This paper describes the purely inertial system that can control its attitude on an axis by the air jet, and gives the results of the adaptive optimum control of the system. It is well known that the time optimum control of the purely inertial system can be accomplished by appling positive and negative maximum torque according to the parabolic switching curve in the phase plane. In order to realize the desired switching curve a nonlinear resistor-varistor-is used in this control system. Keeping the time optimum control loop closed, the small perturbation signal is applied to the system and the inertial moment of the controlled subject is measured by the correlation method. The result of the measurement of the moment is used in the adaptive control system to give the optimum switching curve in the phase plane automatically. By changing the moment, it is experimentally verified that the switching curve is optimally set. The response of the moment measuring system to the sudden change of the moment is described. Also the effects of the test signal on the time optimum control and error of the measured moment are stated.