# L$^1$-optimality conditions for the circular restricted three-body problem

1 McTAO - Mathematics for Control, Transport and Applications
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In this paper, the L1-minimization for the translational motion of a spacecraft in the circular restricted three-body problem (CRTBP) is considered. Necessary conditions are derived by using the Pontryagin Maximum Principle (PMP), revealing the existence of bang-bang and singular controls. Singular extremals are analyzed, recalling the existence of the Fuller phenomenon according to the theories developed in (Marchal in J Optim Theory Appl 11(5):441–486, 1973; Zelikin and Borisov in Theory of Chattering Control with Applications to Astronautics, Robotics, Economics, and Engineering. Birkhäuser, Basal 1994; in J Math Sci 114(3):1227–1344, 2003). The sufficient optimality conditions for the L1-minimization problem with fixed endpoints have been developed in (Chen et al. in SIAM J Control Optim 54(3):1245–1265, 2016). In the current paper, we establish second-order conditions for optimal control problems with more general final conditions defined by a smooth submanifold target. In addition, the numerical implementation to check these optimality conditions is given. Finally, approximating the Earth-Moon-Spacecraft system by the CRTBP, an L1-minimization trajectory for the translational motion of a spacecraft is computed by combining a shooting method with a continuation method in (Caillau et al. in Celest Mech Dyn Astron 114:137–150, 2012; Caillau and Daoud in SIAM J Control Optim 50(6):3178–3202, 2012). The local optimality of the computed trajectory is asserted thanks to the second-order optimality conditions developed.
Keywords :
Document type :
Journal articles

https://hal.archives-ouvertes.fr/hal-01402647
Contributor : Zheng Chen <>
Submitted on : Thursday, November 24, 2016 - 11:23:14 PM
Last modification on : Thursday, February 7, 2019 - 4:41:59 PM

### Citation

Zheng Chen. L$^1$-optimality conditions for the circular restricted three-body problem. Celestial Mechanics and Dynamical Astronomy, Springer Verlag, 2016, 126 (4), pp.461-481. ⟨10.1007/s10569-016-9703-2⟩. ⟨hal-01402647⟩

Record views