Solution of the Probabilistic Lambert's Problem: Optimal Transport Approach
Alexis M. H. Teter, Iman Nodozi, Abhishek Halder
TL;DR
The paper reframes Lambert's problem with probabilistic endpoint constraints as a generalized dynamic optimal transport (Lambertian OMT) problem with cost $\int_{t_0}^{t_1} \int_{\mathbb{R}^3} \left(\frac{1}{2}|\mathbf{v}|^2 - V(\mathbf{r})\right)\rho \; d\mathbf{r} dt$ subject to the Liouville constraint $\partial_t \rho + \nabla_{\mathbf{r}} \cdot (\rho \mathbf{v}) = 0$, $\rho(t_0)=\rho_0$, $\rho(t_1)=\rho_1$. It shows existence, uniqueness, and inverse optimality of the probabilistic solution (LambertOT), and extends to a noisy setting as the Lambertian Schrödinger Bridge (L-SBP) with $\varepsilon>0$, proving uniqueness via a large deviations perspective. The optimality conditions form coupled nonlinear PDEs in $\psi$ and $\rho$, which are linearized via the Hopf–Cole transform into linear reaction-diffusion equations for Schrödinger factors $\widehat{\varphi}_{\varepsilon}$ and $\varphi_{\varepsilon}$, obeying $\partial_t \widehat{\varphi}_{\varepsilon} = (\varepsilon \Delta + \tfrac{1}{2\varepsilon} V) \widehat{\varphi}_{\varepsilon}$ and $\partial_t \varphi_{\varepsilon} = - (\varepsilon \Delta + \tfrac{1}{2\varepsilon} V) \varphi_{\varepsilon}$ with endpoint conditions $\widehat{\varphi}_{\varepsilon}(\cdot,t_0) \varphi_{\varepsilon}(\cdot,t_0) = \rho_0$, $\widehat{\varphi}_{\varepsilon}(\cdot,t_1) \varphi_{\varepsilon}(\cdot,t_1) = \rho_1$. A contractive forward–backward recursion computes the factors and hence $\rho$ and $\mathbf{v}$, with the scheme supported by a large-deviation principle; numerical demonstration provides a nonparametric solution and links classical orbital transfer to modern transport theory.
Abstract
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 Schrödinger 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 Schrödinger bridge. Leveraging these newfound connections, we design a computational algorithm to illustrate the nonparametric numerical solution of the probabilistic Lambert's problem.