At the microscopic scale, locomotion through a fluid follows different laws than at our scale, due to the predominance of the viscous effects (Stokes regime). The research on swimming micro-organisms and the design of swimming micro-robots is motivated in particular by biomedical applications, such as non-invasive surgery or delivery of a molecule to a precise location within the body. Of course, this requires the ability to propel and control these micro-robots precisely and efficiently. Mathematical control theory offers a good theoretical framework to address these issues.
Thus, in this talk, I will present some results on controllability and control of microswimming robots. I will start by a short overview on controllability results for finite-dimensional systems. Then, moving towards applications, I will study a model of elastic robot propelled by an external magnetic field, and show that it is in general not controllable around its equilibrium position. Finally, I will focus on the control of a particle using the variation of the surrounding flow generated by another swimmer; I will discuss the controllability of the associated system and present a motion planning method.