Shooting and numerical continuation methods for computing time-minimal and energy-minimal trajectories in the Earth-Moon system using low propulsion

Gautier Picot

doi:10.3934/dcdsb.2012.17.245

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2012, Volume 17, Issue 1: 245-269. Doi: 10.3934/dcdsb.2012.17.245

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Shooting and numerical continuation methods for computing time-minimal and energy-minimal trajectories in the Earth-Moon system using low propulsion

Gautier Picot^1,

1.
Mathematics Institute, Bourgogne University, 9 avenue Savary, 21078 Dijon

Received: February 28, 2010

Revised: March 31, 2011

Published: December 2011

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Abstract

In this article we describe the principle of computations of optimal transfers between quasi-Keplerian orbits in the Earth-Moon system using low-propulsion. The spacecraft's motion is modelled by the equations of the control restricted 3-body problem and we base our work on previous studies concerning the orbit transfer in the two-body problem where geometric and numeric methods were developed to compute optimal solutions. Using numerical simple shooting and continuation methods connected with fundamental results from control theory, such as the Pontryagin Maxium Principle and the second order optimality conditions related to the concept of conjugate points, we compute time-minimal and energy-minimal trajectories between the geostationary initial orbit and a final circular orbit around the Moon, passing through the neighborhood of the libration point $L_1$. Our computations give simple trajectories, obtained by referring to numerical values of the SMART-1 mission.

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Mathematics Subject Classification: Primary: 49K15, 49M05; Secondary: 70F07.

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