椭圆型限制性三体问题中计算极值曲线的通用伯克霍夫方法
Universal Birkhoff Method for Computing Extremals in the Elliptic Restricted Three-Body Problem
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中文总结 AI 辅助
该研究针对椭圆型限制性三体问题轨迹优化，结合通用伯克霍夫理论与快速无猜测谱算法，实现无需动力学系统辅助的可验证极值计算。
中文摘要 AI 辅助
研究椭圆型限制性三体问题轨迹优化中有限推力极值弧段的计算。采用切比雪夫-高斯-洛巴托点处状态向量采样值的快速傅里叶变换，将平动点轨道近似至接近机器精度。对最小时间和时间约束下的最小推进剂消耗问题应用庞特里亚金原理，推导可验证的最优性条件，包括最优出发与到达点判据。推进剂消耗采用近期开发的计算模型，该模型与推进剂比冲无关，随余弦项的平方倒数变化。结合轨迹优化的通用伯克霍夫理论与快速无猜测谱算法生成候选最优解，通过哈密顿极小化条件和横截条件验证伯克霍夫计算解的极值性。结果表明，该伯克霍夫理论谱算法无需动力学系统理论的任何辅助或初始化即可生成可验证的极值。
英文摘要
The computation of finite-thrust extremal arcs in the elliptic restricted three-body trajectory optimization problem is considered. Libration-point orbits are approximated to near-machine precision using a fast Fourier transform of the sampled values of the state vector at Chebyshev-Gauss-Lobatto points. Checkable optimality conditions are derived by applying Pontryagin's principle to minimum-time and time-constrained minimum-propellant problems. These necessary conditions include criteria for optimal departure and arrival points. For propellant consumption, a recently developed computational model is employed. This model is agnostic to the specific impulse of the propellant and varies as the inverse quadratic of a cosine term. Candidate optimal solutions are generated by combining the universal Birkhoff theory for trajectory optimization with the fast, guess-free spectral algorithm. The extremality of the Birkhoff-computed solution is validated against the Hamiltonian minimization condition and the transversality conditions. It is shown that the Birkhoff-theoretic spectral algorithm can generate verifiable extremals without any assistance or initialization from dynamical systems theory.
