  We address the persistent monitoring problem in two-dimensional mission
spaces where the objective is to control the trajectories of multiple
cooperating agents to minimize an uncertainty metric. In a one-dimensional
mission space, we have shown that the optimal solution is for each agent to
move at maximal speed and switch direction at specific points, possibly waiting
some time at each such point before switching. In a two-dimensional mission
space, such simple solutions can no longer be derived. An alternative is to
optimally assign each agent a linear trajectory, motivated by the
one-dimensional analysis. We prove, however, that elliptical trajectories
outperform linear ones. With this motivation, we formulate a parametric
optimization problem in which we seek to determine such trajectories. We show
that the problem can be solved using Infinitesimal Perturbation Analysis (IPA)
to obtain performance gradients on line and obtain a complete and scalable
solution. Since the solutions obtained are generally locally optimal, we
incorporate a stochastic comparison algorithm for deriving globally optimal
elliptical trajectories. Numerical examples are included to illustrate the main
result, allow for uncertainties modeled as stochastic processes, and compare
our proposed scalable approach to trajectories obtained through off-line
computationally intensive solutions.
