# Question on h-infinity norm of a system

Consider a control system, $$\dot{x}=Ax+Bu\\ y=Cx$$.

Define the transfer function $$G(s)=C(sI-A)^{-1}B$$.

Then it is claimed that the following definitions of induced norm are equivalent.

$$\|G\|_{\infty}=\sup_{s\in \mathbb{C}}\sigma_{max}G(s)\\ \|G\|_{\infty}=\sup_{\omega\in\mathbb{R}}\sigma_{max}G(i\omega)$$.

I am looking for a proof for this claim. Basically it says that it suffices to maximize the leading singular value over purely imaginary $$s$$ rather than searching over all $$s\in \mathbb{C}$$.

• Wouldn't maximizing over $\mathbb{C}$ always lead to infinity if the minimal realization of the system has at least one pole? – fibonatic May 11 at 11:22