{"id":"672a6b46-f350-4170-b4df-4ced2d63cfcf","arxiv_id":"2607.19507","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"An H2 inner product, norm, output bounds, and first-order optimality conditions are derived for bilinear systems with quadratic outputs, and an iteration (BQO-TSIA) is shown to meet the conditions upon convergence.","lead":"Bilinear systems with quadratic outputs (BQO) now have an H2 norm, output bounds, and first-order optimality conditions for model reduction, plus an iterative algorithm (BQO-TSIA) that reaches those conditions on convergence. The framework unifies H2 reduction for bilinear and linear-quadratic-output systems, giving engineers a fast, accuracy-aware way to shrink BQO models.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Stated convergence condition in Thm 3.1/3.4 is too weak: scalar BQO example with ||N||=1.5<2 satisfies it, yet the Volterra-Gramian series diverges and the Lyapunov solution is negative.","rationale":"Good-faith summary: The paper develops an H2 norm and optimality framework for BQO systems. The trace algebra in Theorem 4.2, the Lagrange derivative computation in Theorem 5.1, and the projection argument in Theorem 5.2 are internally consistent once the required Gramians/Sylvester solutions exist; the numerical study provides reproducible code. The weakest point is not the algebraic framework but the stated existence/convergence condition. The reader's weakest_assumption correctly identified the convergence assumptions as load-bearing, but treated the inequality as given. A direct scalar counterexample shows the inequality as stated is insufficient. This is a concrete mathematical error in the main theorem's hypothesis, not merely an unproven strong assumption. Because the error is localized and likely fixable by correcting the condition to ||N||_2<sqrt(2α)/β (or by stating a condition on ||N||_2^2) and re-verifying the reduced-system analogues, I recommend conditional acceptance rather than rejection. The requested revisions in the reader's verdict remain necessary; additionally the convergence inequalities must be corrected and the numerical examples should be checked against the corrected condition.","tokens_in":24892,"tokens_out":37430,"duration_ms":351877,"concrete_test":"Instantiate S=(A,B,C,N,M) with A=-1, B=1, C=1, N=1.5, M=0 (so m=p=1, α=β=1). This satisfies the stated Theorem 3.1 condition ||N||_2=1.5<2α/β^2=2. Compute the partial sums P_ell=Σ_{i=1}^ell P_i (or solve (3.2a) directly). The direct solution is P=-4; the partial sums diverge with ratio N^2/2=1.125>1. Thus the theorem's asserted PSD Gramian does not exist. Re-run the same check for N=1.3, for which the corrected condition ||N||_2<sqrt(2α)/β≈1.414 holds and P≈3.225 is PSD, confirming the fix.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central existence premise, Theorem 3.1/Remark 3.2, asserts that max(||N||_2,||N^T||_2)<2α/β^2 is sufficient for a unique positive semidefinite reachability Gramian P and for convergence of the Volterra-kernel series used in Definition 4.1. This condition is quantitatively wrong. The fixed-point iteration for (3.2a) contracts only if β^2||N||_2^2/(2α)<1, i.e. ||N||_2<sqrt(2α)/β (or, equivalently, the hypothesis should be on ||N||_2^2). Consider the scalar SISO BQO system A=-1, B=1, C=1, N=1.5, M=0. With α=β=1 the stated condition 1.5<2 holds. However P_1=1/2 and P_i=(N^2/2)^{i-1}P_1, so ΣP_i diverges; the generalized Lyapunov equation (3.2a) has the unique solution P=-4, which is not PSD. Theorem 4.2 would then give ||S||_{H2}^2=tr(CPC^T)=-4, while Definition 4.1's series is infinite. The same faulty inequality underlies Theorem 3.4 (the proof's assertion ||L^{-1}_{A,\\hat A}Π_{N,\\hat N}||<1 is not implied) and propagates to Corollary 3.6, Theorem 4.2, and Theorem 5.1. The framework can likely be repaired by replacing the condition with ||N||_2<sqrt(2α)/β (and the analogous reduced-system condition), but as written the hypotheses do not ensure that the Gramians, H2 norm, or optimality conditions exist.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an H2 framework for bilinear systems with quadratic outputs (BQO). It defines an H2 inner product and norm via Volterra kernels, derives explicit formulas in terms of solutions of generalized Lyapunov/Sylvester equations, proves output bounds in terms of the H2 norm, derives first-order necessary optimality conditions for H2-optimal model reduction, and proposes an iterative projection algorithm (BQO-TSIA) that satisfies these conditions upon convergence. The framework is intended to unify existing H2 theories for bilinear systems and for linear systems with quadratic outputs. Two numerical examples compare the proposed algorithm with balanced