Analytic interpolation into the tetrablock and a $\mu$-synthesis problem

TL;DR

This paper establishes a criterion for solving a specific $$-synthesis problem by characterizing when an analytic matrix-valued function exists with certain bounds and interpolation conditions, using a realization theorem for the tetrablock.

Contribution

It provides a new realization theorem for analytic functions into the tetrablock and a solvability criterion for related interpolation problems.

Findings

Established a realization theorem for functions into the tetrablock.

Derived a solvability criterion for a class of $$-synthesis problems.

Connected interpolation conditions with structured singular value bounds.

Abstract

We give a solvability criterion for a special case of the $\mu$-synthesis problem. That is, we prove the necessity and sufficiency of a condition for the existence of an analytic $2 \times 2$ matrix-valued function on the disc subject to a bound on the structured singular value and satisfying a finite set of interpolation conditions. To do this we prove a realization theorem for analytic functions from the disc to the tetrablock. We also obtain a solvability criterion for the problem of analytic interpolation from the disc to the tetrablock.