## Section: New Results

### Fully controlled slender microswimmers

#### The $N$-link micro-swimmer

Participants : François Alouges [École Polytechnique] , Antonio Desimone [SISSA Trieste] , Laetitia Giraldi, Marta Zopello [Univ. di Padova] .

We discussed a reduced model to compute the motion of slender swimmers which propel themselves by changing the curvature of their body. Our approach is based on the use of Resistive Force Theory for the evaluation of the viscous forces and torques exerted by the surrounding fluid, and on discretizing the kinematics of the swimmer by representing its body through an articulated chain of $N$ rigid links capable of planar deformations. The resulting system of ODEs, governing the motion of the swimmer, is easy to assemble and to solve, making our reduced model a valuable tool in the design and optimization of bio-inspired artificial microdevices. We prove that the swimmer composed by almost 3 segments is controllable in the whole plane. As a direct result, there exists an optimal swimming strategy to reach a desired configuration in minimum time. Numerical experiments for in the case of the Purcell swimmer suggest that the optimal strategy is periodic, namely a sequence of identical strokes. Our results indicate that this candidate for an optimal stroke, indeed gives a better displacement speed than the classical Purcell stroke.

This is presented in [36] (accepted as a Book chapter in Multi-scale Models in Mechano and Tumor Biology: Modeling, Homogenization and Applications, Lecture Notes in Computational Science and Engineering. June 2016).

#### Optimal periodic strokes for the Copepod and Purcell micro-swimmers

Participants : Piernicola Bettiol [Uni. Bretagne Ouest] , Bernard Bonnard, Alice Nolot, Jérémy Rouot.

We have analyzed the problem of optimizing the efficiency of the displacement of two micro swimmers with slender links, namely the following two models: the symmetric micro swimmer introduced by Takagi (see [29]); this is a model to describe the locomotion of the micro crustaceans named copepod, and the historical three link Purcell swimmer. The problems are studied in the framework of optimal control theory and SR geometry vs the standard curvature control point of view. Our contributions are to determine the optimal solutions combining geometric analysis and adapted numerical scheme. In particular the nilpotent models introduced in SR geometry allow to make a neat analysis of the problem of determining optimal strokes with small amplitudes and numerical continuation methods are then applied to compute more general stroke. This approach is completely original in optimal control. Also necessary and sufficient optimality conditions are applied to select the topology of optimal strokes (simple loops) and to determine the optimal solution in both cases. For the references see [17] and [27]. Also note that in collaboration with D. Takagi and M. Chyba this approach is currently at the experimental level at the university of Hawaii using a robot micro swimmer mimicking a copepod, see above. More theoretical issues in relation with SR geometry are investigated in the framework of A. Nolot's starting PhD (started August, 2016). Other publication relating these advances are [25], [26], [11].