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I'm currently studying Volterra series as an input-ouput-representation of nonlinear systems. For this, I found a variety of interesting papers. For instance, Lesiak & Krener (1978), Brockett (1976) and Boyd & Chua (1985). But it seems like there are two approaches to the Definition/Derivation of Volterra series.

I like the approach in Lesiak & Krener the most, because there, the authors show their results on existence and uniqueness with mathematical rigour. For an analytic control-affine system $\dot x=f(x)+g(x)u$ there exists an Operator $\psi$ mapping input functions $u$ from a suitable input function space to Outputs $y$ contained in a suitable output function space. This Operator possesses the following form:

$y(t)=(\psi u)(t)=w_0(t)+\int_0^T w_1(t,\tau_1)u(\tau_1)d\tau_1+\int_0^T\int_0^{\tau_1}w_2(t,\tau_1,\tau_2)u(\tau_1)u(\tau_2)d\tau_1 d\tau_2+\ldots$

The second approach is done via the buzzword 'multiple convolution', i.e. for LTI Systems $\dot x=Ax+Bu$, $y=Cx$, $x(0)=x_0$ we have $y(t)=Ce^{At}x_0+\int_0^tCe^{A(t-\tau)}Bu(\tau)d\tau$. If we set $x_0=0$, then we obtain the standard convolution formula $y=w_1\ast u$, where $w_1=Ce^{A(\cdot-\tau)}$ corresponds to the first order Volterra kernel. Authors now extend this approach by adding (probably infinitely) many summands, thus rendering the multi-convolution formula:

$y(t)=(Nu)(t)=h_0+\int h_1(\tau_1)u(t-\tau_1)d\tau_1+\int h_2(\tau_1,\tau_2)u(t-\tau_1)u(t-\tau_2)d\tau_1d\tau_2+\ldots$

Q1: How do these two approaches correspond to each other? That is, how can I transform kernels $w_i(t,\tau_1,\ldots,\tau_i)$ and inputs $u(\tau_1),\ldots,u(\tau_i)$ to time-invariant kernels $h_i(\tau_1,\ldots,\tau_i)$ and inputs with retarded arguments $u(t-\tau_1),\ldots,u(t-\tau_i)$? In the case it does, how does the integration domain change/what does it look like for the operator $N$?

Q2: Being an approximation method that Volterra series are, how can I apply the Stone-Weierstrass-Theorem to a nonlinear system $\dot x=f(x,u)$, $y=h(x)$, if $f,g,h$ are analytic or $k$ times continuously differentiable? This means, how does the derivation look like, if one studies more general systems than control-affine systems as was done in Lesiak & Krener.

I greatly appreciate any help, whether it is a full proof/derivation argument, a helpful hint or another paper with good explanations!

Best regards, Joe

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