The Local Realization of Generating Series of Finite Lie Rank

Realization of nonlinear systems by state-space is a classical problem in control theory. This problem has been completely solved by Kaiman [10] in the case of linear systems. Similarly, it was solved for bilinear systems (see Brockett [2], d’Alessandro, Isidori, Ruberti [1], Fliess [3], Sussmann [12]). In the general case, let us mention the work of Sussmann [13], Hermann, Krener [7] and Jakubczyk [9]: they assume that the solutions are regular at any time and for any inputs. This restriction lead Fliess to study local realization of nonlinear systems [5].