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 laws to stabilize agents at given distances from each other, under the constraint that the dynamics 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 using twice differentiable control laws (this is not the case for three-agent formations), even up to sets of measure zero of initial conditions. This suggests that for four-agent formations one needs to look for control laws that are either not differentiable (or even not continuous) or of higher order in the dynamics. The approach employed is based on bifurcation theory, relating the information flow to singularities in the dynamics of formations. The singularities are shown to create stable configurations that do not satisfy generically the prescribed edge lengths. Finally, it is shown that the communication constraints inherent to decentralized control can make such singularities unavoidable.