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

Abstract

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.

Periodical Body Deformations are Optimal Strategies for Locomotion

“…However what we usually understand by locomotion is a less precise motion and can be described as "move in the x-direction". In that case one has to choose a different criterion and the strategy that is usually proposed is to minimize the mean-time T /x(T ) (or equivalently to maximize the average velocity x(T )/T or the efficiency x(T ) 2 /E(X(•)), see [4,25]. The difficulty with such a criterium is that it does not admit optimal solutions in general, thus we propose to model locomotion strategies for arbitrary large displacements as follows.…”

“…Even if this paper focuses on a generic simple systems, the same type of dynamics classically governs the displacement of micro-swimmer at low Reynolds number. Numerous models of micro-robot are expressed by a similar dynamics, for instance the Copepod model [4,25] the spherical one studied in [15], the Three-sphere swimmer [1,2,20], ciliate model [16] and others fit this framework. Similarly, the displacements of micro-crawlers, which derive their propulsion capabilities from tangential resistance offered by the substrate, are also governed by simple equations in agreement with our framework [8,21].…”