When someone refers 2-norm of system,L2 and H2 are used interchangeably by author and is rather confusing. Even the matlab has different functions for H-infinity norm and L-infinity norm.
as shown in picture the author refers it as L2, and later he refers it as H2
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so what is difference and when someone says 2-norm of system, is he referring to L2 or H2?
I am new, so sorry for not following the norms here.
I am not sure about your application -- and we say the $L^2$ norm of a function and not a system. But for simplicity I will explain the concepts for real valued functions. Consider an open domain $\Omega$ and a function $f:\Omega \to \mathbb{R}$. We say that $f \in L^2(\Omega)$ if $||f||_{L^2(\Omega)} < \infty$ where \begin{equation} ||f||^2_{L^2(\Omega)} = \int_{\Omega}|f(x)|^2 dx. \end{equation} Intuitively, an $L^2$ function is a function whose area under its graph is finite while you allow discontinuities in the function itself. For example if you take a sine wave and create a set of discontinuities of measure zero (delete "some" points on the sine wave), it will still be integrable and hence in $L^2$ (yet it is not continuous anymore).
Now if you consider the space $H^1(\Omega)$, it consists of all functions $f$ such that $||f||_{H^1{\Omega}} < \infty$ where \begin{equation} ||f||^2_{H^1(\Omega)} = \int_{\Omega}|f(x)|^2 + |f'(x)|^2 dx. \end{equation} Intuitively, functions in $H^1$ are functions that are weakly differentiable, that is they are differentiable everywhere except at a set of points of measure 0. That means that $f'$ has "some discontinuity points" and so $f' \in L^2$. (a very nice example is the hat functions)
Finally, using the same logic, functions $f \in H^2(\Omega)$ are those functions that are twice - weakly differentiable and so the same logic of the previous space $H^1$ applies.
