Controllability and Time Optimal Control for Low Reynolds Numbers Swimmers.

Lohéac, Jérôme; Scheid, Jean-François; Tucsnak, Marius

The aim of this paper is to tackle the self-propelling at low Reynolds number by using tools coming from control theory. More precisely we first address the controllability problem: 'Given two arbitrary positions, does it exist 'controls' such that the body can swim from one position to another, with null initial and final deformations?'. We consider a spherical object surrounded by a viscous incompressible fluid filling the remaining part of the three dimensional space. The object is undergoing radial and axi-symmetric deformations in order to propel itself in the fluid. Since we assume that the motion takes place at low Reynolds number, the fluid is governed by the Stokes equations. In this case, the governing equations can be reduced to a finite dimensional control system. By combining perturbation arguments and Lie brackets computations, we establish the controllability property. Finally we study the time optimal control problem for a simplified system. We derive the necessary optimality conditions by using the Pontryagin maximum principle. In several particular cases we are able to compute the explicit form of the time optimal control and to investigate the variation of optimal solutions with respect to the number of inputs.

REYNOLDS number; OPTIMAL control theory; STOKES equations; PONTRYAGIN classes; PERTURBATION theory; BOUNDARY value problems

Acta Applicandae Mathematicae, 2013, Vol 123, Issue 1, p175

0167-8019

Academic Journal

10.1007/s10440-012-9760-9