Low-thrust Lyapunov to Lyapunov and Halo to Halo missions with L²-minimization^∗

¹ Sorbonne Universités, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France.
² Airbus Defence and Space, 66 route de Verneuil, BP 3002, 78133 Les Mureaux cedex, France.
chupin@ljll.math.upmc.fr; emmanuel.trelat@upmc.fr
³ Université d’Orléans, Fédération Denis Poisson, Laboratoire MAPMO, UMR CNRS 7349, route de Chartres, 45067 Orléans cedex 2, France.
thomas.haberkorn@univ-orleans.fr

Received: 7 October 2015
Revised: 6 June 2016
Accepted: 8 June 2016

Abstract

In this work, we develop a new method to design energy minimum low-thrust missions (L²-minimization). In the Circular Restricted Three Body Problem, the knowledge of invariant manifolds helps us initialize an indirect method solving a transfer mission between periodic Lyapunov orbits. Indeed, using the PMP, the optimal control problem is solved using Newton-like algorithms finding the zero of a shooting function. To compute a Lyapunov to Lyapunov mission, we first compute an admissible trajectory using a heteroclinic orbit between the two periodic orbits. It is then used to initialize a multiple shooting method in order to release the constraint. We finally optimize the terminal points on the periodic orbits. Moreover, we use continuation methods on position and on thrust, in order to gain robustness. A more general Halo to Halo mission, with different energies, is computed in the last section without heteroclinic orbits but using invariant manifolds to initialize shooting methods with a similar approach.