MR3090050 Reviewed Belabbas, Mohamed Ali On global stability of planar formations. IEEE Trans. Automat. Control 58 (2013), no. 8, 2148–2153. (Reviewer: Gaetana Gambino) 93A14 (93D05 93D21)

The focus of the paper is planar formation control, i.e. the design of control lawsto stabilize agents at given distances from each other, under the constraint that thedynamics of each agent only depends on a subset of the other agents.The main contribution of the paper is the following:It is shown that a simple four-agent formation cannot be globally stabilized usingtwice differentiable control laws (this is not the case for three-agent formations), evenup to sets of measure zero of initial conditions. This suggests that for four-agentformations one needs to look for control laws that are either not differentiable (or evennot continuous) or of higher order in the dynamics.The approach employed is based on bifurcation theory, relating the information flowto singularities in the dynamics of formations. The singularities are shown to createstable configurations that do not satisfy generically the prescribed edge lengths.Finally, it is shown that the communication constraints inherent to decentralizedcontrol can make such singularities unavoidable.