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\(\mathrm{L}^1\)-optimality conditions for the circular restricted three-body problem

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Abstract

In this paper, the \(\mathrm{L}^1\)-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 \(\mathrm{L}^1\)-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 \(\mathrm{L}^1\)-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.

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Acknowledgments

The author is funded by China Scholarship Council (Grant No. 201306290024).

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Correspondence to Zheng Chen.

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Chen, Z. \(\mathrm{L}^1\)-optimality conditions for the circular restricted three-body problem. Celest Mech Dyn Astr 126, 461–481 (2016). https://doi.org/10.1007/s10569-016-9703-2

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