A PDE-constrained Optimization Approach to Optimal Trajectory Planning under Uncertainty via Reflected Schr\"odinger Bridges

TL;DR

This paper introduces a PDE-constrained optimization method for trajectory planning under uncertainty, utilizing a reflected Schrödinger Bridge formulation to efficiently compute optimal paths in complex geometries.

Contribution

It develops a novel PDE-based framework for trajectory optimization under uncertainty with reflecting boundary conditions, avoiding particle collision detection.

Findings

Demonstrates fast convergence in 3D maze scenarios

Ensures mass conservation in numerical solutions

Validates optimal controls via reflected SDE simulations

Abstract

A computational PDE-constrained optimization approach is proposed for optimal trajectory planning under uncertainty by means of an associated Schroedinger Bridge Problem (SBP). The proposed SBP formulation is interpreted as the mean-field limit associated with the energy-optimal evolution of a particle governed by a stochastic differential equation (SDE) with nonlinear drift and reflecting boundary conditions, constrained to prescribed initial and terminal densities. The resulting mean-field system consists of a nonlinear Fokker-Planck equation coupled with a Hamilton-Jacobi-Bellman equation, subject to two-point boundary conditions in time and Neumann boundary conditions in space. Through the Hopf-Cole transformation, this nonlinear system is recast as a pair of forward-backward advection-diffusion equations, which are amenable to efficient numerical solution via a standard finite element discretization. The weak formulation naturally enforces reflecting boundary conditions without requiring explicit particle-boundary collision detection, thus circumventing the computational difficulties inherent to particle-based methods in complex geometries. Numerical experiments on challenging 3D maze configurations demonstrate fast convergence, mass conservation, and validate the optimal controls computed through reflected SDE simulations.