Stabilization of a fluid-rigid body system

Takéo Takahashi (1) , Marius Tucsnak (1) , George Weiss (2)

1 IECL - Institut Élie Cartan de Lorraine
2 TAU - Tel Aviv University

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).

Mots clés: global solutions; PD controller; Navier-Stokes equations; asymptotic stability; fluid-structure interactions; switching feedback

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Takéo Takahashi, Marius Tucsnak, George Weiss. Stabilization of a fluid-rigid body system. Journal of Differential Equations, 2015, ⟨10.1016/j.jde.2015.07.024⟩. ⟨hal-01076688⟩