  This paper presents a novel partial differential equation (PDE)-based
framework for controlling an ensemble of robots, which have limited sensing and
actuation capabilities and exhibit stochastic behaviors, to perform mapping and
coverage tasks. We model the ensemble population dynamics as an
advection-diffusion-reaction PDE model and formulate the mapping and coverage
tasks as identification and control problems for this model. In the mapping
task, robots are deployed over a closed domain to gather data, which is
unlocalized and independent of robot identities, for reconstructing the unknown
spatial distribution of a region of interest. We frame this task as a convex
optimization problem whose solution represents the region as a
spatially-dependent coefficient in the PDE model. We then consider a coverage
problem in which the robots must perform a desired activity at a programmable
probability rate to achieve a target spatial distribution of activity over the
reconstructed region of interest. We formulate this task as an optimal control
problem in which the PDE model is expressed as a bilinear control system, with
the robots' coverage activity rate and velocity field defined as the control
inputs. We validate our approach with simulations of a combined mapping and
coverage scenario in two environments with three target coverage distributions.
