Extremal holomorphic maps and the symmetrised bidisc

We introduce the class of $n$-extremal holomorphic maps, a class that generalises both finite Blaschke products and complex geodesics, and apply the notion to the finite interpolation problem for analytic functions from the open unit disc into the symmetrised bidisc $\Gamma$. We show that a well-known necessary condition for the solvability of such an interpolation problem is not sufficient whenever the number of interpolation nodes is 3 or greater. We introduce a sequence $\mathcal{C}_\nu, \nu \geq 0,$ of necessary conditions for solvability, prove that they are of strictly increasing strength and show that $\mathcal{C}_{n-3}$ is insufficient for the solvability of an $n$-point problem for $n\geq 3$. We propose the conjecture that condition $\mathcal{C}_{n-2}$ is necessary and sufficient for the solvability of an $n$-point interpolation problem for $\Gamma$ and we explore the implications of this conjecture. We introduce a classification of rational $\Gamma$-inner functions, that is, analytic functions from the disc into $\Gamma$ whose radial limits at almost all points on the unit circle lie in the distinguished boundary of $\Gamma$. The classes are related to $n$-extremality and the conditions $\mathcal{C}_\nu$; we prove numerous strict inclusions between the classes.Comment: 40 page

Carath\'eodory extremal functions on the symmetrized bidisc

We show how realization theory can be used to find the solutions of the Carath\'eodory extremal problem on the symmetrized bidisc $$G \stackrel{\rm{def}}{=} \{(z+w,zw):|z|<1, \, |w|<1\}.$$ We show that, generically, solutions are unique up to composition with automorphisms of the disc. We also obtain formulae for large classes of extremal functions for the Carath\'eodory problems for tangents of non-generic types.Comment: 24 pages, 1 figure. This version contains some minor changes. It is to appear in a volume of Operator Theory: Advamces and Applications, Birkhause

A case of mu-synthesis as a quadratic semidefinite program

We analyse a special case of the robust stabilization problem under structured uncertainty. We obtain a new criterion for the solvability of the spectral Nevanlinna-Pick problem, which is a special case of the $\mu$-synthesis problem of $H^\infty$ control in which $\mu$ is the spectral radius. Given $n$ distinct points \la_1,\dots,\la_n in the unit disc and $2\times 2$ nonscalar complex matrices $W_1,\dots,W_n$, the problem is to determine whether there is an analytic $2\times 2$ matrix function $F$ on the disc such that F(\la_j)=W_j for each $j$ and the supremum of the spectral radius of F(\la) is less than 1 for \la in the disc. The condition is that the minimum of a quadratic function of pairs of positive $3n$-square matrices subject to certain linear matrix inequalities in the data be attained and be zero.Comment: 37 pages, 4 figures. To appear in SIAM J. Control and Optimizatio