The reviewed record of science sign in
Pith

arxiv: 2607.00552 · v1 · pith:XHKZZZIQ · submitted 2026-07-01 · math.OC · math.DS

L2-L2-gain bounds for quadratic output systems

Birgit Hillebrecht This is my paper

Reviewed by Pith2026-07-02 08:45 UTCgrok-4.3pith:XHKZZZIQopen to challenge →

classification math.OC math.DS
keywords quadratic output systemsL2-L2 gainbivariate transfer functionanti-diagonal evaluationlinear matrix equationsmodel reductionport-Hamiltonian systems
0
0 comments X

The pith

For linear systems whose output is purely quadratic in the state, the L2-L2 gain bound equals the L2-norm of the bivariate transfer function evaluated on the anti-diagonal of the frequency domain.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives an explicit bound for the L2-L2 gain of linear time-invariant systems whose output is a quadratic function of the state and the input. When the output depends only quadratically on the state, this bound equals the L2-norm of the bivariate transfer function sampled along the anti-diagonal {(s, -s) | s on the imaginary axis} in the two-dimensional frequency plane. The same bound is obtained by solving linear matrix equations rather than performing a frequency-domain integral. This supplies a direct computational tool for systems that appear in port-Hamiltonian, optimal-control, and stochastic settings, where ordinary linear gain formulas no longer apply. A reader would care because the bound can be checked or minimized without time-domain simulation of the quadratic-output dynamics.

Core claim

In case the output is purely quadratic in the state, the bound equals the L2-norm of the bivariate transfer function evaluated along the anti-diagonal {(s, -s) | s in iR} of the iR x iR frequency domain. Further, the bound can be computed by solving linear matrix equations.

What carries the argument

The bivariate transfer function of the underlying linear system, evaluated along the anti-diagonal in the frequency domain, or equivalently the solution of the associated linear matrix equations.

If this is right

  • The gain of any quadratic-output model can be bounded without simulating trajectories.
  • The same bound applies directly to port-Hamiltonian systems, optimal-control problems, and stochastic models that produce quadratic outputs.
  • Model-reduction procedures for quadratic-output systems can use the bound as an explicit performance certificate.
  • The matrix-equation route replaces numerical integration over the frequency plane with standard linear-algebra operations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The anti-diagonal construction may extend to other polynomial output maps whose leading term is quadratic.
  • The bound could serve as a design criterion when synthesizing controllers that keep quadratic outputs small.
  • For large-scale systems the relative cost of solving the matrix equations versus frequency gridding can be measured directly.
  • The result supplies a concrete test case for whether similar frequency-domain shortcuts exist for non-quadratic nonlinear outputs.

Load-bearing premise

The underlying linear system must be stable enough for the frequency-domain quantities and the linear matrix equations to be well-defined.

What would settle it

For a concrete stable linear system with quadratic state output, compute the true supremum of the output L2-norm over all unit-L2 inputs and check whether the value matches the anti-diagonal L2-norm or the matrix-equation result.

read the original abstract

We derive an explicit bound for the L2-L2-gain of linear time-invariant systems whose output is a quadratic function of the state and the input. Such systems appear naturally in many areas, for example for port-Hamiltonian systems, optimal-control, and stochastic problems. In case the output is purely quadratic in the state, the bound equals the L2-norm of the bivariate transfer function evaluated along the anti-diagonal $\{(s,\,-s)\mid s\in i\mathbb R\}$ of the $i \mathbb R\times i \mathbb R$ frequency domain. Further, we show how the bound can be computed by solving linear matrix equations. This result provides a practical tool for assessing and reducing quadratic-output models.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper derives an explicit bound on the L2-L2 gain for linear time-invariant systems whose output is a quadratic function of the state and input. When the output depends purely quadratically on the state, the bound equals the L2-norm of the associated bivariate transfer function evaluated on the anti-diagonal {(s,-s) | s ∈ iR} of the frequency domain; the bound is further obtained by solving linear matrix equations. The result is presented as a practical computational tool for analysis and reduction of quadratic-output models arising in port-Hamiltonian systems, optimal control, and stochastic settings.

Significance. If the central equality and the matrix-equation route are rigorously established under appropriate hypotheses, the result supplies a frequency-domain and algebraic route to gain bounds that avoids direct simulation or gridding, which would be useful for model assessment in the cited application areas.

major comments (2)
  1. [Abstract / §2] Abstract and §2 (system definition): the central claim that the L2-L2-gain bound equals the indicated anti-diagonal L2-norm and is obtained from linear matrix equations presupposes that the underlying linear system matrix A is Hurwitz, so that the resolvent exists on iR and the Lyapunov-type equations admit unique solutions. No explicit spectral hypothesis on A is stated, yet this condition is load-bearing for both the frequency-domain object and the matrix-equation route to be well-defined and finite.
  2. [Abstract] Abstract: the derivation steps, stability hypotheses, and error analysis that would confirm the claimed equality are not supplied even at the level of the abstract; without them it is impossible to verify that the bound follows from the system equations without additional unstated restrictions on the quadratic output map.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need to make stability assumptions explicit. We address the comments point by point below.

read point-by-point responses

  1. Referee: [Abstract / §2] Abstract and §2 (system definition): the central claim that the L2-L2-gain bound equals the indicated anti-diagonal L2-norm and is obtained from linear matrix equations presupposes that the underlying linear system matrix A is Hurwitz, so that the resolvent exists on iR and the Lyapunov-type equations admit unique solutions. No explicit spectral hypothesis on A is stated, yet this condition is load-bearing for both the frequency-domain object and the matrix-equation route to be well-defined and finite.

