Session 1B: Inaugural Lectures by Fellows/Associates

Musti J Swamy

The role of geometric quotients in a problem in control theory

It is well known, in the theory of control of systems comprising interconnected devices each of whose outputs depend linearly on their inputs, that the ability to stabilize such a system is associated with a Pick--Nevanlinna-type interpolation problem into a classical Cartan domain. It was shown in the 1990s that for a system in which only a few, but not all, of the system parameters are prone to uncertainties, its stabilization is more efficiently understood in terms of a complex-analytic interpolation problem into the ``unit ball’’ determined by a non-negative function called the structured singular value. These ``unit balls’’ are non-hyperbolic, which vitiates the interpolation problem. By the work of Agler and Young, one is led to suspect that the latter type of interpolation problem, whenever the interpolation data are in general position, is equivalent to an interpolation problem on a bounded domain of much lower dimension. Such domains turn out to be categorical quotients of the above mentioned ``unit balls’’ under the action of a classical Lie group. In this talk, we shall elaborate on the above assertions, present the categorical (or GIT) quotients for a family of problems, and establish the equivalence of the two interpolation problems hinted at.