Coordination and geometric optimization via distributed dynamical systems

Abstract. This paper discusses dynamical systems for disk-covering and sphere-packing prob-lems. We present facility location functions from geometric optimization and characterize their differentiable properties. We design and analyze a collection of distributed control laws that are related to nonsmooth gradient systems. The resulting dynamical systems promise to be of use in coordination problems for networked robots; in this setting the distributed control laws correspond to local interactions between the robots. The technical approach relies on concepts from computational geometry, nonsmooth analysis, and the dynamical system approach to algorithms. Key words. distributed dynamical systems, coordination and cooperative control, geometric optimization, disk-covering problem, sphere-packing problem, nonsmooth analysis, Voronoi parti