This paper presents controllers that enable a mobile robot to persistently monitor or sweep a changing environment. The changing environment is modeled as an accumulation function which grows in areas that are not within range of the robot, and decreases in areas that are within range of the robot. The robot must continually move through the environment to prevent the accumulation of any area from growing unbounded. We consider the case in which a predefined path is given for the robot, and we focus on controlling the robot's speed along the path. We characterize necessary and sufficient conditions on the speed controller of the robot for keeping the accumulation function bounded. We then search among the space of speed controllers that are parametrized by a finite set of basis functions. We develop a linear program to compute the optimal speed controller; that which minimizes the accumulation over the environment. Simulation results illustrate the performance of the controllers.