  In this paper we study the locomotion of a shape-changing body swimming in a
two-dimensional perfect fluid of infinite extent. The shape-changes are
prescribed as functions of time satisfying constraints ensuring that they
result from the work of internal forces only: conditions necessary for the
locomotion to be termed self-propelled. The net rigid motion of the body
results from the exchange of momentum between these shape-changes and the
surrounding fluid.
  The aim of this paper is several folds: First, it contains a rigorous frame-
work for the study of animal locomotion in fluid. Our model differs from
previous ones mostly in that the number of degrees of freedom related to the
shape-changes is infinite. .
  Second, we are interested in making clear the connection between shape-
changes and internal forces. We prove that, when the number of degrees of
freedom relating to the shape-changes is finite, both choices are actually
equivalent in the sense that there is a one-to-one relation between
shape-changes and internal forces.
  Third, we show how the control problem consisting in associating to each
shape-change the resulting trajectory of the swimming body can be suitably
treated in the frame of geometric control theory. For any given shape-changes
producing a net displacement in the fluid (say, moving forward), we prove that
there exists other shape-changes arbitrarily close to the previous ones, that
leads to a completely different motion (for instance, moving backward): This
phenomenon will be called Moonwalking. Most of our results are illustrated by
numerical examples.

Optimization and Control