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

Published online by Cambridge University Press:  08 November 2010

Scott W. Hansen  and
Oleg Imanuvilov
Scott W. Hansen
Affiliation:
Department of Mathematics, Iowa State University, Ames, IA 50011, USA.shansen@orion.math.iastate.edu .
Oleg Imanuvilov
Affiliation:
Department of Mathematics, Colorado State University, Ft. Collins, CO 80523, USA.
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Abstract

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.

Keywords

Carleman estimatesexact controllabilitymultilayer plateLamé systemKirchhoff plate
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 4 , October 2011 , pp. 1101 - 1132
Copyright
© EDP Sciences, SMAI, 2010

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References

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).
DiTaranto, R.A., Theory of vibratory bending for elastic and viscoelastic layered finite-length beams. J. Appl. Mech. 32 (1965) 881–886. CrossRef
Hansen, S.W., Several related models for multilayer sandwich plates. Math. Models Methods Appl. Sci. 14 (2004) 1103–1132. CrossRef
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.
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.
L. Hörmander, Linear Partial Differential Equations. Springer-Verlag, Berlin (1963).
Imanuvilov, O.Y., Carleman, On estimates for hyperbolic equations. Asymptotic Anal. 32 (2002) 185–220.
Imanuvilov, O.Y. and Puel, J.P., Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. Int. Math. Res. Not. 16 (2003) 883–913. CrossRef
Imanuvilov, O.Y. and Yamamoto, M., Carleman estimates and the non-stationary Lamé system and the application to an inverse problem. ESIAM: COCV 11 (2005) 1–56.
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.
Imanuvilov, O.Y. and Yamamoto, M., 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. CrossRef
Komornik, V., A new method of exact controllability in short time and applications. Ann. Fac. Sci. Toulouse Math. 10 (1989) 415–464. CrossRef
J. Lagnese, Boundary Stabilization of Thin Plates, SIAM Studies in Applied Mathematics 10. Society for Industrial and Applied Mathematics (1989).
J.E. Lagnese and J.-L Lions, Modelling, Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées RMA 6. Springer-Verlag (1989).
Lasiecka, I. and Triggiani, R., Exact controllability and uniform stabilization of Kirchoff plates with boundary controls only on Δw|Σ. J. Differ. Eqn. 93 (1991) 62–101. CrossRef
Lasiecka, I. and Triggiani, R., Sharp regularity for elastic and thermoelastic Kirchoff equations with free boundary conditions. Rocky Mountain J. Math. 30 (2000) 981–1024. CrossRef
J.L. Lions, Optimal Control of Systems Governed by Partial Differential Equations. Springer-Verlag (1971).
Lions, J.L., Exact controllability, stabilization and perturbations for distributed systems. SIAM Rev. 30 (1988) 1–68. CrossRef
Mead, D.J. and Markus, S., The forced vibration of a three-layer, damped sandwich beam with arbitrary boundary conditions. J. Sound Vibr. 10 (1969) 163–175. CrossRef
Rao, Y.V.K.S. and Nakra, B.C., Vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores. J. Sound Vibr. 34 (1974) 309–326.
Rajaram, R., Exact boundary controllability results for a Rao-Nakra sandwich beam. Systems Control Lett. 56 (2007) 558–567. CrossRef
R. Rajaram and S.W. Hansen, Null-controllability of a damped Mead-Markus sandwich beam. Discrete Contin. Dynam. Syst. Suppl. (2005) 746–755.
C.T. Sun and Y.P. Lu, Vibration Damping of Structural Elements. Prentice Hall (1995).
Tataru, D., Carleman estimates and unique continuation for solutions to boundary value problems. J. Math. Pures. Appl. 75 (1996) 367–408.
Zhang, X., Explicit observability inequalities for the wave equation with lower order terms by means of Carleman inequalities. SIAM J. Control Optim. 39 (2000) 812–834. CrossRef