Feedback stabilization of a simplified 1d fluid–particle system
Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 369-389.

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.

DOI : 10.1016/j.anihpc.2013.03.009
Classification : 74F10, 35Q35, 76D55, 93C20, 93D15
Mots-clés : Feedback stabilization, Fluid–structure interaction, Viscous Burgers equation 
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     author = {Badra, Mehdi and Takahashi, Tak\'eo},
     title = {Feedback stabilization of a simplified 1d fluid{\textendash}particle system},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {369--389},
     publisher = {Elsevier},
     volume = {31},
     number = {2},
     year = {2014},
     doi = {10.1016/j.anihpc.2013.03.009},
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     zbl = {1302.74057},
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     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2013.03.009/}
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Badra, Mehdi; Takahashi, Takéo. Feedback stabilization of a simplified 1d fluid–particle system. Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 2, pp. 369-389. doi : 10.1016/j.anihpc.2013.03.009. http://www.numdam.org/articles/10.1016/j.anihpc.2013.03.009/

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