Pattern formation of the Cucker'Smale type kinetic models based on gradient flow

In this paper, we study the pattern formation of the Cucker'Smale type kinetic models. Two distributed Cucker'Smale type kinetic models for formation control are introduced based on gradient flow. We provide rigorous proof to prove that the above two kinetic models will achieve the desired position with the same velocity over a long time. In particular, the exponential convergence rate of the pattern formation on the corresponding particle models is obtained. Our analysis shows the gradient flow structure of the velocity field is important for establishing the convergence rate results of distributed control kinetic models. Finally, some numerical simulations are performed to illustrate our theoretical results.