Uniform feedback stabilization via boundary moments of a 3-dimensional structural acoustic model

We consider a coupled PDE system arising in noise reduction problems. In a three 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 a piezo-ceramic patch (smart material). The goal is to reduce the acoustic pressure by means of the vibrations of the floor which is modeled by a hyperbolic Kirchhoff equation. These two hyperbolic equations are coupled by appropriate trace operators. This overall model differs from those previously studied in the literature in two ways. The first difference is that the elastic chamber floor is here more realistically modeled by a hyperbolic Kirchhoff 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 second difference is the generalization of the two dimensional acoustic model into a three dimensional one. The main result of this paper is a uniform stabilization of the coupled PDE system by a (physically appealing) boundary dissipation.