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].