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This theoretical study investigates strategies for minimizing Joule losses in resistive random access memory (ReRAM) cells, which are also referred to as memristive devices. Typically, the structure of ReRAM cells involves a nanoscale layer of resistance-switching material sandwiched between two metal electrodes. The basic question that we ask is what is the optimal driving protocol to switch a memristive device from one state to another. In the case of ideal memristors, in the most basic scenario, the optimal protocol is determined by solving a variational problem without constraints with the help of the Euler-Lagrange equation. In the case of memristive systems, for the same situation, the optimal protocol is found using the method of Lagrange multipliers. We demonstrate the advantages of our approaches through specific examples and compare our results with those of switching with constant voltage or current. Our findings suggest that voltage or current control can be used to reduce Joule losses in emerging memory devices.