Conditions for optimality over H

The fundamental optimization problem in worst-case frequency domain design where stability is the key constraint is \[\left( {{\text{OPT}}_{A_N } } \right)\qquad \begin{array}{*{20}c}{{\text{Given}}\,\Gamma \,{\text{a}}\,{\text{map}}\,{\text{from}}\,\mathbb{T} \times \mathbb{C}^N \,{\text{to}}\,\mathbb{R}^ + } \\ {{\text{FIND}}\,\gamma ^ * > 0\,{\text{and}}\,f^ * \in A_N \,{\text{such}}\,{\text{that}}} \\ {\gamma ^ * = \mathop {\inf }\limits_{f \in A_N } \mathop {\sup }\limits_\theta \Gamma \left( {e^{i\theta } ,f\left( {e^{i\theta } } \right)} \right) = \mathop {\sup }\limits_\theta \Gamma \left( {e^{i\theta } ,f^ * \left( {e^{i\theta } } \right)} \right).} \\ \end{array} \] Here $A_N $ denotes the $\mathbb{C}^N $-valued functions on the unit circle $\mathbb{T}$ with analytic continuation to the unit disk $\mathbb{D}$ that are continuous on the closed unit disk. The special case where the sublevel sets of $\Gamma $ in z are “disks” in $\mathbb{C}^N $ is the main mathematical problem in the area called $H...