Lambert's problem concerns with transferring a spacecraft from a given initial to a given terminal position within prescribed flight time via velocity control subject to a gravitational force field. We consider a probabilistic variant of the Lambert problem where the knowledge of the endpoint constraints in position vectors are replaced by the knowledge of their respective joint probability density functions. We show that the Lambert problem with endpoint joint probability density constraints is a generalized optimal mass transport (OMT) problem, thereby connecting this classical astrodynamics problem with a burgeoning area of research in modern stochastic control and stochastic machine learning. This newfound connection allows us to rigorously establish the existence and uniqueness of solution for the probabilistic Lambert problem. The same connection also helps to numerically solve the probabilistic Lambert problem via diffusion regularization, i.e., by leveraging further connection of the OMT with the Schr\"odinger bridge problem (SBP). This also shows that the probabilistic Lambert problem with additive dynamic process noise is in fact a generalized SBP, and can be solved numerically using the so-called Schr\"odinger factors, as we do in this work. We explain how the resulting analysis leads to solving a boundary-coupled system of reaction-diffusion PDEs where the nonlinear gravitational potential appears as the reaction rate. We propose novel algorithms for the same, and present illustrative numerical results. Our analysis and the algorithmic framework are nonparametric, i.e., we make neither statistical (e.g., Gaussian, first few moments, mixture or exponential family, finite dimensionality of the sufficient statistic) nor dynamical (e.g., Taylor series) approximations.

这篇研究论文介绍了Lambert问题的概率变种，将位置矢量的端点约束转化为各自联合概率密度函数的知识，并通过广义最优质量传输问题与现代随机控制和随机机器学习领域的研究相联系，从而严格建立了概率Lambert问题解的存在和唯一性，并提出了用扩散正则化解决这个问题的算法框架。

概率Lambert问题的解决方案：与最优质量运输，薛定谔桥和反应扩散偏微分方程的关联