Avoiding Collisions and Pattern Formation in Flocks

We study the pattern formation problem of the Cucker--Smale (CS) flocking model. A distributed CS type flocking model for formation control is introduced based on our previous collision-avoiding model. We present two sufficient frameworks guaranteeing the emergence of flocking and formation achievement in terms of the initial state and system parameters. In the first framework, we show a threshold phenomenon between unconditional and conditional flocking, embodying all three rules of Reynolds: velocity alignment, cohesion, and collision avoidance. Furthermore, with an additional condition on the initial state and system parameters, we show that the agents also converge to a desired spatial configuration in the second framework. Finally, we perform some numerical simulations to illustrate our theoretical results.