We study the problem of optimally controlling a set of nonideal rechargeable batteries that can be shared to perform a given amount of work over some specified time period. We seek to maximize the minimum residual energy among all batteries at the end of this period by optimally controlling the discharging and recharging process at each battery. Modeling a battery as a dynamic system, we adopt a kinetic battery model and formulate an optimal control problem under the constraint that discharging and recharging cannot occur at the same time. We show that the optimal solution must result in equal residual energies for all batteries as long as such a policy is feasible. This simplifies the task of subsequently deriving explicit solutions for the problem, which is accomplished by first analyzing the 2-battery case and then considering the general N-battery case .
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