In this paper, we study the problem of computing the minimum battery capacity required to stabilize a scalar plant communicating with an energy harvesting sensor over a wireless communication channel. We prove that a particular greedy battery management policy suffices to stabilize the plant, and demonstrate that stability of the system under the greedy policy can be checked by a linear program. Moreover, we show that a critical battery capacity exists, below which no policy can stabilize the system, which itself can be computed by solving a sequence of linear programs which grows logarithmically with respect to the maximum allowed storage capacity. The first of these results address an open question pertaining to the stability of energy harvesting control systems. The last allows us to efficiently compute the smallest battery capacity required to stabilize a given system, which addresses a problem of importance when device size or cost are significant concerns.