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

2025-10-24

AI Summary

This paper addresses the fixed-horizon continuous-time linear-quadratic covariance control problem, aiming to exactly match the terminal-state covariance to a prescribed target. To this end, we innovatively incorporate the Hilbert–Schmidt (Frobenius) norm of the terminal covariance error into the cost functional—its first application in this problem class. This leads to a two-point boundary-value problem governed by coupled matrix Riccati differential equations. Leveraging the state-transition properties of the associated Hamiltonian matrix, we propose a linear fractional transformation-based recursive algorithm. Theoretical analysis integrates optimal control theory, matrix differential equations, and Hamiltonian system theory to rigorously establish existence and uniqueness of the solution. Numerical experiments on two- and six-dimensional systems demonstrate the algorithm’s convergence and high-precision covariance regulation capability, significantly improving terminal distribution matching performance.

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Abstract

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.

Problem

• Steering terminal covariance to desired values
• Solving coupled matrix ODE boundary value problem
• Developing convergent matricial recursive algorithm

Innovation

• Continuous-time covariance steering with Hilbert-Schmidt terminal cost
• Matricial recursive algorithm solving coupled matrix ODEs
• Linear fractional transforms parameterized by Hamiltonian transition matrix