Controllability of 3D Low Reynolds Swimmers

In this article, we consider a swimmer (i.e. a self-deformable body) immersed in a fluid, the flow of which is governed by the stationary Stokes equations. This model is relevant for studying the locomotion of microorganisms or micro robots for which the inertia effects can be neglected. Our first main contribution is to prove that any such microswimmer has the ability to track, by performing a sequence of shape changes, any given trajectory in the fluid. We show that, in addition, this can be done by means of arbitrarily small body deformations that can be superimposed to any preassigned sequence of macro shape changes. Our second contribution is to prove that, when no macro deformations are prescribed, tracking is generically possible by means of shape changes obtained as a suitable combination of only four elementary deformations. Eventually, still considering finite dimensional deformations, we state results about the existence of optimal swimming strategies for a wide class of cost functionals.

💡 Research Summary

The paper addresses the fundamental problem of controllability for self‑deforming bodies (microswimmers) moving in a viscous fluid where inertial effects are negligible, i.e., the flow satisfies the stationary Stokes equations. After a concise motivation—microorganism locomotion and micro‑robotic applications—the authors introduce a rigorous mathematical model. The swimmer’s shape is described by a finite‑dimensional vector of shape parameters (q\in\mathbb{R}^m); a smooth deformation map (\chi(q,\cdot)) determines the instantaneous geometry of the body. The surrounding fluid obeys the Stokes system \