Rendezvous Control Design for the Generalized Cucker-Smale Model on Riemannian Manifolds

In this article, we consider a rendezvous problem for the generalized Cucker-Smale model, which is a double-integrator multiagent system, on complete Riemannian manifolds. With the help of the covariant derivative, parallel transport, and logarithm map on the Riemannian manifold, we design a rendezvous feedback law that enables all agents to converge at a given target in the Riemannian manifold, under some a priori conditions. Furthermore, we consider three concrete complete Riemannian manifolds, such as the unit circle, unit sphere, and hyperboloid, and present the explicit feedback laws for rendezvous on them by calculating the corresponding covariant derivatives, parallel transports, and logarithm maps. Meanwhile, numerical examples are given for the manifolds as mentioned above to verify and illustrate the theoretical results.