In this paper, the steady-state discrete-time optimal LQG control problem is considered when the noise variance of an actuator depends linearly on the variance of the actuator signal, A first-order system is analyzed in the case of noise-free and noisy output measurement. In the standard LQG problem the actuator noise variance is assumed to be known and constant before the design of the feedback gain, Consequently, the standard LQG design can lead to the deterioration of the system performance in the considered case.