Abstract
We consider a coupled PDE system arising in noise reduction problems. In a two dimensional chamber, the acoustic pressure (unwanted noise) is represented by a hyperbolic wave equation. The floor of the chamber is subject to the action of piezo‐ceramic patches (smart materials). The goal is to reduce the acoustic pressure by means of the vibrations of the floor which is modelled by a hyperbolic Kirchoff equation. These two hyperbolic equations are coupled by appropriate trace operators. This overall model differs from those previously studied in the literature in that the elastic chamber floor is here more realistically modeled by a hyperbolic Kirchoff equation, rather than by a parabolic Euler‐Bernoulli equation with Kelvin‐Voight structural damping, as in past literature. Thus, the hyperbolic/parabolic coupled system of past literature is replaced here by a hyperbolic/hyperbolic coupled model. The main result of this paper is a uniform stabilization of the coupled PDE system by a (physically appealing) boundary dissipation.
Highlights
In this paper we study the uniform stabilization of two coupled hyperbolic equations arising in the noise reduction problem for structural acoustic models
The acoustic pressure inside a two dimensional chamber is mathematically represented by a hyperbolic wave equation, whereas a hyperbolic Kirchoff equation models the elastic displacements of the one
Our main goal is to show the uniform stability of the s.c. contraction semigroup eAF t in the space Y0 described in (2.29), corresponding to the coupled damped PDE system with k1 = k2 = 1 and u(t) ≡ 0
Summary
In this paper we study the uniform stabilization of two coupled hyperbolic equations arising in the noise reduction problem for structural acoustic models. Our main goal is to show the uniform stability of the s.c. contraction semigroup eAF t in the space Y0 described in (2.29), corresponding to the coupled damped PDE system with k1 = k2 = 1 and u(t) ≡ 0. Let us note that the analysis of the Kirchoff equation follows closely the technique of [10]. With respect to the Kirchoff equation part of the coupled PDE’s with k1 = k2 = 1 and u(t) ≡ 0 in (1.1), we have the following inequality:
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
