Maximum entropy optimal density control of discrete-time linear systems and Schr\"odinger bridges

TL;DR

This paper develops a maximum entropy optimal density control framework for discrete-time linear systems, incorporating Gaussian density constraints to manage uncertainty, and reveals its connection to Schr"odinger bridges.

Contribution

It derives explicit MaxEnt optimal density control laws with Gaussian constraints and links them to Schr"odinger bridges for linear systems.

Findings

Explicit form of MaxEnt optimal density control derived.

Gaussian density constraints effectively manage state uncertainty.

MaxEnt control characterized as a Schr"odinger bridge for linear systems.

Abstract

We consider an entropy-regularized version of optimal density control of deterministic discrete-time linear systems. Entropy regularization, or a maximum entropy (MaxEnt) method for optimal control has attracted much attention especially in reinforcement learning due to its many advantages such as a natural exploration strategy. Despite the merits, high-entropy control policies induced by the regularization introduce probabilistic uncertainty into systems, which severely limits the applicability of MaxEnt optimal control to safety-critical systems. To remedy this situation, we impose a Gaussian density constraint at a specified time on the MaxEnt optimal control to directly control state uncertainty. Specifically, we derive the explicit form of the MaxEnt optimal density control. In addition, we also consider the case where density constraints are replaced by fixed point constraints. Then, we characterize the associated state process as a pinned process, which is a generalization of the Brownian bridge to linear systems. Finally, we reveal that the MaxEnt optimal density control gives the so-called Schr\"odinger bridge associated to a discrete-time linear system.