  In this paper, the L1-minimization for the translational motion of a
spacecraft in a circular restricted three-body problem (CRTBP) is considered.
Necessary con- ditions are derived by using the Pontryagin Maximum Principle,
revealing the existence of bang-bang and singular controls. Singular extremals
are detailed, re- calling the existence of the Fuller phenomena according to
the theories developed by Marchal in Ref. [14] and Zelikin et al. in Refs. [12,
13]. The sufficient opti- mality conditions for the L1-minimization problem
with fixed endpoints have been solved in Ref. [22]. In this paper, through
constructing a parameterised family of extremals, some second-order sufficient
conditions are established not only for the case that the final point is fixed
but also for the case that the final point lies on a smooth submanifold. In
addition, the numerical implementation for the optimality conditions is
presented. Finally, approximating the Earth-Moon-Spacecraft system as a CRTBP,
an L1-minimization trajectory for the translational motion of a spacecraft is
computed by employing a combination of a shooting method with a continuation
method of Caillau et al. in Refs. [4, 5], and the local optimality of the
computed trajectory is tested thanks to the second-order optimality conditions
established in this paper.
