Local boundary feedback stabilization of a fuid-structure interaction problem under Navier slip boundary conditions with time delay

TL;DR

This paper demonstrates the exponential stabilization of a fluid-structure interaction system with Navier slip boundary conditions using time-delayed boundary feedback control, ensuring system stability around a stationary state.

Contribution

It introduces a novel feedback stabilization method for a fluid-structure system with Navier slip conditions using time delay control and the Fattorini-Hautus criterion.

Findings

Achieved exponential decay of solutions around stationary state.

Established unique continuation property for the adjoint system.

Validated stabilization for any decay rate.

Abstract

We consider a fluid-structure interaction system composed by a three-dimensional viscous incompressible fluid and an elastic plate located on the upper part of the fluid boundary. The fluid motion is governed by the Navier-Stokes system whereas the structure displacement satisfies the damped plate equation. We consider here the Navier slip boundary conditions. The main result of this work is the feedback stabilization of the strong solutions of the corresponding system around a stationary state for any exponential decay rate by means of a time delayed control localized on the fixed fluid boundary. The strategy here is based on the Fattorini-Hautus criterion. Then, the main tool in this work is to show the unique continuation property of the associate solution to the adjoint system.