This paper deals with the stabilization of flutter phenomena of cantilevered pipes conveying fluid with constraints on input energy for control. The controller in this study is designed by using an algorithm which iteratively tunes the weighting matrix of the quadratic performance index in LQG problem. This controller design method takes the performance of the actuator into consideration. The theoretical and experimental results show that the effectiveness of the controller in stabilizing the flutter phenomena of cantilevered pipes conveying fluid varies according to the input variance constraints, i.e., the difference in the dynamic range of an actuator. Furthermore, the critical flow velocity at instability of the pipe can be estimated by the method stated in this paper, where the system loses its stability because of the restriction of control force produced by an actuator.