Periodic solutions for a bidimensional hybrid system arising in the control of noise

Abstract

We consider a simple model arising in the control of noise. We assume that the two-dimensional cavity $\Omega=(0,1)\times (0,1)$ is occupied by an elastic, inviscid, compressible fluid. The potential $\phi$ of the velocity field satisfies the linear wave equation. The boundary of $\Omega$ is divided in two parts, $\Gamma_{0}$ and $\Gamma_{1}$. The first one, $\Gamma_{0}$, is flexible and occupied by a dissipative vibrating string. The transversal displacement of the string, $W$, satisfies a non homogeneous one-dimensional wave equation. On $\Gamma_{0}$ the continuity of the normal velocities of the fluid and the string is imposed. The subset $\Gamma_{1}$ of the boundary is assumed to be rigid and therefore, the normal velocity of the fluid vanishes. This constitutes a non homogeneous dissipative system of two coupled wave equations in dimensions two and one respectively. The non homogeneous term acting on the flexible part of the boundary (an elastic force or an exterior source of noise) is assumed to be periodic. We are interested in the existence of periodic solutions of this system. Due to the localization of the damping term in a relatively small part of the boundary and to the effect of the hybrid structure of the system, the existence of periodic solutions holds for a restricted class of non homogeneous terms. Some resonance-type phenomena are also exhibited.