On the stability of port-Hamiltonian descriptor systems

The candidate solution reproduces the paper’s Section 4 results: (i) the equivalence between behavioral stability, a Lyapunov inequality restricted to the system space, and spectral conditions (regular pencil, spectrum in the closed left half-plane, semisimple imaginary-axis eigenvalues) matches Proposition 4.1 and its proof via canonical forms; (ii) the construction Q := X̂ E P_{V_sys} and D := A Q^{-1} on the system space, with Q^T E ≥ 0 (strictly > 0 on V_sys) and D + D^T ≤ 0, is exactly the rewriting into port-Hamiltonian form given after Proposition 4.1. The definition and properties of the system space and the invertibility of E on V_sys are also consistent with Lemma 3.1 and the discussion in Section 3. The model’s argument uses Weierstrass/Kronecker canonical form reasoning and standard Lyapunov/eigenmode arguments, as in the paper, with only minor stylistic differences and no substantive conflicts with the paper’s claims (see Prop. 4.1 and the subsequent construction of Q and D, and the system space characterization in Section 3 in 2105.08934.pdf ).