Generic Controllability of 3D Swimmers in a Perfect Fluid

TL;DR

This paper proves that 3D shape-changing swimmers in a perfect fluid can approximately follow any trajectory and shape changes through small deformations, demonstrating controllability in ideal fluid dynamics.

Contribution

It establishes the controllability of 3D swimmers in perfect fluids, allowing approximate trajectory tracking with minimal shape modifications and a finite set of basic movements.

Findings

Swimmers can follow arbitrary trajectories with small shape changes.

Controllability is achieved with at most five basic movements.

Results apply to both synchronized and freestyle swimming scenarios.

Abstract

We address the problem of controlling a dynamical system governing the motion of a 3D weighted shape changing body swimming in a perfect fluid. The rigid displacement of the swimmer results from the exchange of momentum between prescribed shape changes and the flow, the total impulse of the fluid-swimmer system being constant for all times. We prove the following tracking results: (i) Synchronized swimming: Maybe up to an arbitrarily small change of its density, any swimmer can approximately follow any given trajectory while, in addition, undergoing approximately any given shape changes. In this statement, the control consists in arbitrarily small superimposed deformations; (ii) Freestyle swimming: Maybe up to an arbitrarily small change of its density, any swimmer can approximately tracks any given trajectory by combining suitably at most five basic movements that can be generically chosen (no macro shape changes are prescribed in this statement).