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

详细信息    查看全文

• 作者：Zheng Chen
• 关键词：Circular restricted three ; body problem ; Low ; thrust ; \(\mathrm{L}^1\) ; minimization ; Conjugate points ; Sufficient optimality conditions
• 刊名：Celestial Mechanics & Dynamical Astronomy
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：126
• 期：4
• 页码：461-481
• 全文大小：1,492 KB
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Astronomy
  Mathematics
  
• 出版者：Springer Netherlands
• ISSN：1572-9478
• 卷排序：126

文摘

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.