Solution of the Probabilistic Lambert Problem: Connections with Optimal Mass Transport, Schrödinger Bridge, and Reaction-Diffusion PDEs

Core Concepts

The probabilistic Lambert problem connects classical astrodynamics with modern stochastic control and machine learning through optimal mass transport and Schrödinger bridge theory.

Abstract

The content discusses the probabilistic variant of the Lambert problem in astrodynamics, where joint probability density functions replace endpoint constraints. It establishes connections between the probabilistic Lambert problem and optimal mass transport (OMT) as well as the Schrödinger bridge problem (SBP). The solution involves solving a system of reaction-diffusion partial differential equations (PDEs) with gravitational potential as the reaction rate. The existence and uniqueness of solutions are rigorously established for both the Lambertian OMT and SBP problems. The analysis leads to numerical algorithms that solve these problems using nonparametric computation methods.

Stats

Parameter µ = 398600.4415 km3/s2 denotes Earth's gravitational constant and mass. Parameter J2 = 1.08263 × 10−3 is the second zonal harmonic coefficient. Radius of Earth REarth = 6378.1363 km.

Quotes

"Our results show that just like problem (1.4) is a generalized OMT, the corresponding problem with (1.5) replaced by (1.6) is a generalized Schrödinger bridge problem." "We demonstrate that these newfound connections with the OMT and the SBP allow us to numerically solve the probabilistic Lambert problem using nonparametric computation." "The recent work also considers an SBP with killing or creation but with unbalanced endpoint marginals, which is not the case for us."

Key Insights Distilled From

Solution of the Probabilistic Lambert Problem

by Alexis M.H. ... at arxiv.org 03-19-2024

https://arxiv.org/pdf/2401.07961.pdf

Deeper Inquiries

How does incorporating stochastic process noise impact the feasibility and optimality of solutions in astrodynamics problems

Incorporating stochastic process noise in astrodynamics problems introduces additional uncertainties into the system. This can impact the feasibility and optimality of solutions by introducing randomness and variability that need to be accounted for in trajectory planning. The presence of process noise can lead to deviations from the expected path due to unpredictable disturbances, affecting the accuracy of spacecraft maneuvers. In terms of feasibility, the inclusion of stochastic process noise may require more robust control strategies to ensure that trajectories remain within acceptable bounds despite these uncertainties.

What implications do these findings have for spacecraft trajectory design in terms of efficiency and accuracy

The findings regarding stochastic process noise in astrodynamics have significant implications for spacecraft trajectory design. By considering probabilistic uncertainties and incorporating optimal mass transport theory, engineers can develop more efficient and accurate trajectory plans. Understanding how to navigate through uncertain environments with varying levels of noise allows for better decision-making processes when it comes to controlling spacecraft movements. This leads to improved efficiency in reaching desired destinations while maintaining a high level of precision in maneuvering.

How can insights from optimal mass transport theory be applied to other fields beyond astrodynamics

Insights from optimal mass transport theory are not limited solely to astrodynamics but can also be applied across various fields beyond this domain. Optimal mass transport provides a powerful framework for understanding how resources or information should be distributed most efficiently between different locations or states. This concept has applications in logistics, economics, data science, image processing, climate modeling, and many other areas where optimizing transportation or transformation processes is essential. By leveraging principles from optimal mass transport theory, researchers and practitioners can enhance their approaches to resource allocation and distribution problems across diverse disciplines.