Controlled Synchronization of Coupled Pendulums by Koopman Model Predictive Control

The paper formulates the FK pendulum-chain synchronization problem clearly and demonstrates KMPC-based closed-loop synchronization to a stable equilibrium, an unstable equilibrium, and a periodic orbit in simulations and hardware. However, it explicitly provides no rigorous stability or performance guarantees and defers such analysis to future work, so it is incomplete from a theoretical standpoint . The candidate model’s solution offers a Lyapunov/passivity-based boundary-control design with claimed global asymptotic results for (a) and (b) and exponential leader tracking for (c). While the passivity setup and Lyapunov constructions are plausible, the key global convexity claim used in (a)–(b) critically relies on making λmin(k(L+D)+k_p e_1 e_1^T)>mgl by “choosing k_p arbitrarily large,” which is generally false as N grows: the minimal Rayleigh quotient of L on vectors with zero at the first node scales like O(1/N), so k(L+D)+k_p e_1 e_1^T can retain arbitrarily small eigenvalues along directions orthogonal to e_1, independent of k_p. Hence the asserted global result without restrictions on N (and independent of k) is not established. Part (c) is condition-based and more defensible, but still leaves important spectral/dimensional dependencies. In short: the paper is methodologically sound but lacks proofs; the model proposes a rigorous path but overclaims globality due to a spectral gap oversight.