  We establish sufficient conditions for the quick relaxation to kinetic
equilibrium in the classic Vicsek-Cucker-Smale model of bird flocking. The
convergence time is polynomial in the number of birds as long as the number of
flocks remains bounded. This new result relies on two key ingredients:
exploiting the convex geometry of embedded averaging systems; and deriving new
bounds on the s-energy of disconnected agreement systems. We also apply our
techniques to bound the relaxation time of certain pattern-formation robotic
systems investigated by Sugihara and Suzuki.
