Complex Laplacians and Applications in Multi-Agent Systems

Complex-valued Laplacians have been shown to be powerful tools in the study of distributed coordination of multi-agent systems in the plane including formation shape control problems and set surrounding control problems. In this paper, we first provide some characterizations of complex Laplacians. As an application, we then establish some necessary and sufficient conditions to ensure that the agents interacting on complex-weighted networks converge to consensus in some sense. These general consensus results are used to discuss some multi-agent coordination problems in the plane.