  This paper studies variational collision avoidance problems for multi-agents
systems on complete Riemannian manifolds. That is, we minimize an energy
functional, among a set of admissible curves, which depends on an artificial
potential function used to avoid collision between the agents. We show the
global existence of minimizers to the variational problem and we provide
conditions under which it is possible to ensure that agents will avoid
collision within some desired tolerance. We also study the problem where
trajectories are constrained to have uniform bounds on the derivatives, and
derive alternate safety conditions for collision avoidance in terms of these
bounds - even in the case where the artificial potential is not sufficiently
regular to ensure existence of global minimizers.
