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Publiée le 06/04/11 à 15h00

Licence Creative Commons CC-By-NC

Researches on animal locomotion in fluid have now a long history. Focusing on the area of Mathematical Physics, the modeling leads to a system of PDEs (governing the fluid flow) coupled with a system of ODEs (driving the rigid motion of the immersed body). The first difficulty mathematicians came up against was to prove the well-posedness of such systems. This task was carried out in different papers.
Once the well-posedness of the fluid-swimmer dynamics has been established, the following step was to investigate its controllability. About this topic, still very few theoretical results are available: the authors prove that a 3D three-sphere mechanism, swimming along a straight line in a viscous fluid is controllable. We prove that a 2D shape changing body swimming in a potential flow can track approximately any given trajectory.
Some authors are rather interested in describing the dynamics of swimming in terms of Geometric Mechanics. We refer to articles for references in this area.
In this note, we consider a 3D shape changing body swimming in a potential flow. Under some symmetry assumptions (the swimmer is alone in the fluid and the fluid-swimmer system fills the whole space) we shall prove a generic controllability result, generalizing and improving what has been obtained for a particular 2D model.