truncation.","tokens_in":25189,"tokens_out":7557,"duration_ms":75046,"significance":"If the existence hypotheses are corrected, this is a valuable contribution. The explicit H2 inner-product/norm formulas (Theorem 4.2), the output bounds (Theorem 4.7, Corollary 4.8), and the first-order optimality conditions (Theorem 5.1) extend and unify prior work on bilinear and LQO systems in a natural way. The trace identities and Lagrange-gradient computations are carefully performed and appear sound. The projection-based characterization (Theorem 5.2) and the BQO-TSIA algorithm are elegant. The paper also provides reproducible code on ZENODO, which is a strength. However, the central existence premise, stated in Theorem 3.1/Remark 3.2 and propagated through Theorem 3.4 and Theorem 4.2, is quantitatively incorrect as written; this currently undermines the H2 norm, the optimality conditions, and the algorithm's theoretical guarantees.","major_comments":[{"comment":"The stated sufficiency condition Γ := max(||N||_2, ||N^T||_2) < 2α/β^2 does not ensure existence of a PSD reachability Gramian. For the scalar SISO BQO system A=-1, B=1, N=1.5, M=0 with α=β=1, the condition holds (1.5<2), yet Eq. (3.2a) has unique solution P=-4, and the Volterra series P=ΣP_i with P_i=(N^2/2)^{i-1}/2 diverges. The correct contraction condition for the fixed-point map is β^2||N||_2^2/(2α)<1, i.e. ||N||_2<√(2α)/β, not the linear condition. This also affects Corollary 3.3, Remark 3.2, and the looser bound in Remark 3.5. Please correct the hypothesis throughout; the scalar example shows that without this correction Theorem 4.2 would give a negative H2 norm via (4.3).","section":"Theorem 3.1 / Remark 3.2"},{"comment":"The proof asserts that ||L^{-1}_{A,\\hat A}Π_{N,\\hat N}||<1 follows from the stated assumptions. This is not implied. Using the standard bound ||L^{-1}||≤β\\hatβ/(α+\\hatα) and ||Π||≤||N||_2||\\hat N||_2, the hypotheses ||N||_2<2α/β^2 and ||\\hat N||_2<2\\hatα/\\hatβ^2 only give an upper bound of order 4α\\hatα/(α+\\hatα), which is generally not less than 1. Thus the fixed-point contraction in the proof of Eq. (3.5) is not established, and uniqueness of X in (3.9a) — and hence Corollary 3.6 and Theorem 4.2 — is not guaranteed. A sufficient condition would be β\\hatβ||N||_2||\\hat N||_2/(α+\\hatα)<1, which is satisfied, for example, if ||N||_2<√(2α)/β and ||\\hat N||_2<√(2\\hatα)/\\hatβ.","section":"Theorem 3.4"}],"minor_comments":[{"comment":"The convergence of Algorithm 5.1 is assumed rather than proved. This is stated explicitly in Theorem 5.3, but the abstract and introduction describe the method as an 'algorithmic solution' without this caveat. Please make the conditional nature of the convergence claim more prominent.","section":"Theorem 5.3 / Algorithm 5.1"},{"comment":"The notation for the multi-indexed input u^{(t_1,...,t_l)}_k(t) in the Volterra series expansion is dense and not fully defined. A short explanation or a reference for this notation would improve readability.","section":"Section 4.1"},{"comment":"The figures and tables are clear, but the text states that BQO_TSIA has a convergence issue for order 8 without further discussion. A brief comment on the cause, or on whether this is related to the existence conditions in Theorem 3.1, would be useful.","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern is valid and lands directly on the paper's central existence hypothesis. The error appears to be repairable by strengthening the norm conditions to ||N||_2<√(2α)/β (and analogous conditions) in all generalized Lyapunov/Sylvester equations, after which the main derivations should go through. For this reason I do not recommend rejection, but the revision must re-derive the contraction conditions and re-check the numerical examples under the corrected hypotheses."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper supplies a genuinely useful H2 theory for bilinear systems with quadratic outputs: an explicit inner product and norm (Theorem 4.2), output bounds (Theorems 4.5-4.7), first-order optimality conditions (Theorem 5.1), and a BIRKA/LQO-TSIA-style algorithm that satisfies those conditions on convergence. The trace identities and the Lagrange-gradient computation check out, and the code and data are provided. For a subfield that has been lacking a unified H2 treatment of this class, this is a solid step forward.