Finite-time stability of nonlinear conformable fractional-order delayed impulsive systems: Impulsive control and perturbation perspectives

The paper’s Theorems 1–2 prove finite-time stability (FTS) for conformable FO delayed impulsive systems under stabilizing (0<β<1) and destabilizing (β≥1) impulses, with settling times T1=(γ^N)^{1/Q}Γ_S0 and T2=(γ^{N0−1})^{1/Q}Γ_S0, respectively. The candidate solution reaches the same conditions and settling times via a discrete majorant construction for U=V^{1−η} and an explicit recursion across flow/impulse segments. The paper’s argument iteratively bounds V^{1−η} on each interval and uses inequalities (3.12)–(3.13) and (3.20)–(3.21); the model’s argument uses a global upper bound p(t) and concavity/subadditivity of x↦x^Q to obtain the same sufficient conditions. Both rely on the system assumptions F(0)=0 and G_j(0)=0 to ensure invariance of the origin. No logical contradictions were found; the proofs are essentially different in style but aligned in conclusions and hypotheses, matching Theorem 1 conditions (3.4)–(3.5) and Theorem 2 conditions (3.17)–(3.18) in the paper.