MULTI-AGENT SYSTEM FOR TARGET TRACKING ON A SPHERE AND ITS ASYMPTOTIC BEHAVIOR

The paper proves asymptotic complete rendezvous (with exponential rate) when the acceleration feed-forward Ui = 2<wγ,qi>(qi×pi) + ẇγ×qi is used, under either cq>σ>0 or a small-energy condition, and asymptotic practical rendezvous when Ui=0 with a quantified bound using a rate D, all for the S2 model with velocity projection P and cp,cq>0. These are stated and proved via a moving-frame (St_γ) decomposition and a carefully designed family of Lyapunov-type functionals, including a reduction to a linear(ized) structural system and an energy dissipation identity (Ek+Ec) that yields the decay of the velocity misalignment term; see the model definition, the role of P, the feed-forward choice (3.7), and Theorems 2–3 with their proof strategy and estimates . The candidate solution reproduces the same results with a different Lyapunov design centered on the velocity-tracking error ei=pi−wγ×qi and a potential Ψ, correctly identifies key identities (e.g., P_{qγ→qi}(pγ)=wγ×qi), and recovers the same dissipation identity V̇=−(cp/N)∑||ei||^2 and the same practical-rendezvous rate template. However, it sketches (rather than proves) the strong convexity/PL-type inequality needed for global exponential convergence in case cq>σ, and it compresses several technical bounds that the paper develops in detail. Overall, both reach the same conclusions; the paper’s argument is complete, whereas the model’s proof is a shorter but higher-level sketch that would need a few missing technical justifications for full rigor.