ON THE EXPONENTIAL SYNCHRONIZATION FOR THE ASYMMETRIC SECOND-ORDER KURAMOTO MODEL

The paper proves exponential frequency synchronization for the inertial Kuramoto model under Assumption (A) (network depth ≤ 2 via constant C in (2.2), small γ and ᾱ, and sufficiently large K per (2.3)–(2.7)) using two diameter-based energy functions E1 and E2 and first-order Grönwall-type inequalities, together with a finite-time entrance into a quarter-circle region Dθ < D∞ and a strict exponential decay bound for Dω via Lemma 3.9; see the setup and main theorem in 2.2 , the diameter definitions and model (1.3) , the structural −C sin Dθ bound used in Lemma 3.2 , the E1 differential inequality and trapping (Lemmas 3.5–3.6) , and the E2-based exponential decay (Lemma 3.9) . The candidate solution’s core comparison step asserts D^+Dω ≤ ω̇_{i^θ} − ω̇_{j^θ}, i.e., it bounds the Dini derivative of the frequency diameter using the derivatives at phase-extreme indices. This is not a valid general bound: the upper Dini derivative of a diameter is bounded by the max–min of the derivatives over the frequency-extreme indices, not over the phase-extreme indices. The paper avoids this pitfall by working with piecewise-analytic intervals and energy functions, and by deriving a separate second-order inequality for Dω inside the invariant region; see (3.29) and Lemma 3.4 for the correct first-order estimate . Consequently, the model’s rectangle invariance and LTI-comparison steps rest on an unjustified inequality, while the paper’s argument is complete and consistent with its assumptions.