\n\nBut there is a load-bearing flaw that the reader's report missed. Theorem 3.1/Remark 3.2 assert that max(||N||,||N^T||) < 2α/β^2 suffices for the reachability Gramian P to exist and for the Volterra-kernel series to converge. That condition is too weak. Take the scalar SISO example A=-1, B=1, C=1, N=1.5, M=0 with α=β=1. The stated condition holds (1.5<2), yet the generalized Lyapunov equation gives P=-4 and the series defining P diverges. The correct contraction condition is ||N||^2 < 2α/β^2 (equivalently ||N|| < sqrt(2α)/β). The same faulty inequality appears in the proof of Theorem 3.4 for the Sylvester equation and propagates to Corollary 3.6, Theorem 4.2, and Theorem 5.1. As written, the hypotheses do not ensure the objects whose properties are derived. This is fixable—replace the condition with the squared version—but it is a genuine error, not a minor gap.\n\nOther soft spots are less serious. Theorem 5.3 is explicitly conditional on convergence of the algorithm, and the numerical results indeed show a convergence failure for order 8 on the RC example. The numerical comparison omits the closest H2-optimal baseline, the QBS method of Benner-Goyal-Gugercin (ref [10]); given that paper's H2-quasi-optimal framework is more general, the 'first complete H2 theory' claim should be toned down. The paper cites its own prior work appropriately, and the self-citation patterns are not problematic.\n\nWho benefits: researchers working on H2-optimal reduction of bilinear and quadratic-output systems. The algebraic core is correct, the algorithm is well motivated, and the counterexample is easy to repair. The paper deserves a serious referee, and I would accept it for peer review with the expectation of a revision that corrects the convergence condition, addresses the QBS comparison, and softens the novelty claim.","headline":"Solid H2 framework for BQO model reduction, but a wrong convergence condition in Theorem 3.1 undermines the existence assumptions; fixable and worth refereeing.","tokens_in":25930,"tokens_out":3116,"would_cite":true,"duration_ms":31210,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93A15","93B40","93C10","93C15"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper develops a complete H2 theory for bilinear systems with quadratic outputs (BQO), giving a computable norm, output error bounds, and a projection algorithm whose converged reduced models provably satisfy first-order optimality con","keywords":["H2 model order reduction","bilinear systems","quadratic outputs","optimality conditions","Petrov–Galerkin projection","generalized Lyapunov equations","Volterra kernels","model reduction"],"falsifier":"Compute the H2 norm of a small BQO system both by numerical quadrature of the Volterra-kernel series and by the closed-form trace formula (4.3); a mismatch for any system satisfying the stated bounds would disprove Theorem 4.2. More sharply, run BQO-TSIA on a system satisfying the hypotheses and check whether the converged reduced matrices satisfy (5.2a)–(5.2e); a violation would disprove Theorem 5.3.","tokens_in":24619,"feed_emoji":"🎛️","tokens_out":8122,"duration_ms":65170,"temperature":0.7,"pith_summary":"This paper develops a complete H2 theory for bilinear systems with quadratic outputs (BQO), a class where both the state dynamics and the measured output depend on quadratic state terms. The central claim is that every sufficiently stable BQO system has an H2 norm that can be computed from finite matrix equations rather than an infinite Volterra series — explicitly, the squared norm equals tr(CPC^T) plus a sum of trace terms involving the quadratic-output matrices, and also equals tr(B^T Q B), where P and Q solve generalized Lyapunov equations. On this foundation the authors build output-error bounds, first-order necessary optimality conditions for the best H2 reduced model, and an iterative two-sided projection algorithm (BQO-TSIA) that, on convergence, provably satisfies those conditions. If the theory holds, it unifies the separate H2 model-reduction frameworks for bilinear systems and for linear systems with quadratic outputs, and gives large-scale simulations a principled way to replace a BQO model by a much smaller one with error control.","feed_headline":"H2 norm theory built for bilinear systems with quadratic outputs","feed_subtitle":"The new H2 framework yields a computable norm, output error bounds, and an algorithm that meets the optimality conditions.","key_machinery":"The load-bearing identity is the generalized Sylvester/Lyapunov trace duality: tr(E^T Y) = tr(F^T X) whenever X and Y solve the paired equations (3.10). It converts the infinite Volterra-kernel definition of the H2 inner product into finite trace formulas involving a cross-Gramian X and a dual variable Pi. The algorithm's engine is the two-sided projection update V = orth(X), W = orth(Pi), where X and Pi are the solutions of (5.6a)–(5.6b), so that the reduced matrices are built as (W^T V)^{-1} W^T A V, W^T B, C V, W^T N_k V, V^T M_j V.","core_discovery":"The paper's central discovery is an explicit, computable formula for the H2 inner product between two BQO systems, and hence