Let N be the number of available sensor sources. Noisy observations of an underlying state process are available for these N sources. We consider the continuous time sensor scheduling problem in which N, of these N sources are to be chosen to collect data at each time point. This sensor scheduling problem (with switching costs and switching constraints) is formulated as a constrained optimal control problem. In this framework, the controls represent the sensors that are chosen at a particular time. Thus, the control variables are constrained to take values in a discrete set, and switchings between sensors can occur in continuous time. By incorporating recent results on discrete valued optimal control, we show that this problem can be transformed into an equivalent continuous optimal control problem. In this way, we obtain the sensor scheduling policy as well as the associated switching times.