Controllability and Time Optimal Control for Low Reynolds Numbers Swimmers
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Controllability and Time Optimal Control for Low Reynolds Numbers Swimmers Jérôme Lohéac · Jean-François Scheid · Marius Tucsnak
Received: 26 October 2011 / Accepted: 20 May 2012 / Published online: 2 June 2012 © Springer Science+Business Media B.V. 2012
Abstract 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. Keywords Stokes equations · Fluid-structure interaction · Controllability · Time optimal control · Low Reynolds number swimming J. Lohéac · J.-F. Scheid () · M. Tucsnak Institut Élie Cartan de Nancy, UMR 7502, Université de Lorraine, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France e-mail: [email protected] J. Lohéac e-mail: [email protected] M. Tucsnak e-mail: [email protected] J. Lohéac · J.-F. Scheid · M. Tucsnak Institut Élie Cartan de Nancy, UMR 7502, CNRS, B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France J. Lohéac · J.-F. Scheid · M. Tucsnak Inria, Villers-lès-Nancy, 54600, France
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Mathematics Subject Classification 93C10 · 93C20 · 93C15 · 93C73 · 93C95 · 76D07
1 Introduction Understanding the motion of solids (such as aquatic microorganisms and micro or nano swimming robots) is a challenging issue since the propelling mechanism should be adapted to very low Reynolds numbers, i.e., it should be essentially based on friction forces, with no role of inertia (which is essential for the swimming mechanism of macroscopic objects, such as fish like swimming). Among the early contributions to the modeling and analysis of these phenomena, we mention the seminal works by Taylor [21], Lighthill [14, 15], and Childress [6]. In several relatively more recent papers (see, for instance, Shapere and Wilczek [20], San Martin Takahashi and Tucsnak [19], Alouges, DeSimone and Lefebvre [2], Alouges, DeSimone and Heltai [3] and Lauga and Michelin [16]) the self-propelling at low Reynolds number has be
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