Stabilization of a hybrid system fo noise reduction

Abstract

We consider a simple model arising in the control of noise. We assume that the two-dimensional cavity Ω = (0, 1)×(0, 1) is occupied by an elastic, inviscid, compressible fluid. The potential φ of the velocity field satisfies the linear wave equation. The boundary of Ω is divided in two parts Γ0 and Γ1. The first one, Γ0 is flexible and occupied by a Bernoulli-Euler beam. On Γ0 the continuity of the normal velocities of the fluid and the beam is imposed. The subset Γ1 of the boundary is assumed to be rigid and therefore, the normal velocity of the fluid vanishes. A dissipative term is assumed to act in the one-dimensional beam equation. We prove the existence and the uniqueness of solutions. Each trajectory is proved to converge to an equilibrium as t → ∞. On the other hand we show that the convergence rate of the energy is not exponential. The proof of this r esult uses a perturbation argument allowing to modify the boundary conditions so that separation of variables applies.