  Recently, a theory for stochastic optimal control in non-linear dynamical
systems in continuous space-time has been developed (Kappen, 2005). We apply
this theory to collaborative multi-agent systems. The agents evolve according
to a given non-linear dynamics with additive Wiener noise. Each agent can
control its own dynamics. The goal is to minimize the accumulated joint cost,
which consists of a state dependent term and a term that is quadratic in the
control. We focus on systems of non-interacting agents that have to distribute
themselves optimally over a number of targets, given a set of end-costs for the
different possible agent-target combinations. We show that optimal control is
the combinatorial sum of independent single-agent single-target optimal
controls weighted by a factor proportional to the end-costs of the different
combinations. Thus, multi-agent control is related to a standard graphical
model inference problem. The additional computational cost compared to
single-agent control is exponential in the tree-width of the graph specifying
the combinatorial sum times the number of targets. We illustrate the result by
simulations of systems with up to 42 agents.
