Skip to main content Accessibility help
×
Home
Hostname: page-component-5c569c448b-b4ls7 Total loading time: 0.186 Render date: 2022-07-05T02:08:42.117Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "useNewApi": true } hasContentIssue true

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

Published online by Cambridge University Press:  22 March 2006

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; borisvk@lncc.br
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; miarab@esiee.fr
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; perla@lncc.br
Get access

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.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Akamatsu, M. and Nakamura, G., Well-posedness of initial-boundary value problems for piezoelectric equations. Appl. Anal. 81 (2002) 129141. 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) 10241065. 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) 817870. 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) 819846.
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) 343357. CrossRef
Lebeau, G. and Zuazua, E., Decay rates for the three-dimensional linear system of thermoelasticity. Archive for Rational Mechanics and Analysis 148 (1999) 179231. CrossRef
Lions, J.-L., Exact controllability, stabilization and perturbation for distributed systems. SIAM Rev. 30 (1988) 168. 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) 267270.
Pazy, A., On the applicability of Lyapunov's theorem in Hilbert space. SIAM J. Math. Anal. 3 (1972) 291294. 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) 199229. CrossRef

Save article to Kindle

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

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

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

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

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *