Collective motion of planar particles and coupled lasers

We consider the problem of controlling coupled identical, planar particles in the polar coordinate to achieve various collective motions. We first introduce a model that allows amplitude and phase coordination by decomposing particles' positions into the amplitude and phase states. Then we present two control algorithms based on this model. The convergence analysis shows that under different coordination strategies and control parameters, the particles achieve rendezvous, circular formation, or rotation. In particular, we show that a coupled laser system can be treated as a special case of the proposed model. Using this interpretation, analytical explanation of the observed phase locking phenomenon is provided for identical lasers. The various collective motions by our proposed control algorithms are verified using Matlab simulations.