    Authors: We agree that the Hurwitz property of A is required for the resolvent to be defined on the imaginary axis and for the linear matrix equations to admit unique solutions. Although the manuscript works throughout in the L2 setting (which implicitly requires stability), we accept that an explicit statement is needed. We will add the hypothesis that A is Hurwitz to both the abstract and Section 2. revision: yes

  2. Referee: [Abstract] Abstract: the derivation steps, stability hypotheses, and error analysis that would confirm the claimed equality are not supplied even at the level of the abstract; without them it is impossible to verify that the bound follows from the system equations without additional unstated restrictions on the quadratic output map.

    Authors: The abstract is deliberately concise and cannot contain full derivations or error bounds. The complete proofs establishing the equality with the anti-diagonal L2-norm, together with the precise hypotheses on the quadratic map, appear in Sections 3 and 4. To improve clarity we will revise the abstract to mention the Hurwitz assumption on A; no further restrictions on the quadratic output beyond those already stated in the paper are required. revision: partial

Circularity Check

0 steps flagged

No circularity; bound expressed via external frequency-domain object and matrix equations

full rationale

The derivation states that for purely quadratic state output the L2-L2-gain bound equals the L2-norm of the bivariate transfer function evaluated on the anti-diagonal and is obtained by solving linear matrix equations. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the abstract or described chain. The result maps the quadratic-output system onto independent frequency-domain and Lyapunov objects without reducing to its own inputs by construction. Stability of A is an external domain assumption required for the resolvent and equations to exist, but does not create circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities. The central claim rests on the unstated assumption that the linear system is asymptotically stable so that the indicated frequency-domain objects exist.

pith-pipeline@v0.9.1-grok · 5639 in / 1244 out tokens · 29472 ms · 2026-07-02T08:45:43.804223+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · 1 internal anchor

  1. [1]

    A. C. Antoulas.Approximation of large-scale dynamical systems. Society for Industrial and Applied Mathematics, 2005

    work page 2005

  2. [2]

    Benner, P

    P. Benner, P. Goyal, and I. P. Duff. Gramians, energy functionals, and balanced truncation for linear dynamical systems with quadratic outputs.IEEE Trans. Automat. Control, 67(2):886–893, 2021

    work page 2021

  3. [3]

    Benner, W

    P. Benner, W. Schilders, S. Grivet-Talocia, A. Quarteroni, G. Rozza, and L. M. Silveira. Volume 1: System- and Data-Driven Methods and Algorithms. Number 1 in Model Order Reduction. De Gruyter, Berlin/Boston, 2021

    work page 2021

  4. [4]

    Chen and R

    C. Chen and R. F. Chiu. New theorems of association of variables in multiple dimensional Laplace transform.Internat. J. Syst. Sci., 4(4):647–660, 1973

    work page 1973

  5. [5]

    L. Debnath. The double Laplace transforms and their properties with applications to functional, integral and partial differential equations.Internat. J. Appl. Comput. Math., 2(2):223–241, 2015

    work page 2015

  6. [6]

    A. N. Diaz, M. Heinkenschloss, I. V. Gosea, and A. C. Antoulas. Interpolatory model reduction of quadratic-bilinear dynamical systems with quadratic-bilinear outputs.Adv. Comput. Math., 49(6):95, 2023

    work page 2023

  7. [7]

    Engwerda and A

    J. Engwerda and A. Weeren. A result on output feedback linear quadratic control.Automatica, 44(1):265–271, 2008

    work page 2008

  8. [8]

    Fuller, B

    S. Fuller, B. Greiner, J. Moore, R. Murray, R. van Paassen, and R. Yorke. The Python control systems library (python-control). In2021 60th IEEE Conference on Decision and Control (CDC), pages 4875–4881. IEEE, 2021

    work page 2021

  9. [9]

    Haasdonk, K

    B. Haasdonk, K. Urban, and B. Wieland. Reduced basis methods for parameterized partial differential equations with stochastic influences using the Karhunen–Loe‘ve expansion.SIAM-ASA J. Uncertain. Quantif., 1(1):79–105, 2013

    work page 2013

  10. [10]

    Holicki, J

    T. Holicki, J. Nicodemus, P. Schwerdtner, and B. Unger. Energy matching in reduced passive and port-Hamiltonian systems.SIAM J. Cont. Optim., 63(3):2154–2176, 2025. [11]R. A. Horn and C. R. Johnson.Topics in Matrix Analysis. Cambridge University Press, 1991. [12]K. B. Petersen and M. S. Pedersen. The matrix cookbook, 2012. Version 20121115

    work page 2025

  11. [11]

    $\mathcal{H}_2$-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation

    S. Reiter, I. V. Gosea, I. P. Duff, and S. Gugercin. H2-optimal model reduction of linear quadratic-output systems by multivariate rational interpolation.arXiv e-print 2505.03057, 2025

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  12. [12]

    van der Schaft

    A. van der Schaft. l2-gain analysis of nonlinear systems and nonlinear state feedbackH∞control. IEEE Trans. Automat. Control, 37(6):770–784, 1992

    work page 1992

  13. [13]

    H. K. Wimmer. Contour integral solutions of Sylvester-type matrix equations.Linear Algebra Appl., 493:537–543, 2016

    work page 2016

  14. [14]

    Zhang and N

    Y. Zhang and N. Wong. Compact model order reduction of weakly nonlinear systems by associated transform.Int J. Circ. Theor. Appl., 44(7):1367–1384, 2015. [17]K. Zhou, J. C. Doyle, and K. Glover.Robust and optimal control. Prentice-Hall, 1996. AppendixA.Proof of Proposition 3.4 Proof. Let the eigenvalues ofAbe given by λ1,...,λn, then every eigenvalue λ⊕ k...

    work page 2015