Feedback stabilization of a 3D fluid flow by shape deformations of an obstacle^*

¹ Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Paul Sabatier, 31062 Toulouse Cedex 9, France.
² Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai, Maharashtra 400076, India.

^** Corresponding author: raymond@math.univ-toulouse.fr

Received: 23 October 2020
Accepted: 31 May 2021

Abstract

We consider a fluid flow in a time dependent domain Ω_f(t)=Ω\Ω_s(t)̅⊂ℝ³, surrounding a deformable obstacle Ω_s(t). We assume that the fluid flow satisfies the incompressible Navier-Stokes equations in Ω_f(t), t > 0. We prove that, for any arbitrary exponential decay rate ω > 0, if the initial condition of the fluid flow is small enough in some norm, the deformation of the boundary ∂Ω_s(t) can be chosen so that the fluid flow is stabilized to rest, and the obstacle to its initial shape and its initial location, with the exponential decay rate ω > 0.