In this paper we show that the output of a nonlinear system with inputs in () whose state satisfies a nonlinear differential equation with standard smoothness conditions can be written as the composition of a nonlinear map with a linear Hilbert-Schmidt operator acting on the input. The result also extends to abstract semi-linear infinite dimensional systems. The approach is via the study of the continuity of the solution in a locally convex topology generated by seminorms of Hilbert-Schmidt operators in Hilbert space. The result reveals an entirely new structure related to nonlinear systems which can lead to useful approximation results.
Mazumdar, R. R., Kannurpatti, R., & Bagchi, A. (1995). On input/output maps for nonlinear systems via continuity in a locally convex topology. Systems and control letters, 24(24), 273-281. https://doi.org/10.1016/0167-6911(94)00031-P