Stabilization of a layered piezoelectric 3-D body by boundary dissipation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 198-215.

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.

DOI : 10.1051/cocv:2005028
Classification : 35Q99, 74F99, 35B40
Mots-clés : distributed systems, boundary control, stabilization, exact controllability 
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     title = {Stabilization of a layered piezoelectric {3-D} body by boundary dissipation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {198--215},
     publisher = {EDP-Sciences},
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Kapitonov, Boris; Miara, Bernadette; Menzala, Gustavo Perla. Stabilization of a layered piezoelectric 3-D body by boundary dissipation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 2, pp. 198-215. doi : 10.1051/cocv:2005028. http://archive.numdam.org/articles/10.1051/cocv:2005028/

[1] M. Akamatsu and G. Nakamura, Well-posedness of initial-boundary value problems for piezoelectric equations. Appl. Anal. 81 (2002) 129-141. | Zbl

[2] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation control and stabilization of waves from the boundary. SIAM J. Control Optim. 30 (1992) 1024-1065. | Zbl

[3] N. Burq and G. Lebeau, Mesures de défaut de compacité, application au système de Lamé. Annals Scientifiques de l'École Normale Supérieure (4) 34 (2001) 817-870. | Numdam | Zbl

[4] T. Duyckaerts, Stabilisation haute frequence d'équations aux dérivées partialles linéaires. Thèse de Doctorat, Université Paris XI-Orsay (2004).

[5] J.N. Eringen and G.A. Maugin, Electrodynamics of continua. Vols. 1, 2, Berlin, Springer (1990).

[6] T. Ikeda, Fundamentals of Piezoelectricity. Oxford University Press (1996).

[7] B.V. Kapitonov and G. Perla Menzala, Energy decay and a transmission problem in electromagneto-elasticity. Adv. Diff. Equations 7 (2002) 819-846. | Zbl

[8] B. Kapitonov, B. Miara and G. Perla Menzala, Boundary observation and exact control of a quasi-electrostatic piezoelectric system in multilayered media. (submitted). | Zbl

[9] V. Komornik, Exact controllability and stabilization, the multiplier method. Masson (1994). | MR | Zbl

[10] J.E. Lagnese, Boundary controllability in problems of transmission for a class of second order hyperbolic systems. ESAIM: COCV 2 (1997) 343-357. | Numdam | Zbl

[11] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity. Archive for Rational Mechanics and Analysis 148 (1999) 179-231. | Zbl

[12] J.-L. Lions, Exact controllability, stabilization and perturbation for distributed systems. SIAM Rev. 30 (1988) 1-68. | Zbl

[13] J.-L. Lions, Controlabilité exacte, perturbations et stabilisation de systèmes distribués. Masson, Paris (1988). | Zbl

[14] B. Miara, Controlabilité d'un corp piézoélectrique. CRAS Paris 333 (2001) 267-270. | Zbl

[15] A. Pazy, On the applicability of Lyapunov's theorem in Hilbert space. SIAM J. Math. Anal. 3 (1972) 291-294. | Zbl

[16] A. Pazy, Semigroup of linear operators and applications to Partial Differential Equations. Springer-Verlag (1983). | MR | Zbl

[17] D.L. Russell, The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region. SIAM J. Control Optim. 24 (1986) 199-229. | Zbl

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