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

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.

💡 Research Summary

The paper addresses the fixed‑horizon linear‑quadratic (LQ) covariance steering problem for continuous‑time linear stochastic systems. The system dynamics are given by
dxₜ = Aₜxₜ dt + Bₜuₜ dt + Bₜdwₜ,
where the same matrix Bₜ multiplies both the control input and the Wiener noise, a setting common in stochastic actuation. The initial state is assumed to be zero‑mean Gaussian with covariance Σ₀, and the desired terminal distribution is also zero‑mean Gaussian with covariance Σ_d. The control law is restricted to a linear state‑feedback form uₜ = Kₜxₜ.

The performance index consists of two parts: (i) an integral cost ∫₀¹ (E‖uₜ‖² + xₜᵀQₜxₜ) dt, which penalises control effort and state deviation, and (ii) a terminal cost ϕ(Σ₁, Σ_d) = ½‖Σ₁ − Σ_d‖_F², i.e., the squared Hilbert‑Schmidt (Frobenius) norm of the covariance mismatch. This terminal cost is the simplest Euclidean distance on the cone of positive‑definite matrices, in contrast to the more geometrically motivated Bures‑Wasserstein distance used in earlier work.

Applying Pontryagin’s minimum principle yields a coupled two‑point boundary‑value problem (BVP) on the cotangent bundle of the positive‑definite cone. The state equation for the covariance Σₜ is a Lyapunov‑type forward ODE, while the costate Pₜ satisfies a backward Riccati ODE. The transversality condition links the terminal costate to the covariance error: P₁ = Σ₁ − Σ_d. Direct numerical integration of this BVP is difficult because the forward ODE depends on the unknown Pₜ and the backward ODE depends on Σₜ.

To decouple the system, the authors introduce the change of variables Hₜ = Σₜ⁻¹ − Pₜ. This transformation converts both forward and backward equations into Riccati differential equations (RDEs) for Pₜ and Hₜ, respectively. Each RDE can be expressed as a linear fractional transformation (LFT) of the initial condition, using the state‑transition matrix Φ(s, t) of the Hamiltonian matrix
Mₜ =