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Riccati Stability Without Auxiliary Matrices

Jeffrey Humpherys

arXiv:2609.06275v1math.OCcs.SYeess.SYmath.FAmath.SP

TL;DR

Verriest’s question was whether Riccati stability could be characterized without auxiliary matrices. The paper defines a scalar covariance radius for the resolvent family and shows that Riccati stability is equivalent to the first matrix being Hurwitz and this radius being less than one.

  • Problem

    Verriest posed the question of characterizing Riccati stability without additional matrices, analogous to how Lyapunov’s theorem replaces a matrix-existence condition with a spectral test when B = 0.

    Source: §1 · p. 2

  • Method

    The paper defines a covariance radius from finite balances of frequency–direction pairs and proves it equals the optimal common ellipsoidal norm of the resolvent family.

    Source: §1 · p. 2, §2 · p. 4, §5 · p. 7

  • Results

    Riccati stability holds exactly when A is Hurwitz and ρR(A, B) < 1.

    Source: §3 · p. 5

  • Takeaways & Limitations

    The scalar invariant resolves Verriest’s elimination problem without the positive-definite matrices used in the original condition.

    Source: §5 · p. 7

    The paper leaves open whether the general n^2 + 1 support bound can be reduced for resolvent families and when a finite-word lower bound is exact.

    Source: §5 · p. 7

Abstract

from arXiv · show

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.

1 Introduction

The introduction frames delay-independent stability as the relevant setting when delays are unknown or time-varying, and revisits Verriest’s request for a Riccati-stability test without auxiliary matrices. It motivates the resolvent family as the frequency-domain object and connects Riccati stability to a common quadratic metric for that family.

1 Introduction

  • Delay-independent criteria guarantee stability for every delay and are appropriate when delays are unknown or time-varying.Delay-independent stability can be conservative when the delay is known to be small.Source: §1 · p. 1
  • Verriest asked for a characterization of Riccati stability without auxiliary matrices, analogous to the eigenvalue-only Lyapunov test available when B = 0.For B = 0, the functional reduces to xT Px and the condition becomes the Lyapunov inequality.Source: §1 · p. 2
  • In feedback with a pure delay, each possible loop gain is a unit-modulus multiple of a member of the resolvent family, whose norms are unchanged by that factor.The frequency-domain construction assumes A is Hurwitz.Source: §1 · p. 2
  • Riccati stability is equivalent to A being Hurwitz and the existence of a quadratic metric under which the entire resolvent family is contractive.The proof uses the strict bounded real lemma for the corresponding realization.Source: §1 · p. 2

2 The covariance radius

The covariance radius measures the strongest aggregate covariance balance across a matrix family and equals its optimal common ellipsoidal norm. Finite-dimensional separation shows that at most n^2 + 1 matrix–direction pairs suffice.

2 The covariance radius

  • For a family, the covariance radius aggregates transformed covariances across multiple matrices and directions, unlike a single direction that may be adapted away by a common metric.The definition has no common metric or similarity transformation; its finite balances capture aggregate amplification.Source: §2 · p. 3
  • At most n^2 + 1 matrix–direction pairs suffice, because the Hermitian matrices form a real vector space of dimension n^2.The bound follows from Carathéodory’s theorem; larger supports are unnecessary.Source: §2 · pp. 3–4
  • The covariance radius equals the optimal common ellipsoidal norm: a metric contracts the family exactly when it separates the origin from the covariance-balance set.When no such metric exists, the origin lies in the convex hull of that set, giving an exact finite balance rather than an upper bound from one metric.Source: §2 · p. 4
  • For one matrix, the covariance radius reduces to its ordinary spectral radius and extends that quantity to the common-metric problem for a family.The best covariance amplification and best ellipsoidal norm coincide for a single matrix.Source: §2 · pp. 3–4
  • Finite chains of directions extend the eigenvector construction to products, and the covariance radius bounds every finite-product spectral quantity.For k = 1, the supremum of the family’s pointwise spectral radii is at most the covariance radius.Source: §2 · p. 4

3 Main characterization and bounds

For real A and B, Riccati stability is characterized by A being Hurwitz and the resolvent family's covariance radius being less than one. The scalar lies between familiar frequency-domain bounds and yields finite certificates of instability.

3 Main characterization and bounds

  • Riccati stability holds exactly when A is Hurwitz and ρR(A, B) < 1.The scalar inequality replaces the existential condition over positive-definite matrix pairs.Source: §3 · p. 5
  • When A is Hurwitz but the pair is not Riccati-stable, failure has a certificate using at most n^2 + 1 frequencies and directions.The certificate is a single semidefinite inequality in data generated by A and B.Source: §3 · p. 5
  • The covariance radius lies between the resolvent family's pointwise spectral-radius bound and its unscaled small-gain bound.The lower bound follows from the one-member case of the finite-product estimate; the identity metric gives the upper bound.Source: §3 · pp. 5–6
  • The characterization sharpens earlier scaling and similarity preservation results to exact invariance, and its lower bound forces A^-1B to be Schur stable.Joint scaling of A and B leaves the resolvent family, and hence ρR, invariant.Source: §3 · p. 6

4 A two-dimensional example

The two-dimensional example makes the covariance-radius bounds strict and evaluates the radius exactly through a particular metric. Scaling B by 5/2 yields strong delay-independent stability without Riccati stability.

4 A two-dimensional example

  • supω r(G_A,B(ω)) < ρ_R(A, B) < supω ∥G_A,B(ω)∥_2.Both outer bounds are strict for the example.Source: §4 · p. 6
  • At every frequency, r(G_A,B(−ω)G_A,B(ω))^1/2 ≤ ρ_R(A, B), giving a lower bound on the covariance radius.The example’s radius is evaluated exactly through a particular metric.Source: §4 · pp. 6–7
  • Scaling B by 5/2 gives a system that is strongly delay-independent stable but not Riccati-stable.The covariance radius thus measures a gap between pointwise spectral stability and existence of one common quadratic metric.Source: §4 · p. 7

5 Discussion

The discussion presents the scalar invariant as a matrix-free resolution of Verriest’s problem, equal to the optimal common ellipsoidal norm and supported by finite covariance balances. It also gives a bisection computation, connects the invariant to an operator-space radius, and identifies open questions about support bounds and finite-word bounds.

5 Discussion

  • The scalar invariant resolves Verriest’s elimination problem without the positive-definite matrices in the Riccati inequality.The covariance quantity equals the optimal common ellipsoidal norm, and every obstruction needs at most n^2 + 1 rank-one frequency-direction terms.Source: §5 · p. 7
  • ρ_R can be computed by bisection on γ, solving the LMI (3) at each step.Source: §5 · p. 7
  • The operator-space spectral radius and finite covariance radius are two descriptions of the same scalar for the natural resolvent function space.The operator-space matrix norm for the resolvent is the supremum over frequencies of its matrix norm.Source: §5 · p. 7
  • Open questions are whether resolvent families need fewer than n^2 + 1 support terms and when a finite-word lower bound is exact.Covariance balances combine several frequency-direction pairs, unlike pointwise spectral tests or finite chains of directions.Source: §5 · p. 7
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