Asymptotic and finite-time cluster synchronization of coupled time-varying delayed neural networks

The paper proves two results: (i) asymptotic cluster synchronization under pinning impulsive control via a Lyapunov–Razumikhin/Halanay inequality and an impulse contraction bound, with rate λ/2, and (ii) finite-time cluster synchronization under a hybrid controller, via a Lyapunov functional whose derivative satisfies V̇ ≤ −c V^α (0<α<1). The candidate solution reproduces the same Lyapunov choices, the same Young/Schur-complement manipulations, the same Razumikhin condition with q ≥ γe^{λτ}, the same impulse inequality V(t_k^+) ≤ η_k V(t_k), and the same finite-time argument (Bhat–Bernstein), matching the paper’s steps almost line-by-line. See Theorem 1 (conditions (8)–(11) and proof steps leading to (18)–(19) and the jump bound (20)–(22), concluding rate λ/2) and Theorem 2 (controller (23), derivative (26), and the application of the finite-time lemmas) in the PDF . Minor issues: in Theorem 1, (11) appears to contain a typographical slip (μ+ευ−(σ+λ)<0) that should read μ+νq−(σ+λ)<0, consistent with the proof using c=μ+νq; and both the paper and model rely on a noncausal term ∫_{t}^{t1} in the hybrid controller, which raises implementability concerns although it does not invalidate the formal Lyapunov proof.