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