Uniform position alignment estimate of a spherical flocking model with inter-particle bonding forces

We present a sufficient condition for asymptotic rendezvous of a Cucker-Smale type model on the unit sphere with an inter-particle bonding force. This second-order dynamical system includes a rotation operator defined on the surface of the three-dimensional unit sphere, and we derive an exponential decay estimate for the diameter of agent positions and demonstrate time-asymptotic flocking for a class of initial data. The sufficient condition for the initial data depends only on the communication rate and inter-particle bonding parameter, independent of the number of agents. The lack of momentum conservation and the presence of a curved space domain pose challenges in applying standard methodologies used in the original Cucker-Smale model. To address this and obtain a uniform position alignment estimate, we employ an energy dissipation property of this system and a transformation from the Cucker-Smale type flocking model into an inhomogeneous system in which the solution contains the position and velocity diameters. The coefficients of the transformed system are controlled by the communication rate and a uniform upper bound of velocities obtained by the energy dissipation.