Feedback Stabilization of the Incompressible Navier–Stokes Equations Coupled with a Damped Elastic System in Two Dimensions

Abstract

In this article we study a system coupling the incompressible Navier–Stokes equations with an elastic structure governed by a damped wave equation in a two dimensional channel with periodic boundary conditions. The elastic structure is located at the upper boundary of the domain occupied by the fluid. The domain occupied by the fluid depends on the displacement of the elastic structure, and therefore it depends on time. We prove that this coupled system may be stabilized around the steady state zero, at any exponential decay rate, by a Dirichlet control acting in the lower boundary of the fluid domain.