Stabilization of a layered piezoelectric 3-D body by boundary dissipation

Boris Kapitonov; Bernadette Miara; Gustavo Perla Menzala

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 2, page 198-215
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Kapitonov, Boris, Miara, Bernadette, and Menzala, Gustavo Perla. "Stabilization of a layered piezoelectric 3-D body by boundary dissipation." ESAIM: Control, Optimisation and Calculus of Variations 12.2 (2006): 198-215. <http://eudml.org/doc/249622>.

@article{Kapitonov2006,
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. },
author = {Kapitonov, Boris, Miara, Bernadette, Menzala, Gustavo Perla},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Distributed systems; boundary control; stabilization; exact controllability.; distributed systems; exact controllability},
language = {eng},
month = {3},
number = {2},
pages = {198-215},
publisher = {EDP Sciences},
title = {Stabilization of a layered piezoelectric 3-D body by boundary dissipation},
url = {http://eudml.org/doc/249622},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Kapitonov, Boris
AU - Miara, Bernadette
AU - Menzala, Gustavo Perla
TI - Stabilization of a layered piezoelectric 3-D body by boundary dissipation
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/3//
PB - EDP Sciences
VL - 12
IS - 2
SP - 198
EP - 215
AB - 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.
LA - eng
KW - Distributed systems; boundary control; stabilization; exact controllability.; distributed systems; exact controllability
UR - http://eudml.org/doc/249622
ER -

References

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  8. B. Kapitonov, B. Miara and G. Perla Menzala, Boundary observation and exact control of a quasi-electrostatic piezoelectric system in multilayered media. (submitted).  
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  11. G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity. Archive for Rational Mechanics and Analysis148 (1999) 179–231.  
  12. J.-L. Lions, Exact controllability, stabilization and perturbation for distributed systems. SIAM Rev.30 (1988) 1–68.  
  13. J.-L. Lions, Controlabilité exacte, perturbations et stabilisation de systèmes distribués. Masson, Paris (1988).  
  14. B. Miara, Controlabilité d'un corp piézoélectrique. CRAS Paris333 (2001) 267–270.  
  15. A. Pazy, On the applicability of Lyapunov's theorem in Hilbert space. SIAM J. Math. Anal.3 (1972) 291–294.  
  16. A. Pazy, Semigroup of linear operators and applications to Partial Differential Equations. Springer-Verlag (1983).  
  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.  

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