  We address the problem of persistent monitoring, where a finite set of mobile
agents has to persistently visit a finite set of targets. Each of these targets
has an internal state that evolves with linear stochastic dynamics. The agents
can observe these states, and the observation quality is a function of the
distance between the agent and a given target. The goal is then to minimize the
mean squared estimation error of these target states. We approach the problem
from an infinite horizon perspective, where we prove that, under some natural
assumptions, the covariance matrix of each target converges to a limit cycle.
The goal, therefore, becomes to minimize the steady state uncertainty. Assuming
that the trajectory is parameterized, we provide tools for computing the steady
state cost gradient. We show that, in one-dimensional (1D) environments with
bounded control and non-overlapping targets, when an optimal control exists it
can be represented using a finite number of parameters. We also propose an
efficient parameterization of the agent trajectories for multidimensional
settings using Fourier curves. Simulation results show the efficacy of the
proposed technique in 1D, 2D and 3D scenarios.
