Controllability and Optimal Strokes for N-link Micro-swimmer

Laetitia Giraldi (1) , Pierre Martinon (1, 2) , Marta Zoppello (3)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique
2 Commands - Control, Optimization, Models, Methods and Applications for Nonlinear Dynamical Systems
3 Dipartimento di Matematica Pura e Applicata [Padova]

In this paper we focus on the N-link swimmer, a generalization of the classical Purcell swimmer. We use the simplification of the Resistive Force Theory to derive the motion equation for the swimmer in a fluid with a low Reynolds number. We prove that the swimmer is controllable in the whole plane when it is composed by more than 3 sticks and for almost every set of sticks length. As a direct result, we show that there exist an optimal swimming strategy which leads to minimize the time to reach a configuration desired. Numerical experiments on the case of the Purcell swimmer suggest that the optimal strategy is periodic, i.e. composed of a sequence of identical strokes. Our results indicate that this candidate for an optimal stroke indeed gives a x-displacement twice better than the classical Purcell stroke.

Mots clés: Low Reynolds number swimming; Self-propulsion; controllability; optimal control

Domaines: Optimisation et contrôle [math.OC]

Laetitia Giraldi, Pierre Martinon, Marta Zoppello. Controllability and Optimal Strokes for N-link Micro-swimmer. 2013. ⟨hal-00798363v1⟩