A novel criterion for global incremental stability of dynamical systems

The paper’s Theorem 1 states and proves (i) uniform exponential contraction between any two trajectories under µ[J_x f(x,t)] ≤ −α(t) ≤ −α0 < 0, and (ii) convergence x(t)→0 provided |f(0,t)+δ(t)|/α(t)→0; the proof uses the mean-value integral for the Jacobian, convexity/Jensen for the logarithmic norm, fundamental-matrix bounds (P4–P5), and an L’Hospital argument for the convolution term, all given explicitly in the text. These steps appear correct and complete in the provided manuscript (see the theorem statement and proof including A1–A2, the use of Lemma 1, and equations leading to the L’Hospital limit) . The candidate model solution follows the same core structure: mean-value Jacobian, matrix-measure bounds, and a variation-of-constants/convolution estimate; it replaces the paper’s fundamental-matrix/L’Hospital step by a standard Dini-derivative inequality and a split-integral estimate under α(t)≥α0. Hence both are correct, with essentially the same approach in spirit (matrix-measure contraction and averaged Jacobian), differing only in the final analytic device used to show the convolution term vanishes.