Lyapunov–Krasovskii functionals for some classes of nonlinear time delay systems

The paper formulates Theorems 2–3 giving (i) an explicit attraction-region estimate via ∆ defined by α1∆^γ + b2∆^{γ+μ−1} + b3∆^{γ+σ−1} = a1(α)δ^γ, and (ii) a solution bound using a comparison inequality dv/dt ≤ −ρ v^{(γ+μ−1)/γ}, both derived from the Lyapunov–Krasovskii functional v(t,ϕ) and Lemmas 1–3 (lower/upper bounds and derivative estimate) . Critically, the paper imposes an explicit auxiliary condition ensuring that x_t remains in S_α so the lower bound (valid only on S_α) applies for all t, namely 1 + ρ̃ h ((μ−1)/γ)(α1 + b2∆^{μ−1} + b3∆^{σ−1})^{(μ−1)/γ}∆^{μ−1} ≤ α^{μ−1} . The candidate solution reproduces the same overall scheme (choice of ∆, comparison argument, ĉ1 = δ/∆, ĉ2 formula) but omits the necessary invariance-of-S_α argument and replaces it with an unsubstantiated continuation claim and a false “density of S_α” assertion. Since Lemma 2’s lower bound is only guaranteed on S_α, the model’s proof is incomplete without the paper’s extra condition. Therefore, the paper’s argument is correct as written, while the model’s is flawed by missing assumptions and a gap in the continuation step .