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We present a sufficient condition of the complete position flocking theoremfor the Cucker-Smale type model on the unit sphere with an inter-particlebonding force. For this second order dynamical system derived in [Choi, S.-H.,Kwon, D. and Seo, H.: Cucker-Smale type flocking models on a sphere. arXivpreprint arXiv:2010.10693, 2020] by using the rotation operator in threedimensional sphere, we obtain an exponential decay estimate for the diameter ofagents' positions as well as time-asymptotic flocking for a class of initialdata. The sufficient condition for the initial data depends only on thecommunication rate and inter-particle bonding parameter but not the number ofagents. The lack of momentum conservation and the curved space domain make itdifficult to apply the standard methodology used in the original Cucker-Smalemodel. To overcome this and obtain a uniform position alignment estimate, weuse an energy dissipation property of this system and transform theCucker-Smale type flocking model into an inhomogeneous system of differentialequations of which solution contains the position and velocity diameters. Thecoefficients of the transformed system are controlled by the communication rateand a uniform upper bound of velocities obtained by the energy dissipation.

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