H∞ model order reduction for quadratic output systems

Items (1), (2), and (4) of Theorem 2.2 match standard facts and are argued plausibly in the paper (norm properties and the frequency-axis characterization via maximum modulus; Frobenius-norm identity when p=1) as stated in Definition 2.1 and Theorem 2.2 items 1, 2, and 4 . However, item (3) (the claimed bound ∥y∥2 ≤ ∥G1∥H∞∥u∥2 + ∥G2∥H∞∥u∥22) is incorrect under the stated L2([0,∞)) setting. The proof line equating ∥H2∗2(u⊗u)∥2 with ∥G2 L[u⊗u]∥2 via Parseval is not valid for the bilinear convolution-to-frequency mapping of a one-dimensional output; the two-variable transform of the kernel does not reduce to a simple pointwise multiplication in the output’s one-dimensional frequency variable. A concrete counterexample (y(t) = u(t)^T P u(t) with P≠0) violates the asserted inequality, showing that L2 alone does not control the quadratic term on infinite horizon. The paper’s derivation hinges on the kernel representation (7) and its multivariate Laplace transform, but the step to the claimed L2 bound misapplies Parseval in this mixed (1D output vs 2D kernel/input) setting (see Theorem 2.2(3) and its proof) . Consequently, the corollary that uses item (3) to bound the model reduction error likewise lacks validity without strengthening assumptions .