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

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.