## Section: New Results

### Swimming at low Reynolds number an optimal control problem

Participants : François Alouges [École Polytechnique] , Luca Berti, Antonio Desimone [SISSA Trieste] , Yacine El Alaoui-Faris, Laetitia Giraldi, Yizhar Or [Technion, Israel] , Christophe Prud'Homme [Univ. de Strasbourg] , Jean-Baptiste Pomet, Stéphane Régnier [Sorbonne Université] , Oren Wiezel [Technion, Israel] .

This part is devoted to study the displacement of micro-swimmers. We attack this problem using numerical tools and optimal control theory. Micro-scale swimmers move in the realm of negligible inertia, dominated by viscous drag forces, the fluid is governed by the Stokes equation. We study two types of models. First, deriving from the PDE system, in [5] we use Feel$++$, a finite elements library in order to simulate the motion of a one-hinged swimmer, which obeys to the scallop theorem. Then, we adress the flagellar microswimmers. In [31] we formulate the leading order dynamics of a $2D$ slender multi-link microswimmer assuming small-amplitude undulations about its straight configuration. The energy optimal stroke to achieve a given prescribed displacement in a given time period is obtained as the largest eigenvalue solution of a constrained optimal control problem. We prove that the optimal stroke is an ellipse lying within a two-dimensional plane in the $(N-1)$ dimensional space of shape variables, where $N$ can be arbitrarily large. If the number of shape variables is small, we can consider the same problem when the prescribed displacement in one time period is large, and not attainable with small variations of the joint angles. The fully non-linear optimal control problem is solved numerically for the cases $N=3$ and $N=5$ showing that, as the prescribed displacement becomes small, the optimal solutions obtained using the small-amplitude assumption are recovered. We also show that, when the prescribed displacements become large, the picture is different. Finally, in [28] we present an automated procedure for the design of optimal actuation for flagellar magnetic microswimmers based on numerical optimization. Using this method, a new magnetic actuation method is provided which allows these devices to swim significantly faster compared to the usual sinusoidal actuation. This leads to a novel swimming strategy which shows that a faster propulsion is obtained when the swimmer is allowed to go out-of-plane. This approach is experimentally validated on a scaled-up flexible swimmer.