The existence and uniqueness of Volterra series for nonlinear systems

Given an input-output map described by a nonlinear control system
\dot{x}=f(x,u)
and nonlinear output
y=h(x)
, we present a simple straightforward means for obtaining a series representation of the output
y(t)
in terms of the input
u(t)
. When the control enters linearly,
\dot{x} =f(x)+ ug(x)
, the method yields the existence of a Volterra series representation. The proof is constructive and explicitly exhibits the kernels. It depends on standard mathematical tools such as the Fundamental Theorem of Calculus and the Cauchy estimates for the Taylor series coefficients of analytic functions. In addition, the uniqueness of Volterra series representations is discussed.