  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.
