Stabilization of a rigid body moving in a compressible viscous fluid

TL;DR

This paper proves the global stabilization of a rigid body in a viscous, compressible fluid using feedback control, ensuring velocities decay to zero and the body reaches a fixed point over time.

Contribution

It introduces a feedback control law for fluid-structure interaction with compressible viscous fluids and proves global existence and stabilization results.

Findings

Global-in-time strong solutions exist under small initial data.

Velocities of fluid and structure decay to zero over time.

The rigid body converges to a fixed point $h_1$ as time approaches infinity.

Abstract

We consider the stabilizability of a fluid-structure interaction system where the fluid is viscous and compressible and the structure is a rigid ball. The feedback control of the system acts on the ball and corresponds to a force that would be produced by a spring and a damper connecting the center of the ball to a fixed point $h_1$. We prove the global-in-time existence of strong solutions for the corresponding system under a smallness condition on the initial velocities and on the distance between the initial position of the center of the ball and $h_1$. Then, we show with our feedback law, that the fluid and the structure velocities go to 0 and that the center of the ball goes to $h_1$ as $t\to \infty$.