A generalized type of H-infinity 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 supplys H-infinity controls with good transients or assures H-infinity controls of robustness against initial-state uncertainty. A necessary and sufficient condition for an H-infinity control to attain the mixed attenuation is given. On the basis of this condition, the robustness of H-infinity controls against initial-state uncertainty is discussed.