Stabilization of a fluid–rigid body system

Takéo Takahashi,Marius Tucsnak,George Weiss

doi:10.1016/j.jde.2015.07.024

Abstract

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).

Highlights

Introduction and main resultsWe consider a coupled system described by nonlinear partial and ordinary differential equations modelling the motion of a rigid body inside a viscous incompressible fluid in a bounded domain Ω
We have studied an infinite-dimensional nonlinear dynamical system coupling the Navier-Stokes equations with the rigid body dynamics, in the presence of a free boundary
We have proposed a PD-type controller which asymptotically steers the rigid body to a prescribed final position, while the velocities of the fluid and of the rigid body tend to zero

Summary

Introduction and main results

The domain occupied by the fluid and the rigid ball is Ω ⊂ R3, a connected open bounded set with C2 boundary. We would like to have a result that tells us something similar to the above theorem, but without the requirement that h0 is close to h1 We achieve this by imposing a more complicated control law, in which the anchor point of the spring and damper is not fixed at h1, but instead it jumps between a finite number of possible points. We can skip the smallness condition that is necessary in dimension 3

Changing variables to a fixed domain

Local existence of solutions

Energy estimates

The bidimensional case

Concluding remarks

10. Appendix: A Barbalat type lemma