Differential and Integral Equations

Sharp regularity of a coupled system of a wave and a Kirchoff equation with point control arising in noise reduction

M. Camurdan and R. Triggiani

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Abstract

We consider a mathematical model of the noise reduction problem, which couples two hyperbolic equations: the wave equation in the interior ("chamber")---which describes the unwanted acoustic waves---and a (hyperbolic) Kirchoff equation ---which models the vibrations of the elastic wall. In past models, the elastic wall was modeled by an Euler-Bernoulli equation with Kelvin-Voight damping (parabolic model). Our main result is a sharp regularity result, in two dual versions, of the resulting system of two coupled hyperbolic P.D.E.'s. With this regularity result established, one can then invoke a wealth of abstract results from [14], [15], [16], [19], etc. on optimal control problems, min-max game theory (and $H^\infty$-problems), etc. The proof of the main result is based on combining technical results from [18] and [11].

Article information

Source
Differential Integral Equations, Volume 12, Number 1 (1999), 101-118.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.die/1367266996

Mathematical Reviews number (MathSciNet)
MR1668545

Zentralblatt MATH identifier
1014.35058

Subjects
Primary: 35L70: Nonlinear second-order hyperbolic equations
Secondary: 35B65: Smoothness and regularity of solutions 49K20: Problems involving partial differential equations 74F10: Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.) 74H30: Regularity of solutions 74H45: Vibrations

Citation

Camurdan, M.; Triggiani, R. Sharp regularity of a coupled system of a wave and a Kirchoff equation with point control arising in noise reduction. Differential Integral Equations 12 (1999), no. 1, 101--118. https://projecteuclid.org/euclid.die/1367266996


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