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The possibility of controlling the resistance of a memristive giant magnetoresistance (GMR) system via current-induced domain-wall motion was investigated. For a narrow spin-valve structure, current-induced domain-wall motion in the free layer can be detected once the current density exceeds a critical threshold. Then, the resistance of the device depends on the position of the domain wall. The GMR system shows a MR ratio of 10% in the as-prepared state. Narrow stripes were fabricated by e-beam lithography and ion-beam etching with a width of 200 nm. The stripes exhibit GMR ratios up to 8% at room temperature. Micromagnetic simulations of the domain-wall motion in the free layer allow an estimation of the time scale of the domain-wall migration in the stripe. Furthermore, the simulations were compared with measured critical current densities in the free layer with and without an applied external field.