We address the persistent monitoring problem in two-dimensional (2D) mission spaces where the objective is to control the movement of multiple cooperating agents to minimize an uncertainty metric. In a one-dimensional (1D) 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 2D mission space, such simple solutions can no longer be derived. An alternative is to optimally assign each agent a linear trajectory, motivated by the 1D analysis. We prove, however, that elliptical trajectories outperform linear ones. Therefore, 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 solution. Numerical examples are included to illustrate the main result and to compare our proposed scalable approach to trajectories obtained through off-line computationally intensive solutions.
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