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

TL;DR

This paper addresses a continuous-time fixed horizon linear quadratic covariance steering problem with a Hilbert-Schmidt terminal cost, proposing a convergent recursive algorithm to solve the resulting coupled matrix ODE boundary value problem.

Contribution

It introduces a novel recursive algorithm for solving the covariance steering problem with Hilbert-Schmidt terminal cost and proves its convergence.

Findings

Algorithm successfully solves the covariance steering problem.

Numerical examples demonstrate the effectiveness of the proposed method.

The approach handles multi-dimensional state spaces effectively.

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.