Unmanned aerial vehicles (UAVs) are well-suited to a wide range of mission scenarios, such as search and rescue, border patrol, and military surveillance. The complex and distributed nature of these missions often requires teams of UAVs to work together. Furthermore, overall mission performance can be strongly influenced by vehicle failures or degradations, so an autonomous mission system must account for the possibility of these anomalies if it is to maximize performance. This paper presents a general health management methodology for designing mission systems that can anticipate the negative effects of various types of anomalies on the future mission state and choose actions that mitigate those effects. The formulation is then specialized to the problem of providing persistent surveillance coverage using a group of UAVs, where uncertain fuel usage dynamics and strong interdependence effects between vehicles must be considered. Finally, the paper presents results showing that the health-aware persistent surveillance planner based on this formulation exhibits excellent performance in both simulated and real flight test experiments.