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We consider the problem of planning a closed walk $\mathcal W$ for a UAV topersistently monitor a finite number of stationary targets with equalpriorities and dynamically changing properties. A UAV must physically visit thetargets in order to monitor them and collect information therein. The frequencyof monitoring any given target is specified by a target revisit time, $i.e.$,the maximum allowable time between any two successive visits to the target. Theproblem considered in this paper is the following: Given $n$ targets and $k\geq n$ allowed visits to them, find an optimal closed walk $\mathcal W^*(k)$so that every target is visited at least once and the maximum revisit time overall the targets, $\mathcal R(\mathcal W(k))$, is minimized. We prove thefollowing: If $k \geq n^2-n$, $\mathcal R(\mathcal W^*(k))$ (or simply,$\mathcal R^*(k)$) takes only two values: $\mathcal R^*(n)$ when $k$ is anintegral multiple of $n$, and $\mathcal R^*(n+1)$ otherwise. This resultsuggests significant computational savings - one only needs to determine$\mathcal W^*(n)$ and $\mathcal W^*(n+1)$ to construct an optimal solution$\mathcal W^*(k)$. We provide MILP formulations for computing $\mathcal W^*(n)$and $\mathcal W^*(n+1)$. Furthermore, for {\it any} given $k$, we prove that$\mathcal R^*(k) \geq \mathcal R^*(k+n)$.

Persistent Monitoring of Dynamically Changing Environments Using an Unmanned Vehicle