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

arXiv: https://arxiv.org/abs/1608.03622 · HTML: https://mlsanity.com/paper/1608.03622/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Aug 2016.

## Built on by (most cited first, 12 of 12)

- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [Maximum entropy optimal density control of discrete-time linear systems and Schr"odinger bridges](https://mlsanity.com/paper/2204.05263/) (2022)
- [Data Assimilation for Sign-indefinite Priors: A generalization of Sinkhorn's algorithm](https://mlsanity.com/paper/2308.11791/) (2023)
- [A Generalized Sinkhorn Algorithm for Mean-Field Schr"odinger Bridge](https://mlsanity.com/paper/2604.06531/) (2026)
- [On the Convexity of Discrete Time Covariance Steering in Stochastic Linear Systems with Wasserstein Terminal Cost](https://mlsanity.com/paper/2103.13579/) (2021)
- [Deep Learning for Mean Field Optimal Transport](https://mlsanity.com/paper/2302.14739/) (2023)
- [Globally Solving Unbalanced Optimal Transport and Density Control for Gaussian Distributions](https://mlsanity.com/paper/2605.04246/) (2026)
- [Nonlocal Mean Field Schr"odinger Bridge with Learned Interactions](https://mlsanity.com/paper/2606.04265/) (2026)
- [Non-Gaussian Uncertainty Minimization Based Control of Stochastic Nonlinear Robotic Systems](https://mlsanity.com/paper/2303.01628/) (2023)
- [Fixed Horizon Linear Quadratic Covariance Steering in Continuous Time with Hilbert-Schmidt Terminal Cost](https://mlsanity.com/paper/2510.21944/) (2025)
- [Lifted Schr"odinger Bridges for Gaussian Mixture Endpoints: Projection Gaps and Path-Space Obstructions](https://mlsanity.com/paper/2605.24795/) (2026)
- [Linear Exponential Quadratic Gaussian Covariance Steering](https://mlsanity.com/paper/2609.12463/) (2026)

## Abstract (arXiv)

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.