FLOCKING FORMATION AND STABILIZER OF BOOSTED COOPERATIVE CONTROL ON A SPHERE

The paper proves global well-posedness, exact energy dissipation, LaSalle-based convergence to a largest invariant set, and a full classification of asymptotic configurations (rendezvous, formation, uniform deployment) with sharp parameter/energy thresholds (Theorems 1–2) and precise characterization of the zero-dissipation set (Proposition 3.7) . It also defines the alignment measure and shows asymptotic velocity alignment via the dissipation identity and LaSalle’s invariance principle . By contrast, the candidate solution broadly mirrors the paper’s structure and most conclusions, but contains a critical gap and a wrong identity in case (iii). Specifically: (1) it attempts to force x̄→0 from a Jensen-type lower bound on EC without showing that every zero-dissipation ω-limit configuration must satisfy x̄=0—whereas the paper correctly uses Lemma 3.8 to prove exactly that when σr ≥ (2N/(N−1))σa or σa=0<σr ; and (2) it incorrectly claims that summing the pairwise forces for σa=0 yields Σi (1/N)Σj≠i σij(…)=−((σr/2)(N−1))x̄ (no such simplification holds due to the variable 1/∥xi−xj∥2 weights). The remaining parts (constraint invariance, well-posedness with collision avoidance for σr>0, dissipation identity, D(t)→0 via bounded-derivative/Barbalat-type reasoning, characterization of zero-dissipation motions as rigid rotations) align with the paper’s results and intent, but the flaw in case (iii) means the candidate proof is not fully correct or complete .