Large Time Behavior Of The Delayed Cucker-Smale System With Attractive Potential

The Cucker-Smale model is a mathematical model describing the movement of a group.The basic idea is that each individual in the group is affected by other individuals,and this effect is based on the relative position and speed between them.This paper will give a comprehensive introduction and analysis of the Cucker-Smale model,aiming to provide important references for researchers in related fields on group motion and time delay.The main research contents of this paper are as follows :A Cucker-Smale model with external force and measurement delay and secondorder continuous-time system consistency is proposed.Considering the special quadratic potential,it is pointed out that the original energy dissipation characteristics are destroyed due to the existence of measurement delay.Therefore,the Lyapunov functional is constructed and the dissipative differential inequality is established,which shows the boundedness of the Lyapunov functional.Furthermore,the boundedness of velocity difference and spatial diameter is pointed out,and the consistency of Cucker-Smale model is also proved.A Cucker-Smale model with external force and reaction delay and second-order continuous-time system consistency is proposed.Considering the more general powerlaw attraction potential,it is shown that the reaction delay also destroys the characteristics of energy dissipation.A new Lyapunov functional is constructed,and the boundedness of the Lyapunov functional is proved by establishing a dissipative differential inequality.It is further pointed out that the Lyapunov functional decays exponentially when the external force is a special quadratic potential,and the Lyapunov functional decays polynomially when the external force is a convex potential.Finally,not only the boundedness of velocity difference and spatial diameter is proved,but also the consistency of Cucker-Smale model is proved,and a more accurate convergence speed is established.