# Optimal steering of a linear stochastic system to a final probability distribution, part II

arXiv: https://arxiv.org/abs/1410.3447 · HTML: https://mlsanity.com/paper/1410.3447/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Oct 2014.

## Built on by (most cited first, 17 of 17)

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- [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)
- [Nonlinear Non-Gaussian Density Steering with Input and Noise Channel Mismatch: Sinkhorn with Memory for Solving the Control-affine Schr"odinger Bridge Problem](https://mlsanity.com/paper/2604.23370/) (2026)
- [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)
- [Distribution-Agnostic Robust Trajectory Optimization via Chance-Constrained Reinforcement Learning](https://mlsanity.com/paper/2606.13605/) (2026)
- [A Graph-based approach to derive the geodesic distance on Statistical manifolds: Application to Multimedia Information Retrieval](https://mlsanity.com/paper/2106.14060/) (2021)
- [Non-Gaussian Uncertainty Minimization Based Control of Stochastic Nonlinear Robotic Systems](https://mlsanity.com/paper/2303.01628/) (2023)
- [Rethinking the Flow-Based Gradual Domain Adaptation: A Semi-Dual Optimal Transport Perspective](https://mlsanity.com/paper/2602.01179/) (2026)
- [Distribution Steering via Sliced Optimal Transport Control](https://mlsanity.com/paper/2608.12828/) (2026)
- [Linear Exponential Quadratic Gaussian Covariance Steering](https://mlsanity.com/paper/2609.12463/) (2026)

## Abstract (arXiv)

We consider the problem of minimum energy steering of a linear stochastic system to a final prescribed distribution over a finite horizon and to maintain a stationary distribution over an infinite horizon. We present sufficient conditions for optimality in terms of a system of dynamically coupled Riccati equations in the finite horizon case and algebraic in the stationary case. We then address the question of feasibility for both problems. For the finite-horizon case, provided the system is controllable, we prove that without any restriction on the directionality of the stochastic disturbance it is always possible to steer the state to any arbitrary Gaussian distribution over any specified finite time-interval. For the stationary infinite horizon case, it is not always possible to maintain the state at an arbitrary Gaussian distribution through constant state-feedback. It is shown that covariances of admissible stationary Gaussian distributions are characterized by a certain Lyapunov-like equation. We finally present an alternative to solving the system of coupled Riccati equations, by expressing the optimal controls in the form of solutions to (convex) semi-definite programs for both cases. We conclude with an example to steer the state covariance of the distribution of inertial particles to an admissible stationary Gaussian distribution over a finite interval, to be maintained at that stationary distribution thereafter by constant-gain state-feedback control.