External boundary control of the motion of a rigid body immersed in a perfect two-dimensional fluid

Abstract

We consider the motion of a rigid body immersed in a two-dimensional irrotational perfect incompressible fluid. The fluid is governed by the Euler equation, while the trajectory of the solid is given by Newton’s equation, the force term corresponding to the fluid pressure on the body’s boundary only. The system is assumed to be confined in a bounded domain with an impermeable condition on a part of the external boundary. The issue considered here is the following: is there an appropriate boundary condition on the remaining part of the external boundary (allowing some fluid going in and out the domain) such that the immersed rigid body is driven from some given initial position and velocity to some final position (in the same connected component of the set of possible positions as the initial position) and velocity in a given positive time, without touching the external boundary? In this paper we provide a positive answer to this question thanks to an impulsive control strategy. To that purpose we make use of a reformulation of the solid equation into an ODE of geodesic form, with some force terms due to the circulation around the body, as used by Glass, Munnier and Sueur (Invent. Math. 214:1 (2018), 171–287), and some extra terms here due to the external boundary control.