Optimal steering of a linear stochastic system to a final probability distribution, Part III

TL;DR

This paper extends the optimal control framework for linear stochastic systems to include quadratic costs in both control input and state, connecting stochastic control with deterministic mass transport in the zero-noise limit.

Contribution

It introduces a method to incorporate quadratic state costs into the stochastic control approach for steering linear systems between distributions.

Findings

Extended the stochastic control framework to include quadratic state costs.

Derived the zero-noise limit as a deterministic mass transport problem.

Provided detailed technical steps for the new formulation.

Abstract

The subject of this work has its roots in the so called Schroedginer Bridge Problem (SBP) which asks for the most likely distribution of Brownian particles in their passage between observed empirical marginal distributions at two distinct points in time. Renewed interest in this problem was sparked by a reformulation in the language of stochastic control. In earlier works, presented as Part I and Part II, we explored a generalization of the original SBP that amounts to optimal steering of linear stochastic dynamical systems between state-distributions, at two points in time, under full state feedback. In these works the cost was quadratic in the control input. The purpose of the present work is to detail the technical steps in extending the framework to the case where a quadratic cost in the state is also present. In the zero-noise limit, we obtain the solution of a (deterministic) mass transport problem with general quadratic cost.