On the data-sparsity of the solution of Riccati equations with applications to feedback control

TL;DR

This work addresses solving large-scale continuous-time Riccati equations when coefficients are quasiseparable, proving that the stabilizing solution is numerically quasiseparable and linking off-diagonal decay to Zolotarev numbers. It develops two scalable solvers: a general divide-and-conquer CARE solver based on hierarchical semiseparable representations and a banded-case inexact Newton–Kleinman method with truncation (tink). The authors derive decay bounds for off-diagonal blocks and TT ranks of the value function, and validate performance on synthetic tests and real control problems including SDRE-based infinite-horizon control for PDEs and agent-based models, achieving substantial speedups over dense solvers. Overall, the paper provides principled structure-exploiting approaches that enable large-scale CAREs to be solved efficiently in practice, with broad applicability to control of PDEs and multi-agent systems.

Abstract

Solving large-scale continuous-time algebraic Riccati equations is a significant challenge in various control theory applications. This work demonstrates that when the matrix coefficients of the equation are quasiseparable, the solution also exhibits numerical quasiseparability. This property enables us to develop two efficient Riccati solvers. The first solver is applicable to the general quasiseparable case, while the second is tailored to the particular case of banded coefficients. Numerical experiments confirm the effectiveness of the proposed algorithms on both synthetic examples and case studies from the control of partial differential equations and agent-based models.

Figures (9)

• Figure 1: Offdiagonal singular values in the solution $X\in\mathbb R^{500\times 500}$ of \ref{['eq:care']}, with $A$ diagonal, $F=I$, and $Q$ tridiagonal symmetric positive definite. On the left, the matrix $A$ has diagonal entries logarithmically spaced in $[-1, -0.001]$; on the right, the diagonal entries of $A$ are equally spaced on the circle of radius $1$ and center $-1.1$.
• Figure 2: Relative offdiagonal singular values of the solution $X\in\mathbb R^{500\times 500}$ of \ref{['eq:care']}, with $A$ diagonal, $F$ diagonal with increasing condition number, and $Q$ tridiagonal symmetric positive definite. On the left, the matrix $A$ has eigenvalues logarithmically spaced in $[-1, -0.001]$; on the right, the eigenvalues of $A$ are on the circle of radius $1$ and center $-1.1$.
• Figure 3: Magnitude of the entries in the first column of the solution $X\in\mathbb R^{300\times 300}$ of a CARE with $A$ diagonal with positive entries, $F=I$, and $Q$ symmetric positive definite and tridiagonal. On the left, it is set $A=$diag(logspace(-1, 0, 300)), on the right $A=$diag(logspace(-3, 0, 300)). For both cases, the matrix $Q$ is chosen as in Example \ref{['ex:decay']}. The upper bound \ref{['eq:decay-band']}, denoted by cyan squares, has been computed via the minimax function of the chebfun toolbox.
• Figure 4: Comparison of the methods in terms of the influence of the line search method and truncation strategy, for $n=2000$.
• Figure 5: Uncontrolled solution (left) and controlled solution via SDRE solved with tink (right) with $n=500$.

Theorems & Definitions (33)

• Definition 2.1
• Theorem 2.2: Theorem 2.1 in beckermann17
• Theorem 2.3
• proof
• Theorem 2.4
• proof
• Remark 1
• Remark 2
• Corollary 2.5
• proof