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 forwhich the inertia effects can be neglected. Our first main contribution is to prove thatany 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 ofarbitrarily small body deformations that can be superimposed to any preassigned sequenceof macro shape changes. Our second contribution is to prove that, when no macrodeformations are prescribed, tracking is generically possible by means of shape changesobtained as a suitable combination of only four elementary deformations. Eventually, stillconsidering finite dimensional deformations, we state results about the existence ofoptimal swimming strategies on short time intervals, for a wide class of costfunctionals.
