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Section: New Results

Optimal control of fully actuated micro-swimmers

A general geometric approach of optimal strokes for driftless micro-swimmers

Participants : Thomas Chambrion [Univ. Lorraine] , Laetitia Giraldi, Alexandre Munnier [Univ. Lorraine] .

In [3], we study the control problem associated to the locomotion of a deformable swimmer. we present a unified geometric approach for optimization of the body deformation of the swimmers in a 3D Stokes flow (case of micro-swimmers) and 2D or 3D potential flow. The latter cases correspond to the analysis of the sphere in a sub-Riemannian space. A general framework is introduced, allowing the complete analysis of five usual nonlinear optimization problems to be carried out. The results are illustrated with examples and with a in-depth study of a swimmer in a 2D potential flow. Numerical tests are also provided.

Optimal periodic strokes for the Copepod and Purcell micro-swimmers

Participants : Piernicola Bettiol [Uni. Bretagne Ouest] , Bernard Bonnard, Alice Nolot, Jérémy Rouot.

We have analyzed the problem of optimizing the efficiency of the displacement of two micro swimmers with slender links, namely the following two models: the symmetric micro swimmer introduced by Takagi (see [43], this model describes the locomotion of the micro crustaceans named copepod), and the historical three link Purcell swimmer. The problems are studied in the framework of optimal control theory and SR geometry vs the standard curvature control point of view. Our contribution is to determine the optimal solutions combining geometric analysis and adapted numerical scheme. In particular the nilpotent models introduced in SR geometry allow to make a neat analysis of the problem of determining optimal strokes with small amplitudes and numerical continuation methods are then applied to compute more general stroke. This approach is completely original in optimal control. Also necessary and sufficient optimality conditions are applied to select the topology of optimal strokes (simple loops) and to determine the optimal solution in both cases, see [16]. Also note that in collaboration with D. Takagi and M. Chyba, this approach is currently at the experimental level at the university of Hawaii using a robot micro swimmer mimicking a copepod, see above. More theoretical issues in relation with SR geometry are investigated in the framework of A. Nolot's starting PhD (started August, 2016) and K. Sérier's PhD (started September, 2017), see [10], [42] and other publications under review.