Geometric and numerical methods in optimal control and applications to the swimming problem at low Reynolds number and to low thrust orbital transfer

Jérémy Rouot 1
1 McTAO - Mathematics for Control, Transport and Applications
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : The first part of this work is devoted to the study of the swimming at low Reynolds number where we consider a2-link swimmer to model the motion of a Copepod and the seminal model of the Purcell Three-link swimmer. Wepropose a geometric and numerical approach using optimal control theory assuming that the motion occursminimizing the energy dissipated by the drag fluid forces related with a concept of efficiency of a stroke. TheMaximum Principle is used to compute periodic controls considered as minimizing control using propertransversality conditions, in relation with periodicity, minimizing the energy dissipated for a fixed displacement ormaximizing the efficiency of a stroke. These problems fall into the framework of sub-Riemannian geometry whichprovides efficient techniques to tackle these problems : the nilpotent approximation is used to compute strokeswith small amplitudes which are continued numerically for the true system. Second order optimality, necessary orsufficient, are presented to select weak minimizers in the framework of periodic optimal controls.In the second part, we study the motion of a controlled spacecraft in a central field taking into account thegravitational interaction of the Moon and the oblateness of the Earth. Our purpose is to study the time minimalorbital transfer problem with low thrust. Due to the small control amplitude, our approach is to define anaveraged system from the Maximum Principle and study the related approximations to the non averaged system.We provide proofs of convergence and give numerical results where we use the averaged system to solve the nonaveraged system using indirect method
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Jérémy Rouot. Geometric and numerical methods in optimal control and applications to the swimming problem at low Reynolds number and to low thrust orbital transfer. Other. Université Côte d'Azur, 2016. English. ⟨NNT : 2016AZUR4103⟩. ⟨tel-01472370v2⟩

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