Actuator power consumption is a limiting factor in mobile robot design. In this paper we introduce the concept of an energy-recycling actuator, which uses an array of springs and clutches to capture and return elastic energy in parallel with an electric motor. Engaging and disengaging clutches appropriately could reduce electrical energy consumption without sacrificing controllability, but presents a challenging control problem. We formulated the optimal control objective of minimizing actuator power consumption as a mixed-integer quadratic program (MIQP) and solved for the global minimum. For a given actuator design and a wide range of simulated torque and rotation patterns, all corresponding to zero net work over one cycle, we compared optimized actuator energy consumption to that of an optimized gear motor with simple parallel elasticity. The simulated energy-recycling actuator consumed less electrical energy: 57% less on average and 80% less in the best case. These results demonstrate an effective approach to optimal control of this type of system, and suggest that energy-recycling actuators could substantially reduce power consumption in some robotics applications.