Schrodinger Bridge over Averaged Systems

TL;DR

This paper develops a robust control framework for Schr"odinger bridge problems involving parameter-perturbed ensembles, revealing that optimal strategies are non-Markovian stochastic feedforward controls, with implications for robust distribution deformation.

Contribution

It introduces a novel approach to Schr"odinger bridge problems with parameter perturbations, showing that optimal controls are non-Markovian stochastic feedforward strategies.

Findings

Optimal control is a non-Markovian stochastic feedforward strategy.

Robust distribution deformation is achievable under parameter perturbations.

The approach extends optimal transport theory to perturbed systems.

Abstract

We consider a Schr\"odinger bridge problem where the Markov process is subject to parameter perturbations, forming an ensemble of systems. Our objective is to steer this ensemble from the initial distribution to the final distribution using controls robust to the parameter perturbations. Utilizing the path integral formalism, we demonstrate that the optimal control is a non-Markovian strategy, specifically a stochastic feedforward control, which depends on past and present noise. This unexpected deviation from established strategies for Schr\"odinger bridge problems highlights the intricate interrelationships present in the system's dynamics. From the perspective of optimal transport, a significant by-product of our work is the demonstration that, when the evolution of a distribution is subject to parameter perturbations, it is possible to robustly deform the distribution to a desired final state using stochastic feedforward controls.