Time-Delay Systems with Delayed Impulses: A Unified Criterion on Asymptotic Stability

The paper’s Theorem 1 and its Appendix A proof establish W(t) ≤ W(t0) e^{-σ N(t,t0) - c(t-t0)} by induction using (i)–(iv), and then convert this to a KL bound via condition (v), with σ defined case-wise by (3)–(6) . The candidate solution follows the same broad structure (flow estimate; jump estimate; use of V2 ∈ V*0; case-wise σ) but commits a sign error in the weighted sliding supremum step: it defines M(t) using e^{σ N(t,r)} and then claims a per-jump contraction M(t_k) ≤ e^{-σ} M(t_k−) for all cases, which is impossible for σ>0 because M(t_k) = max{e^{σ} M(t_k−), U(t_k)} yields expansion by e^{σ} at jumps; this breaks the subsequent claim M(t) ≤ e^{-σ N(t,s)} M(s). Moreover, it momentarily misuses condition (v) by writing e^{σ N(t,s)} e^{-c(t-s)} ≤ e^{µ} e^{-λ(t-s)} instead of the correct e^{-σ N(t,s)} e^{-c(t-s)} ≤ e^{µ} e^{-λ(t-s)} implied by (v) . The paper’s proof avoids this pitfall and is consistent and complete under the stated assumptions, including the role of σ explained in Remark 1 .