Optimal control theory and the efficiency of the swimming mechanism of the Copepod Zooplankton

International audienceIn this article, the model of swimming at low Reynolds number introduced by D. Takagi (2015) to analyze the displacement of an abundant variety of zooplankton is used as a testbed to analyze the motion of symmetric microswimmers in the framework of optimal control theory assuming that the motion occurs minimizing the energy dissipated by the fluid drag forces in relation with the concept of efficiency of a stroke. The maximum principle is used to compute periodic controls candidates as minimizing controls and is a decisive tool combined with appropriate numerical simulations using indirect optimal control schemes to determine the most efficient stroke compared with standard computations using Stokes theorem and curvature control. Also the concept of graded approximations in SR-geometry is used to evaluate strokes with small amplitudes providing a fixed displacement and minimizing the dissipated energy

Optimal Control Theory and the Swimming Mechanism of the Copepod Zooplankton

In this article, the model of swimming at low Reynolds number introduced by D. Takagi (2015) to analyze the displacement of an abundant variety of zooplankton is used as a testbed to analyze the motion of symmetric microswimmers in the framework of optimal control theory assuming that the motion occurs minimizing the energy dissipated by the fluid drag forces in relation with the concept of efficiency of a stroke. The maximum principle is used to compute periodic controls candidates as minimizing controls and is a decisive tool combined with appropriate numerical simulations using indirect optimal control schemes to determine the most efficient stroke compared with standard computations using Stokes theorem and curvature control. Also the concept of graded approximations in SR-geometry is used to evaluate strokes with small amplitudes providing a fixed displacement and minimizing the dissipated energy

Sub-Riemannian geometry and swimming at low Reynolds number: the Copepod case

International audienceBased on copepod observations, Takagi proposed a model to interpret the swimming behavior of these microorganisms using sinusoidal paddling or sequential paddling followed by a recovery stroke in unison, and compares them invoking the concept of efficiency. Our aim is to provide an interpretation of Takagi’s results in the frame of optimal control theory and sub-Riemannian geometry. The Maximum principle is used to select two types of periodic control candidates as minimizers: sinusoidal up to time reparameterization and the sequential paddling, interpreted as an abnormal stroke in sub-Riemannian geometry. Geometric analysis combined with numerical simulations are decisive tools to compute the optimal solutions, refining Takagi computations. A family of simple strokes with small amplitudes emanating from a center is characterized as an invariant of SR-geometry and allow to identify the metric used by the swimmer. The notion of efficiency is discussed in detail and related with normality properties of minimizers.A partir de l'observation du mécanisme de nage d'une famille de microorganismes appelées copépodes, Takagi propose un modèle pour interpréter ces nages. Deux types de contrôles périodiques associés sont proposés : le premier correspond à des contrôles sinusoïdaux formant une courbe de nage lisse et simple et le second est un contrôle constant par morceaux produisant untriangle. Notre objectif est d'interpréter cela dans le cadre du contrôle optimal et de la géométrie sous-Riemannienne. Le principe du Maximum est utilisé pour sélectionner des nages géodésiques de deux types : des nages formant des courbes simples associées à des contrôles sinusoïdaux à une reparamétrisation du temps près et le triangle est interprété comme une nage anormale. L'analysegéométrique combinée avec des simulations numériques permet de générer une famille de nages depetites amplitudes ce qui par continuation permet de calculer la nage la plus efficace. La notion d'efficacité est discutée en détails en relation avec le concept de normalité