On the locomotion and control of a self-propelled shape-changing body in a fluid

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.