OPTIMAL STROKES AT LOW REYNOLDS NUMBER: A GEOMETRIC AND NUMERICAL STUDY OF COPEPOD AND PURCELL SWIMMERS

摘要

In this article we provide a comparative geometric and numerical analysis of optimal strokes for two different rigid links swimmer models at low Reynolds number: the copepod swimmer (a symmetric swimmer recently introduced by Takagi) and the long-standing three-link Purcell swimmer by Purcell. The design of strokes satisfying standard performance criteria leads one to investigate optimal control problems which can be analyzed in the framework of subRiemannian geometry. In this context nilpotent approximations allow one to compute strokes with small amplitudes, which in turn can be used numerically to obtain more general strokes. For the copepod model a detailed analysis of both abnormal and normal strokes is also described. First and second order optimality conditions, combined with numerical analysis, allow us to detect optimal strokes for both the copepod and the Purcell swimmers. C-1-optimality is investigated using the concept of conjugate point. Direct and indirect numerical schemes are implemented in Bocop and HamPath software to perform numerical simulations, which are crucial to complete the theoretical study and evaluate the optimal solutions.