Existence of realizations of nonlinear analytic input-output maps

We are interested in the existence of realizations of nonlinear input-output maps when they are defined for positive times, and when the set of control values is finite. First, we give examples that point out the nature of the problems which may appear in the attempt to find global results in realization theory. Then, we state a necessary and sufficient condition for the existence of realizations and we give a sketch of the proof.