{"id":"a04e7825-e440-4ac6-a905-8bc16d5bbec4","arxiv_id":"2607.00552","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Derives explicit L2-L2-gain bound for quadratic-output LTI systems; equals L2-norm of bivariate transfer function on anti-diagonal when output is purely state-quadratic, and is obtained by solving linear matrix equations.","lead":"The paper derives an explicit bound on the L2-L2 gain for linear time-invariant systems whose output is a quadratic function of state and input. This supplies a computable tool for model assessment in control and related domains.","discovery_kind":"new_method","skeptic_critique":{"model":"grok-4.3","headline":"Stability of the underlying linear system is required for anti-diagonal evaluation and matrix equations to be defined","rationale":"The reader's weakest_assumption already isolates the stability prerequisite as the point at which the frequency-domain claim is least secure; no additional internal inconsistency is visible from the abstract-level claim.","tokens_in":1668,"tokens_out":311,"duration_ms":65770,"concrete_test":"Extract the precise stability hypothesis used in the derivation of the anti-diagonal formula and the linear matrix equations (likely in the section stating the main theorem); then substitute a single marginally stable pole (Re(λ)=0) into a low-order example, recompute both the frequency-domain L2-norm and the matrix-equation solution, and check whether either remains finite or matches the actual L2-gain.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim states that when the output is purely quadratic in the state the L2-L2-gain bound equals the L2-norm of the bivariate transfer function restricted to the anti-diagonal {(s,-s) | s ∈ iR} and that this quantity is obtained by solving linear matrix equations. Both the frequency-domain object and the matrix equations are only well-defined when the linear system matrix A is Hurwitz (so that the resolvent exists on iR and the Lyapunov-type equations admit unique solutions). The abstract invokes this premise without stating explicit spectral conditions on A or showing that the quadratic output map preserves the domain on which the bound remains finite.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1799,"tokens_out":402,"duration_ms":20861,"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":[{"comment":"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.","section":"Abstract / §2"},{"comment":"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.","section":"Abstract"}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[{"response":"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_made":"yes","referee_comment":"[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."},{"response":"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_made":"partial","referee_comment":"[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."}],"tokens_in":1301,"tokens_out":383,"duration_ms":24754,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The central result is that when the output depends only quadratically on the state, the L2-L2 gain bound equals the L2 norm of the bivariate transfer function restricted to the anti-diagonal {(s, -s) | s on iR}, and this quantity is obtained by solving linear matrix equations. That explicit reduction and the matrix-equation route look like the actual new piece.\n\nThe work is useful for the narrow setting it targets. Systems with quadratic outputs show up in port-Hamiltonian models, certain optimal-control problems, and stochastic settings, so a direct computational path for the gain bound saves time on model assessment and reduction. The abstract frames the claim clearly and ties it to existing frequency-domain objects without introducing extra fitted parameters.\n\nThe main soft spot is the stability assumption. Both the anti-diagonal evaluation and the linear matrix equations require the underlying linear system to be stable (A Hurwitz) so that the resolvent exists on the imaginary axis and the Lyapunov-type equations have unique solutions. The abstract invokes this without stating the spectral condition explicitly or checking whether the quadratic output map changes the domain where the bound stays finite. If the full paper does not add a clear statement or a short proof that the bound remains valid under the stated hypotheses, readers will have to fill that gap themselves.\n\nThis is a paper for control theorists and engineers who already work with quadratic-output or bilinear-like models and need a practical bound rather than a general theory overhaul. A reader in that niche will get a usable shortcut; outsiders will not find much to take away.\n\nIt is worth sending to peer review. The claim is concrete and falsifiable once the stability conditions are written down, and the matrix-equation route is the sort of thing referees can check directly.","headline":"The paper gives a clean explicit bound for L2-L2 gain in purely quadratic-output LTI systems via anti-diagonal evaluation of the bivariate transfer function plus linear matrix equations, but leaves the stability hypothesis on A implicit.","tokens_in":2314,"tokens_out":444,"would_cite":false,"duration_ms":13691,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"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.","keywords":["quadratic output systems","L2-L2 gain","bivariate transfer function","anti-diagonal evaluation","linear matrix equations","model reduction","port-Hamiltonian systems"],"falsifier":"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.","tokens_in":2541,"feed_emoji":"","tokens_out":711,"duration_ms":20724,"temperature":0.7,"pith_summary":"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.","feed_headline":"Quadratic-output LTI systems have explicit L2-gain bounds via anti-diagonal","feed_subtitle":"The bound equals the L2 norm of the bivariate transfer function on the anti-diagonal and is obtained by solving linear matrix equations.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Anti-diagonal bounds L2-gain for quadratic-output LTI systems","L2-norm on anti-diagonal gives quadratic output gain bound","Linear matrix equations yield quadratic-output L2-gain bounds","Quadratic outputs have explicit L2 bounds via anti-diagonal"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The underlying linear system must be stable enough for the frequency-domain quantities and the linear matrix equations to be well-defined.","fun_headline_variants_meta":{"raw":{"variants":["Anti-diagonal bounds L2-gain for quadratic-output LTI systems","L2-norm on anti-diagonal gives quadratic output gain bound","Linear matrix equations yield quadratic-output L2-gain bounds","Quadratic outputs have explicit L2 bounds via anti-diagonal"]},"model":"grok-4.3","cost_usd":0.005244,"raw_usage":{"total_tokens":2485,"prompt_tokens":560,"num_sources_used":0,"completion_tokens":67,"cost_in_usd_ticks":52437000,"prompt_tokens_details":{"text_tokens":560,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1858,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":560,"tokens_out":67,"duration_ms":19353,"temperature":1.0,"reasoning_tokens":1858,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T08:45:43.804223+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}