On some Riemannian aspects of two and three-body controlled problems

Jean-Baptiste Caillau (1) , Bilel Daoud (1) , Joseph Gergaud (2)

1 IMB - Institut de Mathématiques de Bourgogne [Dijon]
2 IRIT - Institut de recherche en informatique de Toulouse

The flow of the Kepler problem (motion of two mutually attracting bodies) is known to be geodesic after the work of Moser [20], extended by Belbruno and Osipov [2, 21]: Trajectories are reparameterizations of minimum length curves for some Riemannian metric. This is not true anymore in the case of the three-body problem, and there are topological obstructions as observed by McCord et al. [19]. The controlled formulations of these two problems are considered so as to model the motion of a spacecraft within the influence of one or two planets. The averaged flow of the (energy minimum) controlled Kepler problem with two controls is shown to remain geodesic. The same holds true in the case of only one control provided one allows singularities in the metric. Some numerical insight into the control of the circular restricted three-body problem is also be given.

Mots clés: Two and three-body problems; geodesic flow; optimal control; cut and conjugate loci

Domaines: Optimisation et contrôle [math.OC]

Jean-Baptiste Caillau, Bilel Daoud, Joseph Gergaud. On some Riemannian aspects of two and three-body controlled problems. 2009. ⟨hal-00432635v1⟩