The Lambert problem originated in orbital mechanics. It concerns with determining the initial velocity for a boundary value problem involving the dynamical constraint due to gravitational potential with additional time horizon and endpoint position constraints. Its solution has application in transferring a spacecraft from a given initial to a given terminal position within prescribed flight time via velocity control. 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 a generalized SBP, and can be solved numerically using the so-called Schr\"odinger factors, as we do in this work. Our analysis leads to solving a system of reaction-diffusion PDEs where the gravitational potential appears as the reaction rate.

翻译：Lambert问题起源于轨道力学。该问题关注在包含引力势动力学约束的边值问题中，结合给定的时间范围与端点位置约束，确定初始速度。其解可用于通过速度控制在规定飞行时间内将航天器从给定初始位置转移至指定终端位置。我们考虑Lambert问题的一个概率变体，其中端点位置向量的确定性约束被替换为已知其联合概率密度函数。我们证明具有端点联合概率密度约束的Lambert问题是一个广义最优质量输运问题，从而将这一经典航天动力学问题与现代随机控制和随机机器学习的新兴研究领域联系起来。这一新发现的关联使我们能够严格证明概率Lambert问题解的存在唯一性。该关联同时有助于通过扩散正则化数值求解概率Lambert问题，即借助最优质量输运与薛定谔桥问题的进一步联系。这也表明具有加性动态过程噪声的概率Lambert问题是一个广义薛定谔桥问题，并可采用所谓的薛定谔因子进行数值求解，正如本文所实现的方法。我们的分析最终归结为求解一个反应-扩散偏微分方程组，其中引力势以反应速率的形式出现。