Asymptotic Tracking of a Point Cloud Moving on Riemannian Manifolds

We present two Cucker–Smale type models for the asymptotic tracking of a point cloud moving on complete, connected, and smooth Riemannian manifolds. For each model, we provide a sufficient framework in terms of a moving target point cloud, system parameters, and initial data. In the proposed framework, we show asymptotic flocking, collision avoidance, and asymptotic tracking to a given point cloud. We employ a dissipative energy estimate and the LaSalle invariance principle to extract desired qualitative properties as key ingredients. We also offer several numerical examples and compare them with analytical results.