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

Alexis M. H. Teter, Iman Nodozi, Abhishek Halder

Abstract

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 Schrödinger 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 Schrödinger 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.

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

Abstract

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 Schrödinger 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 Schrödinger 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.
Paper Structure (18 sections, 6 theorems, 76 equations, 5 figures)

This paper contains 18 sections, 6 theorems, 76 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Background on OMT and SBP
  3. Optimal Mass Transport (OMT)
  4. Schrödinger Bridge Problem (SBP)
  5. Lambertian Optimal Mass Transport
  6. Formulation
  7. Existence and uniqueness of Solution
  8. Stochastic regularization: Lambertian Schrödinger Bridge
  9. Formulation and existence-uniqueness of Solution
  10. Conditions for optimality
  11. Algorithm
  12. Overall Approach
  13. Integral representation for the Schrödinger factors
  14. Left Riemann sum approximation
  15. Numerical results
  16. ...and 3 more sections

Key Result

Theorem 3.3

\newlabelthm:existenceuniqueness0 Let $\rho_0,\rho_1\in\mathcal{P}_{2}(\mathbb{R}^{3})$, and the gravitational potential $V$ is negative and lower bounded as in defPotential. Then the minimizing tuple $\left(\rho^{\rm{opt}},\bm{v}^{\rm{opt}}\right)$ for problem LambertOT exists and is unique.

Figures (5)

  • Figure 1: Fixed point recursion over the endpoint Schrödinger factor pair $(\widehat{\varphi}_{\varepsilon,0},\varphi_{\varepsilon,1})$ to solve the boundary-coupled system of PDEs \ref{['FactorPDEs']}-\ref{['factorBC']}. The arrows with integrals denote marching the respective IVP solution over time.
  • Figure 1: Optimally controlled closed loop state sample paths for the numerical simulation in \ref{['sec:Numerical']}.
  • Figure 2: Sample paths for the components $v_x,v_y,v_z$ of the optimal control for the numerical simulation in \ref{['sec:Numerical']}.
  • Figure 3: The filled circles show the mean position snapshots for the 50 optimally controlled sample paths in $\mathbb{R}^3$, as detailed in \ref{['sec:Numerical']}. The translucent ellipsoids denote one standard deviation around each mean position.
  • Figure 4: Univariate $x,y,z$ marginals for the optimally controlled joint $\rho_{\varepsilon}^{\text{opt}}(\bm{r},t)$ at $t = 0, 0.25, 0.50, 0.75, 1$ hours.

Theorems & Definitions (20)

  • Remark 3.1
  • Definition 3.2: Superlinear function
  • Theorem 3.3: Existence-uniqueness of solution for L-OMT
  • Proof 1
  • Remark 3.4
  • Definition 4.1
  • Theorem 4.2
  • Proof 2
  • Remark 4.3
  • Proposition 4.4: Conditions for optimality for L-SBP
  • ...and 10 more