  This work shows the existence of optimal control laws for persistent
monitoring of mobile targets in a one-dimensional mission space and derives
explicit solutions. The underlying performance metric consists of minimizing
the total uncertainty accumulated over a finite mission time. We first
demonstrate that the corresponding optimal control problem can be reduced to a
finite-dimensional optimization problem, and then establish existence of an
optimal solution. Motivated by this result, we construct a parametric
reformulation for which an event based gradient descent method is utilized with
the goal of deriving (locally optimal) solutions. We additionally provide a
more practical parameterization that has attractive properties such as
simplicity, flexibility, and robustness. Both parameterizations are validated
through simulation.
