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The propagation of a magnetic domain wall (DW) in a submicron magnetic wire consisting of a magnetic/nonmagnetic/magnetic trilayered structure with asymmetric notches was investigated by utilizing the giant magnetoresistance effect. The propagation direction of a DW was controlled by a pulsed local magnetic field, which nucleates the DW at one of the two ends of the wire. It was found that the depinning field of the DW from the notch depends on the propagation direction of the DW.