Cucker–Smale flocking under rooted leadership with free-will agents

Motivated by the recent works of Cucker–Smale and Shen, we study the emergent behavior of discrete-time Cucker–Smale flocking under rooted leadership with free-will agents. We prove that, if the free-will accelerations of the agents relative to the leader are summable, then the individual velocities will converge to the velocity of leader, under suitable initial conditions. In particular, we prove that the unconditional convergence is true if the decay exponent of the communication weight is below some critical value which depends on the interaction topology. We also present some numerical examples to illustrate the main result.