Feedback stabilization of a fluid--rigid body interaction system

Abstract

We study the feedback stabilization of a system composed by an incompressible viscous fluid and a rigid body. We stabilize the position and the velocity of the rigid body and the velocity of the fluid around a stationary state by means of a Dirichlet control, localized on the exterior boundary of the fluid domain and with values in a finite dimensional space. Our first result concerns weak solutions in the two-dimensional case, for initial data close to the stationary state. Our method is based on general arguments for stabilization of nonlinear parabolic systems combined with a change of variables to handle the fact that the fluid domain of the stationary state and of the stabilized solution are different. This additional difficulty leads to the assumption that the initial position of the rigid body is the position associated to the stationary state. Without this hypothesis, we work with strong solutions, and to deal with compatibility conditions at the initial time, we use finite dimensional dynamical controls. We again prove that for initial data close to the stationary state, we can stabilize the position and the velocity of the rigid body and the velocity of the fluid. In the three dimensional case, we also obtain the local stabilization of strong solutions with finite dimensional dynamical controls.