Integral gains for non-autonomous Wazewski systems

The paper’s Theorem 1 states a small‑gain stability estimate for non‑autonomous Wazewski systems using a matrix‑valued Lyapunov function P(t) that solves operator differential/integral equations; the result is given with explicit h11, h22, h12 and an exponential rate involving q/(2h) (Theorem 1 and its data are spelled out, including (8)–(12) and (14)–(15) ). The proof sketch constructs v(t)=⟨x1,P11x1⟩+2⟨x1,P12x2⟩+⟨x2,P22x2⟩, obtains v̇=−q1∥x1∥2−q2∥x2∥2, then derives v≤h∥x∥2 and v≥φ∥x∥2, leading to the estimate in K and, via nonflatness of K, on all of H with factor 2aK . The candidate solution proves the same qualitative estimate and the same stability consequences via a different “tail‑energy” Lyapunov functional V(t)=∑i∫t∞qi(s)∥xi(s)∥2ds, obtaining V′≤−q∥x∥2 and a linear 2×2 inequality for (V1,V2) that is solved under rσ(Π(t0))<1 to reproduce the paper’s h11,h22,h12 formulas. It matches the paper’s structure—including the roles of (8)–(12), Π(t0), and the small‑gain hypothesis—and correctly recovers the exponential factor with q/(2h). Two caveats: (i) the model’s derivation of the lower state bound uses φ∥x∥2≤2(V+I) and then sets I≲V, which yields an extra factor 2 in K (hence 4aK on H) unless further sharpening is supplied; the paper’s P‑based Lyapunov construction avoids that slack and achieves the better constant 2aK . (ii) The step “rσ(Π)<1 ⇒ πkl≤1” asserted in the model is not generally valid for nonnegative matrices; this only affects constants, not the qualitative conclusions. Net: both arguments establish the same theorem up to a constant factor; the paper is sharper and internally consistent, and the model gives a correct alternate route to the same class of estimates and stability conclusions.