Uniqueness of weakly minimal analytic realizations

After Crouch [2], we clarify the relation between minimal realizations Sussmann [7]-[11] and weakly minimal realizations Hermann and Krener [3] of a given input-output mapping in the analytic case. For a given input-output mapping, all the weakly minimal analytic realizations are classified. Our result, applied to realization theory, is exactly Crouch's one, and can be derived from Sussmann's paper [8]. Otherwise, it is closely related to other works of Krener [4] and Hermann and Krener [3]. Our contribution is essentially a presentation of the above-mentioned results in a general unified manner.