This paper considers a two agent scenario containing an observer and a non-maneuvering target. The observer is maneuverable but is slower than the course-holding target. In this scenario, the observer is endowed with a nonzero radius of observation within which he strives at keeping the target for as long as possible. Using the calculus of variations, we pose and solve the optimal control problem, solving for the heading of the observer which maximizes the amount of time the target remains inside the radius of observation. Utilizing the optimal observer heading we compute the exposure time based upon the angle by which the target is initially captured. Presented, along with an example, are the zero-time of exposure heading, maximum time of observation heading, and proof that observation is persistent under optimal control.