Balanced Truncation of Descriptor Systems with a Quadratic Output

The paper proves the L∞ output-error bounds by (i) writing the proper–proper and improper–proper outputs as convolutions with matrix kernels, (ii) applying Cauchy–Schwarz/Hilbert–Schmidt inequalities to separate input and kernel factors, (iii) evaluating the kernel L2-energies exactly via proper/improper controllability Gramians and mixed cross-Gramians, (iv) using an input inequality that requires u ∈ C^{ν−1}∩L2, and (v) combining symmetry (ypi≡yip) with the fact that improper states are not truncated to obtain the total bound (Eq. (29)). The candidate solution follows the same structure and arrives at the same trace expressions and bounds for ypp, yip, and hence for the total error. No substantive logical gaps or mismatches with the paper’s assumptions were found. Key steps match Lemma 5.1–5.4, Theorem 5.1, Theorem 5.2, Theorem 5.3, and Eq. (29) of the paper.