Stabilization of a fluid-rigid body system

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:259

Article

Takahashi, Takeo^1,2  Tucsnak, Marius³  Weiss, George⁴ 

[1] Inria, F-54600 Villers Les Nancy, France

[2] Univ Lorraine, F-54506 Vandoeuvre Les Nancy, France

[3] Univ Lorraine, Inst Elie Cartan, F-54506 Vandoeuvre Les Nancy, France

[4] Tel Aviv Univ, Dept EE Syst, IL-69978 Ramat Aviv, Israel

关键词: Fluid-structure interactions;    Navier-Stokes equations;    PD controller;    Global solutions;    Asymptotic stability;    Switching feedback;   

DOI  :  10.1016/j.jde.2015.07.024

来源: Elsevier

【 摘 要 】

We consider the mathematical model of a rigid ball moving in a viscous incompressible fluid occupying a bounded domain Omega, with an external force acting on the ball. We investigate in particular the case when the external force is what would be produced by a spring and a damper connecting the center of the ball h to a fixed point h(1) epsilon Omega. If the initial fluid velocity is sufficiently small, and the initial h is sufficiently close to hi, then we prove the existence and uniqueness of global (in time) solutions for the model. Moreover, in this case, we show that h converges to h1, and all the velocities (of the fluid and of the ball) converge to zero. Based on this result, we derive a control law that will bring the ball asymptotically to the desired position h I even if the initial value of h is far from hi, and the path leading to h1 is winding and complicated. Now, the idea is to use the force as described above, with one end of the spring and damper at h, while other end is jumping between a finite number of points in Omega, that depend on h (a switching feedback law). (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free