# How to apply the Stone-Weierstrass-Theorem to nonlinear systems? How to integrate Volterra kernels for TI systems?

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