A control system may be viewed as effecting a transformation, under fixed initial conditions, of the input, the control, to the output; the controls are to be chosen from a given family of functions, and consequently the outputs are members of a set of functions. This viewpoint may be extended in various ways. For example, the initial conditions may be allowed to vary within a given set. Alternatively, one may start with a set of possible outputs, and try to find one or more allowed controls and permissible initial conditions, which will yield the set of outputs. Put concretely, one can ask when a spacecraft, starting at one of a set of initial conditions, and “steered” by one of a set of prescribed controls, can be made to follow any one of a family of curves, with preassigned . velocity. This, roughly, is what is meant by “reproducibility.” Reproducibility has been studied extensively by a number of investigators [ 1-5, 10, 111. Here we obtain necessary and sufficient conditions for the reproducibility of linear time-varying, as well as time-invariant input-output systems, and
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