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In this paper, we consider a scenario where an energy harvesting sensorcontinuously monitors a system and sends time-stamped status updates to adestination. The destination keeps track of the system status through thereceived updates. We use the metric Age of Information (AoI), the time that haselapsed since the last received update was generated, to measure the"freshness" of the status information available at the destination. We assumeenergy arrives randomly at the sensor according to a Poisson process, and eachstatus update consumes one unit of energy. Our objective is to design optimalonline status update policies to minimize the long-term average AoI, subject tothe energy causality constraint at the sensor. We consider three scenarios,i.e., the battery size is infinite, finite, and one unit only, respectively.  For the infinite battery scenario, we adopt a best-effort uniform statusupdate policy and show that it minimizes the long-term average AoI. For thefinite battery scenario, we adopt an energy-aware adaptive status updatepolicy, and prove that it is asymptotically optimal when the battery size goesto infinity. For the last scenario where the battery size is one, we first showthat within a broadly defined class of online policies, the optimal policyshould have a renewal structure, i.e., the status update epochs form a renewalprocess, and the length of each renewal interval depends on the first energyarrival over that interval only. We then focus on a renewal interval, and provethat if the AoI in the system is below a threshold when the first energyarrives, the sensor should store the energy and hold status update until theAoI reaches the threshold, otherwise, it updates the status immediately. Weanalytically characterize the long-term average AoI under such athreshold-based policy, and explicitly identify the optimal threshold.

Optimal Status Update for Age of Information Minimization with an Energy Harvesting Source