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We present an optimal control framework for persistent monitoring problems where the objective is to control the movement of multiple cooperating agents to minimize an uncertainty metric in a given mission space. In a 1-D mission space, we show that the optimal solution is for each agent to move at maximal speed from one switching point to the next, possibly waiting some time at each point before reversing its direction. Thus, the solution is reduced to a simpler parametric optimization problem: determining a sequence of switching locations and associated waiting times at these switching points for each agent. This amounts to a hybrid system which we analyze using infinitesimal perturbation analysis (IPA) to obtain a complete on-line solution through a gradient-based algorithm. We also show that the solution is robust with respect to the uncertainty model used. This establishes the basis for extending this approach to a two-dimensional mission space.