the H2 norm of one. The norm equals ||S||^2_{H2} = tr(CPC^T) + sum_{j=1}^p tr(PM_j P M_j) = tr(B^T Q B), where P is the reachability Gramian solving AP + PA^T + sum N_k P N_k^T + BB^T = 0 and Q solves the dual equation A^T Q + Q A + sum N_k^T Q N_k + sum M_j P M_j + C^T C = 0. This makes the norm finite and computable. From the same formulas the authors derive first-order necessary optimality conditions for an H2-optimal reduced BQO system, and show that any reduced system meeting them is equivalent to a Petrov–Galerkin projection using V = X P-hat^{-1} and W = Pi Psi^","pith_inferences":["The structure of the optimality conditions (5.2) closely parallels the generalized tangential interpolation conditions of linear and bilinear H2 model reduction, so one may be able to prove a matching interpolation theorem for BQO systems without needing the fixed-point iteration — a direct, non-iterative construction of optimal reduced models.","The existence bound ||N|| < 2 alpha / beta^2 is the real bottleneck: the theory cannot certify systems with strong bilinear coupling, and the input-rescaling trick used in the experiments only shifts the effective coupling, so extending the norm definition (e.g., through finite-horizon or density-based arguments) would substantially broaden the class of tractable systems.","Since the final projection is determined by X and Pi alone, a per-iteration complexity analysis of BQO-TSIA would likely show a cost of O(n r^2 m) for solving the Sylvester equations via fixed-point iteration, plus the cost of the projection — the paper does not analyze this, but it suggests the method can scale to the large n typical of discretized PDEs."],"forward_implications":["For any BQO system satisfying the stability conditions, the H2 norm and H2 distance become computable quantities — just solve two generalized Lyapunov equations — so error measurement no longer requires time-domain simulation of the full model.","The output bound of Theorem 4.7 gives a rigorous worst-case guarantee: if a reduced model is close to the full system in H2 norm, its output error is provably small for bounded L2 inputs, which licenses H2-optimal reduction as the right objective.","The optimality conditions (5.2) provide an explicit target: a locally H2-optimal reduced BQO system must satisfy the five matrix equations; any algorithm that reaches them is provably at a first-order necessary optimum.","The framework recovers the bilinear and LQO H2 theories as special cases (M_j = 0 or N_k = 0), so both research lines become instances of one BQO theory with a single algorithm.","Numerical tests on a Carleman-bilinearized RC circuit and a heat-transfer MIMO problem show the algorithm can out-perform balanced truncation in accuracy and speed, and the global-GMRES variant is the fastest of the four methods compared."],"fun_headline_variants":["Explicit H2 norm for bilinear-quadratic output systems","H2 inner product and norm for BQO systems","First-order optimality conditions for H2 BQO reduction","H2 model reduction optimality for bilinear-quadratic outputs","New explicit H2 norm for BQO systems"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole theory rests on the assumption that the bilinear coupling matrices N (and the analogous reduced-system matrices) are small enough that the infinite series defining the Gramians and the H2 norm converge — explicitly ||N||, ||N^T|| < 2 alpha / beta^2 — otherwise the generalized Lyapunov equations have no unique solution and the H2 norm simply does not exist.","fun_headline_variants_meta":{"raw":{"variants":["Explicit H2 norm for bilinear-quadratic output systems","H2 inner product and norm for BQO systems","First-order optimality conditions for H2 BQO reduction","H2 model reduction optimality for bilinear-quadratic outputs","New explicit H2 norm for BQO systems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000541,"raw_usage":{"total_tokens":2448,"prompt_tokens":779,"completion_tokens":1669,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":523,"completion_tokens_details":{"reasoning_tokens":1594}},"tokens_in":523,"tokens_out":1669,"duration_ms":11248,"temperature":1.0,"reasoning_tokens":1594,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-01T12:35:18.642285+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the H2 norm of a small BQO system both by numerical quadrature of the Volterra-kernel series and by the closed-form trace formula (4.3); a mismatch for any system satisfying the stated bounds would disprove Theorem 4.2. More sharply, run BQO-TSIA on a system satisfying the hypotheses and check whether the converged reduced matrices satisfy (5.2a)–(5.2e); a violation would disprove Theorem 5.3.","supporting_citations":[],"review_version":1}