Boundary Asymptotic Stabilizability of a Nonlinear Fluid Structure Interaction

We consider a model of fluid-structure interaction in a bounded domain Omega is an element of R-2 where Omega is comprised of two open adjacent domains occupied, respectively, by the solid and the fluid. This leads to a study of Navier Stokes equation coupled on the boundary to the dynamic system of elasticity. We shall consider models when the elastic body exhibits small but rapid oscillations. These are established models arising in engineering applications when the structure is immersed in a viscous flow of liquid. The goal of the work is to present recent results on asymptotic stabilizability of the interactive structure. It will be shown that under suitable geometric conditions imposed on the domain the model is asymptotically stabilizable with a boundary feedback acting as a force on the interface. The required geometric conditions result from the presence of the pressure acting upon the solid.