$L^1$-optimality conditions for circular restricted three-body problems

TL;DR

This paper develops second-order optimality conditions for L1-minimization in the circular restricted three-body problem, providing theoretical insights and numerical methods to compute and verify optimal spacecraft trajectories.

Contribution

It introduces new second-order sufficient conditions for L1-optimality in CRTBP, extending previous fixed endpoint results to submanifolds, and demonstrates their application through numerical trajectory optimization.

Findings

Derived necessary conditions using Pontryagin Maximum Principle.

Established second-order sufficient conditions for optimality.

Computed and verified a spacecraft trajectory in Earth-Moon system.

Abstract

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.