  The deterministic variant of the Lambert's problem was posed by Lambert in
the 18th century and its solution for conic trajectory has been derived by
many, including Euler, Lambert, Lagrange, Laplace, Gauss and Legendre. The
solution amounts to designing velocity control for steering a spacecraft from a
given initial to a given terminal position subject to gravitational potential
and flight time constraints. In recent years, a probabilistic variant of the
Lambert's problem has received attention in the aerospace community where the
endpoint position constraints are softened to endpoint joint probability
distributions over the respective positions. Such probabilistic specifications
account for the estimation errors, modeling uncertainties, etc. Building on a
deterministic optimal control reformulation via analytical mechanics, we show
that the probabilistic Lambert's problem is a generalized dynamic optimal mass
transport problem where the gravitational potential plays the role of an
additive state cost. This allows us to rigorously prove the
existence-uniqueness of the solution for the probabilistic Lambert problem both
with and without process noise. In the latter case, the problem and its
solution correspond to a generalized Schr\"odinger bridge, much like how
classical Schrodinger bridge can be seen as stochastic regularization of the
optimal mass transport. We deduce the large deviation principle enjoyed by the
Lambertian Schr\"odinger bridge. Leveraging these newfound connections, we
design a computational algorithm to illustrate the nonparametric numerical
solution of the probabilistic Lambert's problem.
