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

arXiv: https://arxiv.org/abs/1408.2222 · HTML: https://mlsanity.com/paper/1408.2222/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Aug 2014.

## Built on by (most cited first, 22 of 22)

- [Building the Bridge of Schr"odinger: A Continuous Entropic Optimal Transport Benchmark](https://mlsanity.com/paper/2306.10161/) (2023)
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- [Go With the Flow: Fast Diffusion for Gaussian Mixture Models](https://mlsanity.com/paper/2412.09059/) (2024)
- [Advancing Trajectory Optimization with Approximate Inference: Exploration, Covariance Control and Adaptive Risk](https://mlsanity.com/paper/2103.06319/) (2021)
- [Efficient Stochastic Optimal Control through Approximate Bayesian Input Inference](https://mlsanity.com/paper/2105.07693/) (2021)
- [Data Assimilation for Sign-indefinite Priors: A generalization of Sinkhorn's algorithm](https://mlsanity.com/paper/2308.11791/) (2023)
- [Learning non-equilibrium diffusions with Schr"odinger bridges: from exactly solvable to simulation-free](https://mlsanity.com/paper/2505.16644/) (2025)
- [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)
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- [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)
- [Reachability and asymptotics of Gaussian Transformer dynamics](https://mlsanity.com/paper/2606.07600/) (2026)
- [Mode Reduction for Markov Jump Systems](https://mlsanity.com/paper/2205.02697/) (2022)
- [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)
- [Mutual Information Optimal Density Control of Linear Systems and Generalized Schr"odinger Bridges with Reference Refinement](https://mlsanity.com/paper/2605.09349/) (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 to steer a linear dynamical system with full state observation from an initial gaussian distribution in state-space to a final one with minimum energy control. The system is stochastically driven through the control channels; an example for such a system is that of an inertial particle experiencing random "white noise" forcing. We show that a target probability distribution can always be achieved in finite time. The optimal control is given in state-feedback form and is computed explicitely by solving a pair of differential Lyapunov equations that are coupled through their boundary values. This result, given its attractive algorithmic nature, appears to have several potential applications such as to active control of nanomechanical systems and molecular cooling. The problem to steer a diffusion process between end-point marginals has a long history (Schr"odinger bridges) and therefore, the present case of steering a linear stochastic system constitutes a Schr"odinger bridge for possibly degenerate diffusions. Our results, however, provide the first implementable form of the optimal control for a general Gauss-Markov process. Illustrative examples of the optimal evolution and control for inertial particles and a stochastic oscillator are provided. A final result establishes directly the property of Schr"odinger bridges as the most likely random evolution between given marginals to the present context of linear stochastic systems.