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In this paper, we analyze a hybrid electric vehicle structure and propose a simple and computationally cheap yet close-to-optimal energy-management strategy. The underlying hybrid configuration is of parallel type. Furthermore, the parallel hybrid does not include a clutch between the electric motor and the engine. The considered vehicle uses a six-speed dual-clutch transmission (DCT). The proposed supervisory controller considers both the torque split management and the gear-shifting strategy. Results show that the proposed approach achieves fuel consumption (FC) within 1% of the global optimum for most driving cycles. In addition, using the proposed controller, the number of gears in the gearbox can be reduced to three while maintaining the FC within 2% of what is achieved with a six-speed gearbox. Moreover, the results shown in this paper are not sensitive to limitations and energy losses associated with gear shifting.