  Recently, ergodic control has been suggested as a means to guide mobile
sensors for information gathering tasks. In ergodic control, a mobile sensor
follows a trajectory that is ergodic with respect to some information density
distribution. A trajectory is ergodic if time spent in a state space region is
proportional to the information density of the region. Although ergodic control
has shown promising experimental results, there is little understanding of why
it works or when it is optimal. In this paper, we study a problem class under
which optimal information gathering trajectories are ergodic. This class relies
on a submodularity assumption for repeated measurements from the same state. It
is assumed that information available in a region decays linearly with time
spent there. This assumption informs selection of the horizon used in ergodic
trajectory generation. We support our claims with a set of experiments that
demonstrate the link between ergodicity, optimal information gathering, and
submodularity.
