Time optimal swimmers and Brockett integrator

The aim of this work is to compute time optimal controls for micro-swimmers. The action of swimming is seen as a control problem. More precisely, given an initial position and a target position, can the swimmer move to the target by changing its shape. The motion of the swimmer in the fluid results from the fluid-structure interaction. For micro-swimmers, the fluid equations in consideration are the stationnary Stokes equations. The way of swimming can be described by the following steps: 1. the swimmer modifies its shape, 2. this modification creates a velocity field in the fluid, 3. the fluid velocity acts on the swimmer as a force, 4. the fluid force moves the swimmer. Of course things are not so distinct and the swimming is a highly coupled nonlinear control problem. In this note, we present some key results for a fast numerical method to compute time optimal controls for axi-symmetric micro-swimmers. This numerical method is based on explicit formulae of time optimal controls for the Brockett integrator which is a system approaching the dynamic of the swimmer.