This paper is concerned with the response of a controlled system after an initiating disturbance has died out. Such a transient is obtained, for example, when the load on a prime mover is suddenly rejected or the speed setting of an engine governor is instantly switched to a new value. It is assumed that the rate of change of the controlling variable with respect to time is bounded, and that the maximum rate of change can be obtained arbitrarily. Thus the speed of a hydraulic governor servo is limited. The best return to equilibrium (minimum over or under-swing, minimum duration of the transient, and so on) can be obtained under rather general conditions by having the servo or its equivalent travel only at maximum or zero speed. Control functions exist which give the optimum transients. These functions are nonlinear. The results of theoretical studies to enable the control designer to obtain optimum or nearly optimum transients are given here along with practical compromises. All results have been verified in the laboratory with physical devices (governors) of various kinds and automatically controlled systems.