Riccati Stability Without Auxiliary Matrices
Abstract
In the 2004 collection \emph{Unsolved Problems in Mathematical Systems and Control Theory}, Erik Verriest posed the problem of characterizing Riccati stability ``without invoking additional matrices.'' We give such a characterization through a scalar invariant of the resolvent family. The invariant is defined by covariance balances using at most n2+1 frequency--direction pairs. A finite-dimensional separation argument shows that this covariance radius equals the optimal common ellipsoidal norm of the resolvent family. Combined with the strict bounded real lemma, this yields a necessary and sufficient condition for Riccati stability involving only the Hurwitz property of the first matrix and a single scalar inequality. The invariant reduces to the ordinary spectral radius for one matrix, lies between the pointwise spectral-radius and unscaled small-gain levels of the resolvent family, and coincides with the operator-space spectral radius of Shalit and Shamovich for the natural resolvent function space. The main results have been formally verified in the Lean proof assistant.
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BibTeX
@misc{humpherys2026riccati,
title = {{Riccati Stability Without Auxiliary Matrices}},
author = {Jeffrey Humpherys},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.06275},
eprint = {2609.06275},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06275},
note = {CC0},
}
RIS
TY - UNPB
TI - Riccati Stability Without Auxiliary Matrices
AU - Jeffrey Humpherys
PY - 2026
DA - 2026/10/10
PB - arXiv
DO - 10.48550/arxiv.2609.06275
UR - https://arxiv.org/abs/2609.06275
AB - In the 2004 collection \emph{Unsolved Problems in Mathematical Systems and Control Theory}, Erik Verriest posed the problem of characterizing Riccati stability ``without invoking additional matrices.'' We give such a characterization through a scalar invariant of the resolvent family. The invariant is defined by covariance balances using at most n^2+1 frequency--direction pairs. A finite-dimensional separation argument shows that this covariance radius equals the optimal common ellipsoidal norm of the resolvent family. Combined with the strict bounded real lemma, this yields a necessary and sufficient condition for Riccati stability involving only the Hurwitz property of the first matrix and a single scalar inequality. The invariant reduces to the ordinary spectral radius for one matrix, lies between the pointwise spectral-radius and unscaled small-gain levels of the resolvent family, and coincides with the operator-space spectral radius of Shalit and Shamovich for the natural resolvent function space. The main results have been formally verified in the Lean proof assistant.
ER -