# Linear Exponential Quadratic Gaussian Covariance Steering

arXiv: https://arxiv.org/abs/2609.12463 · HTML: https://mlsanity.com/paper/2609.12463/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 16 references that are in the map.
- First posted on arXiv 11 Sep 2026.

## Builds on (most cited first, 5 of 5)

- [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)
- [Optimal steering of a linear stochastic system to a final probability distribution, Part III](https://mlsanity.com/paper/1608.03622/) (2016)
- [Schr"odinger Bridge with Quadratic State Cost is Exactly Solvable](https://mlsanity.com/paper/2406.00503/) (2024)
- [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)

## Abstract (arXiv)

We formulate and analyze the linear exponential quadratic Gaussian (LEQG) covariance steering problem in continuous time over a given deadline (finite time horizon). The solution for this problem can be seen as a risk-sensitive Schr"odinger bridge between Gaussian endpoints in the linear quadratic setting. Unlike the risk-neutral case, the LEQG covariance steering controller--still a linear state feedback--can no longer be written in closed form. We show that the optimal controller is parameterized by a symmetric matrix solving an algebraic equation that encodes the implicit dependence on the risk-sensitivity parameter. We explain how the structure of this optimal controller significantly generalizes the existing results for the risk-neutral case. Building on these results, for the matched noise and input channel case, we prove the existence-uniqueness of solution for the LEQG covariance steering problem in the neighborhood of the known risk-neutral optimal solution. We give an illustrative numerical example.