A relaxed small-gain theorem for infinite networks

The paper’s Theorem 23 proves that, under Assumptions 18–19, V(ξ)=sup_i W_i(ξ_i) is an M-step ISS Lyapunov function, giving (i) the M-step decay and (ii) bounds ω(|ξ|_A) ≤ V(ξ) ≤ ω̄(|ξ|_A). This matches the candidate’s Steps 1–2 exactly (paper equations (38)–(39) and the construction V(ξ)=sup_i W_i(ξ_i)) . The paper then invokes a finite-step Lyapunov-to-ISS implication (Proposition 16) to conclude ISS of Σ, which is what the candidate’s Step 3 uses as well (though the paper states an explicit K-boundedness requirement for f in Proposition 16) . Finally, converting the bound from V to |x|_A via the comparison inequalities is also identical to the paper’s approach, using the identity |ξ|_A = sup_i |ξ_i|_{A_i} (Lemma 6) and monotonicity of K∞-functions (used in the paper’s proof of Theorem 23) . Minor differences: the candidate did not state the K-boundedness assumption that the paper explicitly uses to pass from Σ^M to Σ (Proposition 13) and asserted γ̃=γ_u without caveats, whereas the paper’s passage from Σ^M to Σ uses an intermediate lifting step that may alter comparison functions (see Proposition 13 proof) . These are small presentational omissions; the core logic and conclusions coincide.