Generic Controllability of 3D Swimmers in a Perfect Fluid
Provides theoretical controllability guarantees for 3D swimmers in ideal fluids, addressing a fundamental problem in geometric mechanics and control theory.
The paper proves that any 3D shape-changing swimmer in a perfect fluid can approximately track any trajectory, either by superimposing small deformations (synchronized swimming) or by combining at most five generic basic movements (freestyle swimming), up to an arbitrarily small density change.
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).