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Reconfigurable system-on-chip (RSoC) devices are being used to implement many battery-operated systems, where energy efficiency is a major concern. RSoCs incorporate may different components, such as processor core, reconfigurable logic, memory, etc. various power management techniques can be applied to these components. Tasks within an application can be mapped onto different components for execution. The communication and reconfiguration costs incurred under different mappings significantly impact the overall system energy dissipation. In order to achieve energy-efficient designs on RSoCs, we develop (a) a performance model to abstract a general class of RSoC architectures for application development, (b) a mathematical formulation of the energy-efficient mapping problem for a class of applications, and (c) a dynamic programming algorithm that minimizes the system energy dissipation. We illustrate our approach by mapping two beamforming applications onto Xilinx Virtex-II Pro. For these two applications, our approach leads to an average 52% energy reduction over a greedy algorithm.