ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - April 2006 - Volume 12, Issue 02  

Research Article

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

Kapitonov, Borisa1, Miara, Bernadettea2 and Menzala, Gustavo Perlaa3

a1 Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences, Russia, Visiting Researcher at the National Laboratory of Scientific Computation (LNCC/MCT), Brasil; borisvk@lncc.br

a2 Laboratoire de Modélisation et Simulation numérique, École Supérieure d'Ingénieurs en Électrotechnique et Électronique, 2 Boulevard Blaise Pascal, 93160 Noisy-le-Grand, France; miarab@esiee.fr

a3 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; perla@lncc.br

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.

(Received July 3 2004)

(Revised January 4 2005)

(Online publication March 22 2006)

Key Words:

  • Distributed systems;
  • boundary control;
  • stabilization;
  • exact controllability.

Mathematics Subject Classification:

  • 35Q99;
  • 74F99;
  • 35B40