Vol. 3, No. 4, 2021

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Remote trajectory tracking of rigid bodies immersed in a two-dimensional perfect incompressible fluid

Olivier Glass, József J. Kolumbán and Franck Sueur

Vol. 3 (2021), No. 4, 613–652

We consider the motion of several rigid bodies immersed in a two-dimensional incompressible perfect fluid, the whole system occupying a bounded simply connected domain. The external fixed boundary is impermeable except on an open nonempty part where one controls both the normal velocity, allowing some fluid to go in and out of the domain, and the entering vorticity. The motion of the rigid bodies is given by the Newton laws with forces due to the fluid pressure and the fluid motion is described by the incompressible Euler equations. We prove that it is possible to exactly achieve any noncolliding smooth motion of the rigid bodies by the remote action of a controlled normal velocity on the outer boundary which takes the form of state-feedback, with zero entering vorticity. The proof relies on a nonlinear method to solve linear perturbations of nonlinear equations associated with a quadratic operator.

trajectory tracking, fluid-solid interaction, coupled ODE-PDE system, fluid mechanics, Euler equation, control problem
Mathematical Subject Classification
Primary: 76B75, 93C20
Received: 9 July 2020
Revised: 20 May 2021
Accepted: 2 September 2021
Published: 12 February 2022
Olivier Glass
Université de Paris-Dauphine
PSL Research University
József J. Kolumbán
Institut für Mathematik
Universität Leipzig
Franck Sueur
Institut de Mathématiques de Bordeaux, UMR CNRS 5251
Université de Bordeaux
Institut Universitaire de France