Optimal partitioning for spatiotemporal coverage in a drift field

We consider the problem of partitioning an area in the plane populated by a team of aerial/marine vehicles into a finite collection of non-overlapping sets. The sets of this partition are in an one-to-one correspondence with the vehicles under the following rule: each point in the given set of the partition can be reached by the corresponding vehicle in this set faster than any other vehicle in the presence of a spatiotemporal drift field. Consequently, a Voronoi-like partition results, which encodes the proximity relations between the vehicles and arbitrary points in the plane with respect to the minimum time-to-go. The construction of this Voronoi-like partition is based on its interpretation as the intersection of a forest of the cost (minimum time-to-go) surfaces emanating from each generator with their common lower envelope. The characterization of each cost surface is achieved by means of an efficient expansion scheme of the level sets of the minimum time-to-go function, which utilizes, in turn, the structure of the optimal synthesis of the minimum-time problem without resorting to exhaustive numerical techniques, e.g., fast marching methods. We examine the topological characteristics of the partition by using control/system theoretic concepts and tools. The theoretical developments are illustrated with a number of numerical examples.