  We consider the problem of planning a closed walk $\mathcal W$ for a UAV to
persistently monitor a finite number of stationary targets with equal
priorities and dynamically changing properties. A UAV must physically visit the
targets in order to monitor them and collect information therein. The frequency
of 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. The
problem 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 over
all the targets, $\mathcal R(\mathcal W(k))$, is minimized. We prove the
following: 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 an
integral multiple of $n$, and $\mathcal R^*(n+1)$ otherwise. This result
suggests 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
$\mathcal R^*(k) \geq \mathcal R^*(k+n)$.
