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.