6533b872fe1ef96bd12d3a9b

RESEARCH PRODUCT

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

Bilel DaoudJean-baptiste CaillauJoseph Gergaud

subject

[ MATH.MATH-OC ] Mathematics [math]/Optimization and Control [math.OC]Work (thermodynamics)Geodesic010102 general mathematicsMathematical analysisMotion (geometry)[MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]Optimal control01 natural sciencesOptimal controlsymbols.namesakeFlow (mathematics)Kepler problemCut and conjugate loci0103 physical sciencesMetric (mathematics)symbolsGeodesic flowTwo and three-body problems49K15 53C20 70Q05Gravitational singularity[MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC]0101 mathematics010303 astronomy & astrophysicsMathematics

description

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.

https://hal.archives-ouvertes.fr/hal-00432635v2/file/caillau.pdf