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A branch of control theory where the performance of a dynamical system (cf. [[Automatic control theory|Automatic control theory]]) is appraised in terms of the  $  H  ^  \infty  $-
norm. The Banach space  $  H  ^  \infty  $(
named after G.H. Hardy, cf. [[Hardy classes|Hardy classes]]) consists of all complex-valued functions of a complex variable which are analytic and of bounded modulus in the open right half-plane. The norm of such a function is the supremum modulus:

$$ 
\| F \| _  \infty  = \sup _ { \mathop{\rm Re}  s > 0 }  | F( s) | .
$$

By a theorem of Fatou (cf. [[Fatou theorem|Fatou theorem]]), such a function has a boundary value  $  F( i \omega ) $
for almost-all  $  \omega $, 
and, moreover,

$$ 
\| F \| _  \infty   =   \mathop{\rm esssup} _  \omega   | F( i \omega ) | .
$$

The theory of  $  H  ^  \infty  $
control was initiated by G. Zames [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], who formulated a basic feedback problem as an optimization problem with an operator norm, in particular, an  $  H  ^  \infty  $-
norm. Relevant contemporaneous works are those of J.W. Helton [[#References|[a4]]] and A. Tannenbaum [[#References|[a5]]].

The theory treats dynamical systems represented as integral operators of the form

$$ 
y( t)  =  \int\limits _ { 0 } ^ { t }  g( t- \tau ) x( \tau )  d \tau .
$$

Here  $  g $
is sufficiently regular to make the input-output mapping  $  x \mapsto y $
a bounded operator on  $  L _ {2} [ 0 , \infty ) $. 
Taking Laplace transforms gives  $  Y( s)= G( s) X( s) $. 
The function  $  G $
is called the [[Transfer function|transfer function]] of the system and it belongs to  $  H  ^  \infty  $
because the integral operator is bounded. Moreover, the  $  H  ^  \infty  $-
norm of  $  G $
equals the norm of the integral operator, i.e.,

$$ \tag{a1 }
\| G \| _  \infty  = \sup _ {\| x \| _ {2} \leq  1 } \ 
\| y \| _ {2} .
$$

There are two prototype problems giving rise to an optimality criterion with the  $  H  ^  \infty  $-
norm. The first is the problem of robust stability of the feedback system

<img src="https://www.encyclopediaofmath.org/legacyimages/common_img/h046020a.gif" style="border:1px solid;"/>

Figure: h046020a

Here  $  P $
and  $  C $
are transfer functions in  $  H  ^  \infty  $, 
and  $  X _ {1} $, 
$  X _ {2} $, 
$  Y _ {1} $, 
$  Y _ {2} $
are Laplace transforms of signals;  $  P $
represents a  "plant" , the dynamical system which is to be controlled, and  $  C $
represents the  "controller"  (cf. also [[Automatic control theory|Automatic control theory]]). The figure stands for the two equations

$$ 
Y _ {1}  =  X _ {1} + PY _ {2} ,\  Y _ {2}  =  X _ {2} + CY _ {1} ,
$$

which can be solved to give

$$ 
\left [ \begin{array}{c}
Y _ {1} \\
 Y _ {2} 
\end{array}
 \right ]  = \ 
\left [ 
\begin{array}{cc}

\frac{1 }{1- PC }
  &amp;
\frac{P }{1- PC }
  \\

\frac{C }{1- PC }
  &amp;
\frac{1 }{1- PC }
  \\
\end{array}

\right ]
\left [ \begin{array}{c}
X _ {1} \\
 X _ {2} 
\end{array}
 \right ] .
$$

Therefore, the input-output mapping for the feedback system has four transfer functions. The feedback system is said to be internally stable if these four transfer functions are all in  $  H  ^  \infty  $. 
A simple sufficient condition for this is  $  \| PC \| _  \infty  < 1 $.

Internal stability is robust if it is preserved under perturbation of  $  P $. 
There are several possible notions of perturbation, typical of which is additive perturbation. So suppose  $  P $
is perturbed to  $  P+ \Delta P $, 
with  $  \Delta P $
in  $  H  ^  \infty  $. 
About  $  \Delta P $
it is assumed that only a bound on  $  | \Delta P( i \omega ) | $
is known, namely,

$$ 
| \Delta P( i \omega ) |  <  | R( i \omega ) | ,\  \textrm{ a.a.  } \ 
\omega ,
$$

where  $  R \in H  ^  \infty  $. 
J.C. Doyle and G. Stein [[#References|[a6]]] showed that internal stability is preserved under all such perturbations if and only if

$$ \tag{a2 }
\| RC( 1- PC) ^ {- 1 } \| _  \infty   <  1 .
$$

This leads to the robust stability design problem: Given  $  P $
and  $  R $, 
find  $  C $
so that the feedback system is internally stable and (a2) holds.

The second problem relates to the same feedback system. Suppose  $  X _ {2} = 0 $, 
$  X _ {1} $
represents a disturbance signal, and the objective is to reduce the effect of  $  X _ {1} $
on the output  $  Y _ {1} $. 
The transfer function from  $  X _ {1} $
to  $  Y _ {1} $
equals  $  ( 1- PC) ^ {- 1 } $. 
Suppose, in addition, that the disturbance is not a fixed signal, but can be the output of another system with any input in  $  L _ {2} [ 0 , \infty ) $
of unit norm; let this latter system have transfer function  $  W $
in  $  H  ^  \infty  $. 
Then, in view of (a1), the supremal  $  L _ {2} [ 0 , \infty ) $-
norm of  $  y _ {1} $
over all such disturbances equals  $  \| W( 1- PC) ^ {- 1 } \| _  \infty  $. 
This leads to the disturbance attenuation problem: Given  $  P $
and  $  R $, 
find  $  C $
to achieve internal stability and minimize  $  \| W( 1- PC) ^ {- 1 } \| _  \infty  $.

The above two problems are special cases of the more general standard  $  H  ^  \infty  $
control problem. It can be solved by reduction to the Nehari problem of approximating a function in  $  L _  \infty  $(
bounded functions on the imaginary axis) by one in  $  H  ^  \infty  $. 
A summary of this theory is in [[#References|[a7]]], and a detailed treatment is in [[#References|[a8]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  G. Zames,   "Feedback and complexity, Special plenary lecture addendum" , ''IEEE Conf. Decision Control'' , IEEE  (1976)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  G. Zames,   "Optimal sensitivity and feedback: weighted seminorms, approximate inverses, and plant invariant schemes" , ''Proc. Allerton Conf.'' , IEEE  (1979)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  G. Zames,   "Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses"  ''IEEE Trans. Auto. Control'' , '''AC-26'''  (1981)  pp. 301–320</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  J.W. Helton,   "Operator theory and broadband matching" , ''Proc. Allerton Conf.'' , IEEE  (1979)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  A. Tannenbaum,   "On the blending problem and parameter uncertainty in control theory"  ''Techn. Report Dept. Math. Weizmann Institute''  (1977)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  J.C. Doyle,   G. Stein,   "Multivariable feedback design: concepts for a classical modern synthesis"  ''IEEE Trans. Auto. Control'' , '''AC-26'''  (1981)  pp. 4–16</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  B.A. Francis,   J.C. Doyle,   "Linear control theory with an $H ^ { \infty }$ optimality criterion"  ''SIAM J. Control and Opt.'' , '''25'''  (1987)  pp. 815–844</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  B.A. Francis,   "A course in $H _ { \infty }$ control theory" , ''Lect. notes in control and inform. science'' , '''88''' , Springer  (1987)</td></tr></table>