Energetic solutions to rate-independent systems allow for an effective modeling of many physical systems displaying hysteretic effects, e.g., phase transformations in shape-memory alloys, elastoplasticity, and ferroelectricity. For some engineering applications, optimal control of such systems is desirable. We establish existence results for these continuous-time optimal control problems by a combination of the direct method with Gamma-convergence arguments. Applicability to the common situation of a controlled external loading is demonstrated and a concrete partial differential inclusion as well as an academic model of the foreign exchange market including trading costs are investigated.