Selective Convergence of Followers to Multiple Leaders With Repulsion and Cohesion on Riemannian Manifolds

Abstract

We study the long-term dynamics of followers that selectively follow one of multiple leaders on Riemannian manifolds, where the leaders interact through repulsive forces while remaining cohesively bounded. We propose a multileader–follower multiagent system defined on Riemannian manifolds. In our model, each follower chooses exactly one leader among several leaders and follows it, while the leaders interact with each other through repulsive forces that prevent collisions but do not allow excessive dispersion. Through follower–leader interactions, each group of followers converges to its corresponding leader. For the theoretical analysis, we review the Rauch Comparison Theorem and the main geometric concepts related to it. We also introduce the classical Barbalat Lemma. Furthermore, we extend the Barbalat Lemma to a manifold setting so that it can be applied to vector fields on Riemannian manifolds. We then present several sufficient conditions on the initial data, system parameters, and kernel functions. Using a suitable energy function, we obtain several energy estimates, which in particular guarantee collision avoidance among the leaders and leader cohesion in the sense that the interleader distances remain uniformly bounded. By combining these energy estimates with the Rauch Comparison Theorem, the classical Barbalat Lemma, and its manifold extension developed in this paper, we rigorously prove that each selectively assigned group of followers asymptotically converges to its corresponding leader on Riemannian manifolds. Finally, we provide numerical simulations to validate and illustrate our theoretical results.