Stochastic Bridges over Ensemble of Linear Systems

TL;DR

This paper develops a method to generate stochastic bridges over an ensemble of linear systems with both internal and external randomness, revealing that optimal control strategies are non-Markovian and more complex than traditional approaches.

Contribution

It introduces a novel approach for constructing stochastic bridges using non-Markovian control strategies, contrasting with typical Markovian methods in the literature.

Findings

Optimal control for stochastic bridges is non-Markovian.

Ensemble of systems modeled with internal parameter randomness.

External white noise influences the trajectory shaping.

Abstract

We consider particles that are conditioned to initial and final states. The trajectory of these particles is uniquely shaped by the intricate interplay of internal and external sources of randomness. The internal randomness is aptly modelled through a parameter varying over a deterministic set, thereby giving rise to an ensemble of systems. Concurrently, the external randomness is introduced through the inclusion of white noise. Within this context, our primary objective is to effectively generate the stochastic bridge through the optimization of a random differential equation. As a deviation from the literature, we show that the optimal control mechanism, pivotal in the generation of the bridge, does not conform to the typical Markov strategy. Instead, it adopts a non-Markovian strategy, which can be more precisely classified as a stochastic feedforward control input. This unexpected divergence from the established strategies underscores the complex interrelationships present in the dynamics of the system under consideration.