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.

翻译：兰伯特问题涉及通过在引力场作用下进行速度控制，在预定飞行时间内将航天器从给定初始位置转移到给定终端位置。我们考虑兰伯特问题的一个概率变体，其中端点位置向量的约束知识被其各自的联合概率密度函数所替代。我们证明，具有端点联合概率密度约束的兰伯特问题是一个广义最优质量传输（OMT）问题，从而将这一经典天体动力学问题与现代随机控制和随机机器学习领域的新兴研究方向联系起来。这一新发现的联系使我们能够严格确立概率兰伯特问题解的存在性和唯一性。同样，这种联系也有助于通过扩散正则化（即进一步利用OMT与薛定谔桥问题（SBP）的联系）来数值求解概率兰伯特问题。这也表明，具有加性动态过程噪声的概率兰伯特问题实际上是一个广义SBP，并可以通过所谓的薛定谔因子进行数值求解，正如我们在本文中所做的那样。我们解释了由此产生的分析如何导致求解一个边界耦合的反应扩散偏微分方程组，其中非线性引力势作为反应速率出现。我们为此提出了新颖算法，并展示了说明性数值结果。我们的分析和算法框架是非参数化的，即我们既不做统计假设（如高斯分布、前几阶矩、混合或指数族、充分统计量的有限维度），也不做动力学近似（如泰勒级数）。