Controlling artificial devices that mimic the motion of real microorganisms, is attracting increasing interest, both from the mathematical point of view and applications. A model for a magnetically driven slender micro-swimmer, mimicking a sperm cell is presented, supported by two examples showing how to steer it. Using the Resistive Force Theory (RTF) approach [J. Gray and J. Hancock, J. Exp. Biol. 32, 802 (1955)] to describe the hydrodynamic forces, the micro-swimmer can be described by a driftless affine control system where the control is an external magnetic field. Moreover we discuss through at first an asymptotic analysis and then by numerical simulations how to realize different kinds of paths
