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

Dante Kalise; Wenxin Liu

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arXiv · 2511.14355

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

Dante Kalise·Wenxin Liu

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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.

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Dante Kalise, Wenxin Liu. 2025-11-18. A PDE-constrained Optimization Approach to Optimal Trajectory Planning under Uncertainty via Reflected Schr\"odinger Bridges. https://arxiv.org/abs/2511.14355

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