We have developed controllers that enable mobile robots to carry out persistent monitoring and cleaning tasks over 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. The robots travel on given closed paths, and the speed of the robot along the path must be controlled so as to prevent the field from growing too large at any location. The field can be thought of as either a material that must be cleaned---such as oil in the ocean, or dirt on the floor---or as the uncertainty of the robot's estimate of the environment at each point. We have developed controllers which are guaranteed to keep the field bounded. The controllers can be computed using a simple linear program. We can incorporate realistic limitations on robot performance such as speed bounds and sensing bounds.