Minimum fuel control of the planar circular restricted three-body problem

The circular restricted three-body problem is considered to model the dynamics of an artificial body submitted to the attraction of two planets. Minimization of the fuel consumption of the spacecraft during the transfer, e.g. from the Earth to the Moon, is considered. In the light of the controllability results of Caillau and Daoud (SIAM J Control Optim, 2012), existence for this optimal control problem is discussed under simplifying assumptions. Thanks to Pontryagin maximum principle, the properties of fuel minimizing controls is detailed, revealing a bang-bang structure which is typical of L-1-minimization problems. Because of the resulting non-smoothness of the Hamiltonian two-point boundary value problem, it is difficult to use shooting methods to compute numerical solutions (even with multiple shooting, as many switchings on the control occur when low thrusts are considered). To overcome these difficulties, two homotopies are introduced: One connects the investigated problem to the minimization of the L-2-norm of the control, while the other introduces an interior penalization in the form of a logarithmic barrier. The combination of shooting with these continuation procedures allows to compute fuel optimal transfers for medium or low thrusts in the Earth-Moon system from a geostationary orbit, either towards the L (1) Lagrange point or towards a circular orbit around the Moon. To ensure local optimality of the computed trajectories, second order conditions are evaluated using conjugate point tests.