On Spatial Cohesiveness of Second-Order Self-Propelled Swarming Systems

The study of emergent behavior of swarms is of great interest for applied sciences. One of the most fundamental questions for self-organizing swarms is whether the swarms disperse or remain in a spatially cohesive configuration. In this paper we study dissipativity properties and spatial cohesiveness of the swarm of self-propelled particles governed by the model , where , , and is a symmetric positive semidefinite matrix. The self-propulsion term is assumed to be continuously differentiable and to grow faster than , that is, as . We establish that the velocity and acceleration of the particles are ultimately bounded. We show that when is trivial, the positions of the particles are also ultimately bounded. For systems with , we show that, while the system might infinitely drift away from its initial location, the particles remain within a bounded distance from the generalized center of mass of the system, which geometrically coincides with the weighted average of agent positions. The weights are determined by the coefficients of the projection matrix onto . We also discuss the ultimate boundedness for systems with bounded coupling, including the Morse potential systems, and systems governed by power-law potentials with strong repulsion properties. We show that the former systems are ultimately bounded in the velocity-acceleration domain, whereas the models based on the power-law potentials are not.