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Section: New Results

Modelization and Controllability of “Magneto-elastic” Micro-swimmers

Participants : François Alouges [École Polytechnique] , Antonio Desimone [SISSA Trieste] , Laetitia Giraldi, Pierre Lissy [Univ. Paris Dauphine] , Clément Moreau [ENS Cachan and York University] , Jean-Baptiste Pomet, Marta Zopello [Univ. di Padova] .

It is not realistic for artificial micro-swimmers built as micro-robots, to have an actuator at each joint. A possibility is as follows: each link of the swimmer bears is magnetized and the movement is controlled via an exterior magnetic field. These models also bear an internal elastic force, that can be modelled as a torsional spring at each joint and tends to asymptotically restore the straight shape in the absence of other forces.

Control strategies for these models have been proved successful numerically. It can also be proved mathematically via an asymptotic analysis that it is possible to steer the swimmer along a chosen direction with some well chosen oscillating magnetic field, provided some obstruction, like symmetries, are avoided. This is exposed in [23] for a Purcell magnetic swimmer (3 links).

For the smallest magneto-elastic micro-swimmer (2 links), we have been able to prove a strong local controllability result (weaker than STLC) around the straight position of the swimmer, again except for values of the parameters that correspond to symmetries preventing controllability. This is exposed in [8], and a note is under preparation, that shows that STLC is indeed not satisfied. This analysis is difficult because the straight position corresponds to the equilibria but is very degenerate from the control point of view.

To avoid this degeneracy, a possibility is to “twist” one of the torsional springs so that the equilibria no longer occur for a straight shape. This is exposed in [34] for a 3-link magnetic microswimmer (local controllability has not been proved for this system without the twist). A local partial controllability result around the equilibrium is proved in that case and a constructive method to find the magnetic field that allows the swimmer to move along a prescribed trajectory is described.