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Stabilization of a layered piezoelectric 3-D bodyby boundary dissipation

Published online by Cambridge University Press:  22 March 2006

Boris Kapitonov ,
Bernadette Miara  and
Gustavo Perla Menzala
Boris Kapitonov
Affiliation:
Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Russia, Visiting Researcher at the National Laboratory of Scientific Computation (LNCC/MCT), Brasil; [email protected]
Bernadette Miara
Affiliation:
Laboratoire de Modélisation et Simulation numérique, École Supérieure d'Ingénieurs en Électrotechnique et Électronique, 2 Boulevard Blaise Pascal, 93160 Noisy-le-Grand, France; [email protected]
Gustavo Perla Menzala
Affiliation:
National Laboratory of Scientific Computation LNCC/MCT, Rua Getulio Vargas 333, Quitandinha, Petropolis 25651-070, RJ, Brasil and Institute of Mathematics Federal University of Rio de Janeiro, RJ, P.O. 68530, Rio de Janeiro, RJ, Brasil; [email protected]
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Abstract

We consider a linear coupled system of quasi-electrostatic equations which govern the evolution of a 3-D layered piezoelectric body. Assuming that a dissipative effect is effective at the boundary, we study the uniform stabilization problem. We prove that this is indeed the case, provided some geometric conditions on the region and the interfaces hold. We also assume a monotonicity condition on the coefficients. As an application, we deduce exact controllability of the system with boundary control via a classical result due to Russell.

Keywords

Distributed systemsboundary controlstabilizationexact controllability.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 12 , Issue 2 , April 2006 , pp. 198 - 215
Copyright
© EDP Sciences, SMAI, 2006

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References

Akamatsu, M. and Nakamura, G., Well-posedness of initial-boundary value problems for piezoelectric equations. Appl. Anal. 81 (2002) 129–141. CrossRef
Bardos, C., Lebeau, G. and Rauch, J., Sharp sufficient conditions for the observation control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024–1065. CrossRef
Burq, N. and Lebeau, G., Mesures de défaut de compacité, application au système de Lamé. Annals Scientifiques de l'École Normale Supérieure (4) 34 (2001) 817–870. CrossRef
T. Duyckaerts, Stabilisation haute frequence d'équations aux dérivées partialles linéaires. Thèse de Doctorat, Université Paris XI-Orsay (2004).
J.N. Eringen and G.A. Maugin, Electrodynamics of continua. Vols. 1, 2, Berlin, Springer (1990).
T. Ikeda, Fundamentals of Piezoelectricity. Oxford University Press (1996).
Kapitonov, B.V. and Perla Menzala, G., Energy decay and a transmission problem in electromagneto-elasticity. Adv. Diff. Equations 7 (2002) 819–846.
B. Kapitonov, B. Miara and G. Perla Menzala, Boundary observation and exact control of a quasi-electrostatic piezoelectric system in multilayered media. (submitted).
V. Komornik, Exact controllability and stabilization, the multiplier method. Masson (1994).
Lagnese, J.E., Boundary controllability in problems of transmission for a class of second order hyperbolic systems. ESAIM: COCV 2 (1997) 343–357. CrossRef
Lebeau, G. and Zuazua, E., Decay rates for the three-dimensional linear system of thermoelasticity. Archive for Rational Mechanics and Analysis 148 (1999) 179–231. CrossRef
Lions, J.-L., Exact controllability, stabilization and perturbation for distributed systems. SIAM Rev. 30 (1988) 1–68. CrossRef
J.-L. Lions, Controlabilité exacte, perturbations et stabilisation de systèmes distribués. Masson, Paris (1988).
Miara, B., Controlabilité d'un corp piézoélectrique. CRAS Paris 333 (2001) 267–270.
Pazy, A., On the applicability of Lyapunov's theorem in Hilbert space. SIAM J. Math. Anal. 3 (1972) 291–294. CrossRef
A. Pazy, Semigroup of linear operators and applications to Partial Differential Equations. Springer-Verlag (1983).
Russell, D.L., The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region. SIAM J. Control Optim. 24 (1986) 199–229. CrossRef