Controllability properties of a class of systems modeling swimming microscopic organisms

TL;DR

This paper analyzes the controllability of a mathematical model for swimming microscopic organisms, showing conditions under which their motion can be fully controlled based on surface cilia activity and shape.

Contribution

It proves generic controllability for systems with at least three control dimensions and characterizes controllability for spherical organisms and density-matched conditions.

Findings

System is generically controllable with 3+ control dimensions

Complete controllability characterization for spherical organisms

Controllability depends on density and shape assumptions

Abstract

We consider a finite-dimensional model for the motion of microscopic organisms whose propulsion exploits the action of a layer of cilia covering its surface. The model couples Newton's laws driving the organism, considered as a rigid body, with Stokes equations governing the surrounding fluid. The action of the cilia is described by a set of controlled velocity fields on the surface of the organism. The first contribution of the paper is the proof that such a system is generically controllable when the space of controlled velocity fields is at least three-dimensional. We also provide a complete characterization of controllable systems in the case in which the organism has a spherical shape. Finally, we offer a complete picture of controllable and non-controllable systems under the additional hypothesis that the organism and the fluid have densities of the same order of magnitude.