Model reduction for second-order systems with inhomogeneous initial conditions

The paper’s Proposition 5.1 proves that, for the first-order descriptor realization with augmented input matrix B = [0 X0 0; B 0 MV0], the upper-left (position) block P1 of the first-order controllability Gramian equals the combined second-order Gramian Pc = PSO + Px0 + Pv0. The proof proceeds by applying a Schur complement to Γ(iω) = (A − iωE)−1 to identify the relevant blocks and then recognizing the three resulting integral terms as PSO, Px0, and Pv0, respectively . The candidate solution reproduces the same argument in slightly different notation: it selects the position block via Π, rewrites the integrand as F(iω)F(−iω)T with F(s) = Π(sE − A)−1B, and uses a Schur-complement elimination to obtain the three position mappings RSO, Rx0, Rv0; summing their outer products yields Pc. This matches the paper’s definitions of the component Gramians (PSO: Definition 2.2 and Proposition 2.1; Px0: Definition 4.1 and Proposition 4.1; Pv0: Definition 4.2 and Proposition 4.2) and the combined Gramian statement in Section 5.2 . Both rely on the standard asymptotic stability assumption to ensure the frequency-domain integrals converge .