TY - JOUR

T1 - Stabilization of a fluid-rigid body system

AU - Takahashi, Takéo

AU - Tucsnak, Marius

AU - Weiss, George

N1 - Publisher Copyright:
© 2015 Elsevier Inc.

PY - 2015/12/5

Y1 - 2015/12/5

N2 - We consider the mathematical model of a rigid ball moving in a viscous incompressible fluid occupying a bounded domain Ω, 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 h1∈Ω. If the initial fluid velocity is sufficiently small, and the initial h is sufficiently close to h1, 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 h1 even if the initial value of h is far from h1, 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 Ω, that depend on h (a switching feedback law).

AB - We consider the mathematical model of a rigid ball moving in a viscous incompressible fluid occupying a bounded domain Ω, 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 h1∈Ω. If the initial fluid velocity is sufficiently small, and the initial h is sufficiently close to h1, 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 h1 even if the initial value of h is far from h1, 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 Ω, that depend on h (a switching feedback law).

KW - Asymptotic stability

KW - Fluid-structure interactions

KW - Global solutions

KW - Navier-Stokes equations

KW - PD controller

KW - Switching feedback

UR - http://www.scopus.com/inward/record.url?scp=84941802392&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2015.07.024

DO - 10.1016/j.jde.2015.07.024

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AN - SCOPUS:84941802392

SN - 0022-0396

VL - 259

SP - 6459

EP - 6493

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 11

M1 - 7957

ER -