In the linear fixed-end-point optimal control it is usually necessary to give the system sudden input changes at the initial and the terminal time in order to satisfy the boundary conditions. This paper proposes a method to realize the linear optimal control by means of a time-invariant control law and a compensation input without giving those sudden input changes to the system, and shows the procedure to obtain the compensation input. In this paper, the problem of the step transition of a carriage on which an inverted pendulum is mounted is treated as an example, and the results of simulation and experiment are given. As a result of the experiment, it is shown that the accuracy of the end state by the proposed method is better than by the method without considering the control of input changes. Because of this feature, the proposed method can be an effective means of control for problems like the biped walking locomotion in which the same sequence of motion is repeated.