A complex geometry approach to a problem in H∞ control theory

Abstract: 

$\mu$-synthesis is a part of the theory of robust control of systems comprising interconnected devices each of whose outputs depend linearly on the inputs. It has long been known that the ability to stabilize such a system, given uncertainty in its parameters, is associated to a Pick--Nevanlinna-type interpolation problem into a classical Cartan domain. It was realized in the 1990s that for a system in which only a few of the governing parameters are prone to uncertainties, its stabilization is more efficiently understood in terms of an interpolation problem into the ``unit ball'' relative to a homogeneous functional called the structured singular value. These ``unit balls'' are unbounded (in fact, they are non-hyperbolic) which vitiates the interpolation problem. However, by the work of Agler--Young from the early-2000s, one is led to suspect that the latter type of interpolation problem, whenever the Pick--Nevanlinna-type data are in general position, is equivalent to an interpolation problem on a bounded domain of much lower dimension. Moreover, one conjectures that the latter domain is a categorical quotient of the ``unit ball'' under the action of a classical Lie group. In this talk, we shall compute these categorical quotients for a family of problems using absolutely elementary methods, and establish the conjectured equivalence between the pertinent interpolation problems.