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Lambert's problem concerns with transferring a spacecraft from a giveninitial to a given terminal position within prescribed flight time via velocitycontrol subject to a gravitational force field. We consider a probabilisticvariant of the Lambert problem where the knowledge of the endpoint constraintsin position vectors are replaced by the knowledge of their respective jointprobability density functions. We show that the Lambert problem with endpointjoint probability density constraints is a generalized optimal mass transport(OMT) problem, thereby connecting this classical astrodynamics problem with aburgeoning area of research in modern stochastic control and stochastic machinelearning. This newfound connection allows us to rigorously establish theexistence and uniqueness of solution for the probabilistic Lambert problem. Thesame connection also helps to numerically solve the probabilistic Lambertproblem via diffusion regularization, i.e., by leveraging further connection ofthe OMT with the Schr\"odinger bridge problem (SBP). This also shows that theprobabilistic Lambert problem with additive dynamic process noise is in fact ageneralized SBP, and can be solved numerically using the so-calledSchr\"odinger factors, as we do in this work. We explain how the resultinganalysis leads to solving a boundary-coupled system of reaction-diffusion PDEswhere the nonlinear gravitational potential appears as the reaction rate. Wepropose novel algorithms for the same, and present illustrative numericalresults. Our analysis and the algorithmic framework are nonparametric, i.e., wemake neither statistical (e.g., Gaussian, first few moments, mixture orexponential family, finite dimensionality of the sufficient statistic) nordynamical (e.g., Taylor series) approximations.

Solution of the Probabilistic Lambert Problem: Connections with Optimal Mass Transport, Schr\"odinger Bridge and Reaction-Diffusion PDEs