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

arXiv: https://arxiv.org/abs/2204.05263 · HTML: https://mlsanity.com/paper/2204.05263/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 43 references that are in the map.
- First posted on arXiv 11 Apr 2022.

## Builds on (most cited first, 21 of 21)

- [Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor](https://mlsanity.com/paper/1801.01290/) (2018)
- [Generative Adversarial Imitation Learning](https://mlsanity.com/paper/1606.03476/) (2016)
- [Reinforcement Learning with Deep Energy-Based Policies](https://mlsanity.com/paper/1702.08165/) (2017)
- [Reinforcement Learning and Control as Probabilistic Inference: Tutorial and Review](https://mlsanity.com/paper/1805.00909/) (2018)
- [A survey of the Schr"odinger problem and some of its connections with optimal transport](https://mlsanity.com/paper/1308.0215/) (2013)
- [Maximum Entropy RL (Provably) Solves Some Robust RL Problems](https://mlsanity.com/paper/2103.06257/) (2021)
- [On the relation between optimal transport and Schr"odinger bridges: A stochastic control viewpoint](https://mlsanity.com/paper/1412.4430/) (2014)
- [Wasserstein Proximal Algorithms for the Schr"odinger Bridge Problem: Density Control with Nonlinear Drift](https://mlsanity.com/paper/1912.01244/) (2019)
- [Optimal steering of a linear stochastic system to a final probability distribution](https://mlsanity.com/paper/1408.2222/) (2014)
- [Optimal steering of a linear stochastic system to a final probability distribution, part II](https://mlsanity.com/paper/1410.3447/) (2014)
- [Stochastic bridges of linear systems](https://mlsanity.com/paper/1407.3421/) (2014)
- [Optimal steering of a linear stochastic system to a final probability distribution, Part III](https://mlsanity.com/paper/1608.03622/) (2016)
- [Robust transport over networks](https://mlsanity.com/paper/1603.08129/) (2016)
- [Optimal Covariance Control for Stochastic Systems Under Chance Constraints](https://mlsanity.com/paper/1804.02829/) (2018)
- [Nonlinear Covariance Control via Differential Dynamic Programming](https://mlsanity.com/paper/1911.09283/) (2019)
- [Probabilistic design of optimal sequential decision-making algorithms in learning and control](https://mlsanity.com/paper/2201.05212/) (2022)
- [Kullback-Leibler control for discrete-time nonlinear systems on continuous spaces](https://mlsanity.com/paper/2203.12864/) (2022)
- [Finite-Horizon Covariance Control of Linear Time-Varying Systems](https://mlsanity.com/paper/1707.04729/) (2017)
- [Optimal Transport over Deterministic Discrete-time Nonlinear Systems using Stochastic Feedback Laws](https://mlsanity.com/paper/1804.03108/) (2018)
- [Greedy Finite-Horizon Covariance Steering for Discrete-Time Stochastic Nonlinear Systems Based on the Unscented Transform](https://mlsanity.com/paper/2003.03679/) (2020)
- [Covariance Steering for Nonlinear Control-affine Systems](https://mlsanity.com/paper/2108.09530/) (2021)

## Built on by (most cited first, 6 of 6)

- [Risk-sensitive control as inference with R'enyi divergence](https://mlsanity.com/paper/2411.01827/) (2024)
- [Mutual Information Optimal Control of Discrete-Time Linear Systems](https://mlsanity.com/paper/2507.04712/) (2025)
- [On Policy Stochasticity in Mutual Information Optimal Control of Linear Systems](https://mlsanity.com/paper/2507.21543/) (2025)
- [Globally Solving Unbalanced Optimal Transport and Density Control for Gaussian Distributions](https://mlsanity.com/paper/2605.04246/) (2026)
- [Tsallis Entropy Regularization for Linear Quadratic Regulator and Kullback-Leibler Control](https://mlsanity.com/paper/2403.01805/) (2024)
- [Mutual Information Optimal Density Control of Linear Systems and Generalized Schr"odinger Bridges with Reference Refinement](https://mlsanity.com/paper/2605.09349/) (2026)

## Abstract (arXiv)

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.