# Feedback stabilization of a simplified 1d fluid–particle system

### Mehdi Badra

IMT, UMR CNRS 5219, Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France, LMAP, UMR CNRS 5142, UNIV PAU & PAYS ADOUR, 64013 Pau Cedex, France### Takéo Takahashi

Inria, Villers-lès-Nancy, F-54600, France, Université de Lorraine, IECN, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France, CNRS, IECN, UMR 7502, Vandoeuvre-lès-Nancy, F-54506, France

## Abstract

We consider the feedback stabilization of a simplified 1d model for a fluid–structure interaction system. The fluid equation is the viscous Burgers equation whereas the motion of the particle is given by the Newton's laws. We stabilize this system around a stationary state by using feedbacks located at the exterior boundary of the fluid domain. With one input, we obtain a local stabilizability of the system with an exponential decay rate of order $σ<σ_{0}$. An arbitrary order for the exponential decay rate can be proved if a unique continuation result holds true or if two inputs are used to stabilize the system. 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 domains of the stationary state and of the stabilized solution are different.

## Cite this article

Mehdi Badra, Takéo Takahashi, Feedback stabilization of a simplified 1d fluid–particle system. Ann. Inst. H. Poincaré Anal. Non Linéaire 31 (2014), no. 2, pp. 369–389

DOI 10.1016/J.ANIHPC.2013.03.009