In periodic breathing, there are repeated cycles of bursts of breaths separated by pauses several seconds long. We consider the mechanics of periodic breathing in human infants using calibrated traces of tidal volume and esophageal pressure recorded during the first few days after delivery. Each cycle of periodic breathing was analyzed in terms of the inspiratory time and beginning and end inspiratory volumes for each breath, the number of breaths in the cycle, and the total observed inspiratory work. A simple model was used to characterize the mechanics of the lung during inspiration, and the recordings were used to calculate the parameters of this model. These varied from breath to breath. A theoretical formula is derived for the sum of external work performed during inspiration for each burst. It is shown mathematically that there exists a local minimum in the calculated work as the values of the individual tidal volumes in this formula are allowed to vary, with the constraint that the sum of the ventilation during the cycle is as measured. The measured values of inspiratory timing, the starting volume and pressure, and the mechanical parameters for each inspiration are also used. We show that during each cycle of periodic breathing, the total of the observed external work is highly correlated with this theoretical minimum work. In addition, during the cycle, there is a pattern of overshoot and subsequent undershoot in the work with respect to the theoretical optimum, which suggests a control process operating during the cycle to minimize the work.