Matrix nearness problems and eigenvalue optimization

The paper proves ∥H∥∞ = 1/ε⋆ by characterizing spectral value sets via the transfer matrix and showing that iω ∈ Λε(A,B,C,D) ⇔ ∥H(iω)∥ ≥ 1/ε, then using the minimal ε⋆ for which αε = 0 to conclude ∥H∥∞ = 1/ε⋆ (Theorem 1.6, supported by Theorem 1.2 and Corollary 1.5) . The candidate solution derives the same equality via a Schur-complement determinant identity, an eigenvalue-equivalence between M(Δ) and H(λ), and two bounding directions; it matches the paper’s assumptions (A Hurwitz, ε∥D∥<1) and handles the borderline ∥H∥∞=∥D∥ case consistently with the paper’s discussion . The arguments are logically consistent and correct; the approaches are equivalent in substance but follow different technical routes.