# Fixed Horizon Linear Quadratic Covariance Steering in Continuous Time with Hilbert-Schmidt Terminal Cost

arXiv: https://arxiv.org/abs/2510.21944 · HTML: https://mlsanity.com/paper/2510.21944/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 36 references that are in the map.
- First posted on arXiv 24 Oct 2025.

## Builds on (most cited first, 7 of 7)

- [Optimal steering of a linear stochastic system to a final probability distribution, Part III](https://mlsanity.com/paper/1608.03622/) (2016)
- [Gradient Flows in Filtering and Fisher-Rao Geometry](https://mlsanity.com/paper/1710.00064/) (2017)
- [Trajectory Distribution Control for Model Predictive Path Integral Control using Covariance Steering](https://mlsanity.com/paper/2109.12147/) (2021)
- [Minimum energy density steering of linear systems with Gromov-Wasserstein terminal cost](https://mlsanity.com/paper/2402.15942/) (2024)
- [Formation Shape Control using the Gromov-Wasserstein Metric](https://mlsanity.com/paper/2503.21538/) (2025)
- [On the Convexity of Discrete Time Covariance Steering in Stochastic Linear Systems with Wasserstein Terminal Cost](https://mlsanity.com/paper/2103.13579/) (2021)
- [Distributed Model Predictive Covariance Steering](https://mlsanity.com/paper/2212.00398/) (2022)

## Abstract (arXiv)

We formulate and solve the fixed horizon linear quadratic covariance steering problem in continuous time with a terminal cost measured in Hilbert-Schmidt (i.e., Frobenius) norm error between the desired and the controlled terminal covariances. For this problem, the necessary conditions of optimality become a coupled matrix ODE two-point boundary value problem. To solve this system of equations, we design a matricial recursive algorithm and prove its convergence. The proposed algorithm and its analysis make use of the linear fractional transforms parameterized by the state transition matrix of the associated Hamiltonian matrix. To illustrate the results, we provide two numerical examples: one with a two dimensional and another with a six dimensional state space.