Vol. 13, No. 3, 2020

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

Olivier Glass, József J. Kolumbán and Franck Sueur

Vol. 13 (2020), No. 3, 651–684
DOI: 10.2140/apde.2020.13.651
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.

Keywords
fluid-solid interaction, impulsive control, geodesics, coupled ODE/PDE system, fluid mechanics, Euler equation, control problem, external boundary control
Mathematical Subject Classification 2010
Primary: 76B75, 93C15, 93C20
Milestones
Received: 13 July 2017
Revised: 23 October 2018
Accepted: 18 April 2019
Published: 15 April 2020
Authors
Olivier Glass
CEREMADE, UMR CNRS 7534
Université de Paris-Dauphine
PSL Research University
Paris
France
József J. Kolumbán
Institut für Mathematik
Universität Leipzig
Leipzig
Germany
Franck Sueur
Institut Universitaire de France
Bordeaux INP
Institut de Mathématiques de Bordeaux, UMR 5251
Université de Bordeaux
Talence
France