This paper deals with a generalized H∞ control attenuating initial-state uncertainties. An H∞ control problem, which treats a mixed attenuation of disturbance and initial-state uncertainty for linear time-invariant systems in the infinite-horizon case, is examined. The mixed attenuation supplies H∞ controls with good transients and assures H∞ controls of robustness against initial-state uncertainty. We derive a necessary and sufficient condition of the generalized mixed attenuation problem. Furthermore we apply this proposed method to a magnetic suspension system, and evaluate attenuation property of the proposed generalized H∞ control approach.