Stability analysis of a swarm model with rooted leadership

In this paper, we consider the stability of a swarm model, which is typically associated with the phenomenon of maintaining cohesion. In the swarm model, each individual has its own intrinsic nonlinear dynamics and the interaction between individuals follows a rooted leadership topology. We prove that, if the corresponding real symmetric matrix is negative definite, then the swarm is stable in the sense that all individuals will eventually form a cohesive swarm. In addition, we obtain the bounds of the swarm size. Numerical simulations were conducted to validate the theoretical results.