We present controllers that enable mobile robots to persistently monitor or
sweep a changing environment. The changing environment is modeled as a field
which grows in locations that are not within range of a robot, and decreases in
locations that are within range of a robot. We assume that the robots travel on
given closed paths. The speed of each robot along its path is controlled to
prevent the field from growing unbounded at any location. We consider the space
of speed controllers that can be parametrized by a finite set of basis
functions.