ON INPUT-OUTPUT MAPS FOR NONLINEAR-SYSTEMS VIA CONTINUITY IN A LOCALLY CONVEX TOPOLOGY

In this paper we show that the output of a nonlinear system with inputs in (L2[0, T]; R(m)) 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.