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, FranceTakéo 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 . 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