Optimal partitioning for task assignment of spatially distributed vehicles based on quadratic performance criteria

We consider the problem of characterizing an optimal Voronoi-like partition aimed at providing a baseline solution to a special class of target assignment problems involving teams of spatially distributed vehicles. It is assumed that the motion of each vehicle is described by a second order mechanical system with time-varying linear or affine dynamics. The Voronoi-like partition encodes information regarding the proximity relations between the vehicles and arbitrary target points in the plane, which are induced, in turn, by a non-symmetric (generalized) distance function that incorporates the vehicle dynamics. In particular, the generalized distance is taken to be the minimum square integral control associated with the transition of a vehicle to an arbitrary target point with (approximately) zero terminal velocity at a fixed final time. The space we wish to partition corresponds to the union of all the terminal positions that can be attained by each vehicle with approximately zero terminal velocity using finite control effort. Consequently, the partition is computed over a space with a lower dimension than the state space of each vehicle. We present an efficient computational scheme for the characterization of the Voronoi-like partition, which utilizes the one-to-one correspondence between the level sets of the cost-to-go and the Euclidean distance functions. Simulation results that illustrate the theoretical developments are presented.