triggiani, Exact controllability and uniform stabilization of Kirchoff plates with boundary controls only on Δw|Σ. J. Differ. Eqn. 93 (1991) 62-101. | MR 1122306 | Zbl 0748.93010

[16] 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. | MR 1797827 | Zbl 0983.35032

[17] J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag (1971). | MR 271512 | Zbl 0203.09001

[18] J.L. Lions, Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30 (1988) 1-68. | MR 931277 | Zbl 0644.49028

[19] 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. | Zbl 0195.26903

[20] 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. | Zbl 0282.73034

[21] R. Rajaram, Exact boundary controllability results for a Rao-Nakra sandwich beam. Systems Control Lett. 56 (2007) 558-567. | MR 2332008 | Zbl 1118.93012

[22] R. Rajaram and S.W. Hansen, Null-controllability of a damped Mead-Markus sandwich beam. Discrete Contin. Dynam. Syst. Suppl. (2005) 746-755. | MR 2192735 | Zbl 1143.93309

[23] C.T. Sun and Y.P. Lu, Vibration Damping of Structural Elements. Prentice Hall (1995). | Zbl 0840.73001

[24] D. Tataru, Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures. Appl. 75 (1996) 367-408. | MR 1411157 | Zbl 0896.35023

[25] X. Zhang, Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control Optim. 39 (2000) 812-834. | MR 1786331 | Zbl 0982.35059