## Differential and Integral Equations

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

#### 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

https://projecteuclid.org/euclid.die/1367266996

Mathematical Reviews number (MathSciNet)
MR1668545

Zentralblatt MATH identifier
1014.35058

#### 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