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For a given scenario in the context of control system, I'm trying to investigate how the $H_\infty$ norm can be calculated for a transfer function as follows:
$$G(s)= \frac{w_n^2}{s^2 +2\zeta w_ns +w_n^2 }$$

$$\left \| G \right \| _\infty = \max \limits _{\omega} |G(j\omega)|$$

$$\left \| G \right \| _\infty = \sup \limits _{\omega} |G(j\omega)|$$

I know how to deal with such problems when the case is specific one, for example the frequency value is known. However, I get confused with this general case. any idea

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    $\begingroup$ Can you try to formulate a specific question? I'm not sure what you're asking us. $\endgroup$ – Marcus Müller Feb 27 at 9:25
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    $\begingroup$ So, what is the specific question you're asking? I'm really not sure what you need help with! What's the first step that you don't know how to do? $\endgroup$ – Marcus Müller Feb 27 at 11:51
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    $\begingroup$ That's not specific. $\endgroup$ – Marcus Müller Feb 27 at 12:27
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    $\begingroup$ Can you say what you don't understand about your max and sup formulas? I mean, the answer for "what do I need to calculate" is given by these formulas, explicitly. $\endgroup$ – Marcus Müller Feb 27 at 12:39
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    $\begingroup$ @Rwy5 See en.wikipedia.org/wiki/Maxima_and_minima $\endgroup$ – MBaz Feb 27 at 16:12

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