Optimal dwell times for persistent monitoring of a finite set of targets

This paper considers the design of a periodic schedule for one or more agents moving around a finite number of targets, repeatedly visiting them to collect information and reduce uncertainty about the target. This data collection takes time and thus the design involves both the sequence of targets to be visited by each of the agents and the amount of time each agent should spend at each target. For a given visiting sequence, the problem is translated into a discrete-time dynamic system with the targets' sampled uncertainty level as the state vector and the dwell time at each target as the input vector. In the one-agent case we show that under a mild assumption and with a constant input this discrete-time system converges to an asymptotically stable steady state and that the underlying continuous dynamics converge to a periodic cycle with a fixed period. We show further that the sampled uncertainty, the peak uncertainty, and the period are all minimized under the policy that the agent switches to the next target in its sequence as soon as the uncertainty of the current target is reduced to zero. Finally, we show that if the average uncertainty over the steady state period is taken as a measure of performance, then the same policy is optimal under the additional assumption that the targets are homogeneous.