Optimal control theory, sub-Riemannian geometry and swimming of copepod

In [17], based on the observation of copepods, Takagi proposed a model to analyze the swimming of microorganisms using sinusoidal paddling or sequential paddling followed by a recovery stroke in unison, and they are compared with the concept of efficiency. Our aim is to provide an interpretation 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 repa-rameterization 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.