Partitioning algorithms for multi-agent systems based on finite-time proximity metrics

We address a generalized Voronoi partitioning problem for a team of mobile agents with nonlinear dynamics with respect to a state-dependent proximity metric. In particular, the proximity (pseudo-) metric corresponds to the reduction of a generalized energy metric that occurs during the transfer of an agent to an arbitrary destination with zero terminal velocity, in finite time. The realization of every finite-time state transition takes place by means of a class of continuous feedback control laws that render the closed loop dynamics of each mobile agent non-Lipschitzian. The arrival time also turns out to be a state-dependent quantity, whose functional description is not prescribed a priori. We show that the partitioning problem studied in this work can admit a decentralized solution, that is, each agent can compute its own cell independently from its teammates provided that is aware of the positions and velocities of its neighboring agents. Numerical simulations that illustrate the theoretical developments are also presented.