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 the paper we study dissipativity properties and spatial cohesiveness of the swarm of self-propelled particles governed by the model $\ddot r_k = -p_k(|\dot r_k|)\dot r_k - \sum_m a_{k,m}r_m$, where $r_k\in \mathbb R^d$, $k=1,\ldots,n$, and $A = \{a_{k,m}\}$ is a symmetric positive-semidefinie matrix. The self-propulsion term is assumed to be continuously differentiable and to grow faster than $1/z$, that is, $p_k(z)z\to\infty$ as $z\to\infty$. We establish that the velocity and acceleration of the particles are ultimately bounded. We show that when $\ker (A)$ is trivial, the positions of the particles are also ultimately bounded. For systems with $\ker (A)\neq \{0\}$, 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 $\ker (A)$. We also include the proof of the ultimate boundedness of velocities and accelerations for systems with bounded coupling, including systems coupled via the Morse potential. In our proof we switch to the velocity-acceleration coordinates and focus on the study of dissipativity properties for a more general class of Li\'enard systems $\ddot x_k = -\mathbb F_k(x_k)\cdot \dot x_k -\sum_{m} a_{k,m}x_m$, $k=1,\ldots,n$, $\mathbb F_k(x) = \nabla F_k(x)$ with $F_k: \mathbb R^d\rightarrow \mathbb R^d$ given by $F_k(x) = p_k(|x|)x$.