In recent years it has been shown that low Reynolds number swimming can be considered as a control problem which is linear in the control, and without drift.
We focus on the system of the N-link swimmer, a generalization of the classical Purcell swimmer , that was introduced in . One of the main difficulties in exploiting Control Theory in order to solve effectively motion planning or control problems is the complexity of the hydrodynamic forces exerted by the fluid on the swimmer as a reaction to its shape changes. Whereas the N-link swimmer model introduced is simplified and explicit since Resistive Force Theory is used to couple the fluid and the swimmer.
After proving that the swimmer is controllable in the whole plane when it is composed by more than 3 links, we address the minimum time problem to reach a target position. Moreover adding elasticity and magnetism we present a model for a magnetically driven slender micro-swimmer, mimicking a sperm cell (see ) . The micro-swimmer can be described by a driftless affine control system where the control is an external magnetic field. Moreover we discuss through numerical simulations how to realize different kind of paths.
 F. Alouges, A. DeSimone, L. Giraldi, and M. Zoppello. Self-propulsion of slender micro- swimmers by curvature control: N-link swimmers. to appear in International Journal of Non-Linear Mechanics, (2013).
 L. Giraldi, P. Martinon, M. Zoppello. Controllability and Optimal Strokes for N-link Microswimmer. Preprint in Hal (hal-00798363), CDC 2013.
 L. Giraldi, P. Martinon, M. Zoppello. Optimal Design of the Three-link Purcell Swim- mer. .Phys. Rev. E 91, 023012 (2015)
 F. Alouges, A. DeSimone, L. Giraldi, and M. Zoppello. Can magnetic multilayers propel micro-swimmers mimicking sperm cells? Soft Robotics, 2(3):117–128, (2015).
 E. M. Purcell: Life at low Reynolds number. American Journal of Physics, 45 (1977), 3–11.
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