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 realisticly 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.