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

top
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

top

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

top
  1. M. Akamatsu and G. Nakamura, Well-posedness of initial-boundary value problems for piezoelectric equations. Appl. Anal.81 (2002) 129–141.  Zbl1080.74025
  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.  Zbl0786.93009
  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.  Zbl1043.35009
  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. Equations7 (2002) 819–846.  Zbl1052.93025
  8. B. Kapitonov, B. Miara and G. Perla Menzala, Boundary observation and exact control of a quasi-electrostatic piezoelectric system in multilayered media. (submitted).  Zbl1143.35378
  9. V. Komornik, Exact controllability and stabilization, the multiplier method. Masson (1994).  Zbl0937.93003
  10. J.E. Lagnese, Boundary controllability in problems of transmission for a class of second order hyperbolic systems. ESAIM: COCV2 (1997) 343–357.  Zbl0899.93003
  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.  Zbl0939.74016
  12. J.-L. Lions, Exact controllability, stabilization and perturbation for distributed systems. SIAM Rev.30 (1988) 1–68.  Zbl0644.49028
  13. J.-L. Lions, Controlabilité exacte, perturbations et stabilisation de systèmes distribués. Masson, Paris (1988).  Zbl0653.93002
  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.  Zbl0242.47028
  16. A. Pazy, Semigroup of linear operators and applications to Partial Differential Equations. Springer-Verlag (1983).  Zbl0516.47023
  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.  Zbl0594.49026

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.