When optimal sensor interrogation is rhythmic

We define a problem involving the optimal collection of data from multiple sensors with bandwidth constraints. The sensor network has a tree structure with communication constraints on the branches. We postulate a stationary model for the sensed data. Under these circumstances, the best use of the network depends on the processing capabilities available to the nodes at intermediate levels of the tree. If those intermediate nodes have no processing ability, we show that the best approach involves a time-invariant constant ratio sharing of the communication channels. In contrast, and this is the more interesting situation, when the intermediate levels have the capability to processes and summarize data, the optimal policy may involve a rhythmic allocation of the communication resources. The study of this situation involves an analysis of singular controls and of the second variation evaluated at these trajectories.