H-infinity methods in control theory and Hardy space. Sorry this is very simple but I do not know. Why the H-infinity methods that are used in control theory are said to work on Hardy space? If the question is not appropriate then how H-infinity methods are related to Hardy space? May I request few comments? Sorry if the question is not clear:It refers to the Wikipedia link:
http://en.wikipedia.org/wiki/Hardy_space
In the page it is written as:
Hardy spaces have a number of applications in mathematical analysis itself, as well as in control theory (such as H∞ methods) and in scattering theory.
So my question was what is the best way to understand how Hardy space is utilized by  control theory. Hope it is clear now.
 A: We need some mathematical descriptions that are very useful to fully describe our problems and at the same time they can give us enormous results. The Hardy spaces are very appropriate to characterize  both systems and signals. So we can use the results of this profound mathematics in order to solve our problems that are encountered in control systems analysis. 
First of all control systems can be characterized by operators and their inputs outputs are signals. $\mathcal{H}_\infty$ space can be used to describe certain kind of systems which are stable. These operators can be used as a mapping between certain kind of signal spaces that is $\mathcal{L}_2[0,\infty)$, i.e. the energy signals.
The most important property of the operators lie in $\mathcal{H}_\infty$ is the sub-multiplicative property. If $T,\Delta\in\mathcal{H}_\infty$  then 
$$\|T\Delta\|_{\mathcal{H}_\infty}\leq\|T\|_{\mathcal{H}_\infty}\|\Delta\|_{\mathcal{H}_\infty}$$
which is an important property to be used to study the stability of systems with uncertain operator $\Delta$. All we need to know about the uncertain operator its $\mathcal{H}_\infty$ norm instead of trying to describe it exactly. 
Also the $\mathcal{H}_\infty$ describes the energy gain of the system. Loosely speaking if the $\|T\Delta\|_{\mathcal{H}_\infty}<1$ then we gurantee the stability of the uncertain system when its inputs (disturbances) belong to the Hardy-Hilbert space $\mathcal{H}_2$ which is a contraction property. 
But if the operators are defined using $\mathcal{H}_2$ we can not use the useful sub-multiplicative analysis so it is not appropriate for uncertain systems.
The main concern in $\mathcal{H}_\infty$ control is minimize the norm of the $\|T\|_{\mathcal{H}_\infty}$ such that if $\Delta\leq\gamma$ the following propoerty holds 
$$\|T\Delta\|_{\mathcal{H}_\infty}< 1$$
this holds if $\|T\|_{\mathcal{H}_\infty}<1/\gamma$.
To sum up, it is somehow enough to deal with energy signals that are described by the Hardy-Hilbert Space $\mathcal{H}_2$. As well as the Hardy space $\mathcal{H}_\infty$  is very useful to describe a broad range of uncertain linear systems.
