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

TL;DR

This work addresses fixed-horizon covariance steering for continuous-time linear systems with a Hilbert-Schmidt terminal cost, formulating necessary optimality conditions that couple the terminal covariance and its costate. It introduces a novel matrix-valued recursive algorithm built from linear fractional transforms of Riccati-type ODE solutions, and proves the recursion converges to a unique fixed point for almost all initial guesses. The method is demonstrated on two numerical examples (2D double integrator and 6D Clohessy–Wiltshire) showing fast convergence and terminal covariances close to the targets. The approach provides a practical, provably convergent tool for continuous-time covariance control and suggests avenues for extending to other terminal metrics and weighted costs.

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.

Figures (10)

• Figure 1: The proposed fixed point recursion $\bm{P}_{0}\mapsto \left(\bm{P}_{0}\right)_{\text{next}}$ as a composition of four mappings.
• Figure 2: The phase portrait of the recursion $\bm{P}_{0}\mapsto \left(\bm{P}_{0}\right)_{\text{next}}$ proposed in Sec. \ref{['sec:RecrsiveAlgorithm']}. The three subplots (a)-(c) are for three randomly generated problem data $(\bm{A},\bm{B},\bm{Q},\bm{\Sigma}_0,\bm{\Sigma}_{d})$ with controllable $(\bm{A},\bm{B})\in\mathbb{R}^{2\times 2}\times\mathbb{R}^{2\times 1}$, $\bm{Q}\in\mathbb{S}^{2}_{+}$, and $\bm{\Sigma}_0,\bm{\Sigma}_{d}\in\mathbb{S}^{2}_{++}$. For fixed problem data, each subplot shows the convergence of the proposed recursion for random initial guesses $\bm{P}_{0}\in\mathbb{S}^{2}$ highlighted as red circular markers. In each subplot, the black circular markers show $1000$ iterates, and the green square is the converged $\bm{P}_{0}\in\mathbb{S}^{2}$.
• Figure 3: Convergence of the $3$ triangular elements of $\bm{P}_0\in\mathbb{S}^{2}$ for the numerical example in Sec. \ref{['subsec:DI']}.
• Figure 4: $500$ optimally controlled covariance snapshots (gray ellipses) and $5$ closed-loop state sample paths for the numerical example in Sec. \ref{['subsec:DI']}. The hollow circular markers denote the initial conditions for these sample paths. The red-lined ellipse shows the desired terminal covariance $\bm{\Sigma}_{d}$.
• Figure 5: Optimal input paths corresponding to the state sample paths in Fig. \ref{['fig:EllipsoidPlotDI']}.

Theorems & Definitions (26)

• Proposition 1
• proof
• Remark 1
• Remark 2
• Proposition 2
• proof
• Lemma 1
• Theorem 1
• proof
• Theorem 2