Schubert calculus in the control of linear systems Special Geometry Lecture Frank Sottile University of Wisconsin & University of Massachusetts Friday, March 31 2:30-3:30 PM,   1634 LGRT

The purpose of this talk will be to describe how geometry is used to study a specific problem in control theory. While intended for graduate students, all are welcome, and the talk will only assume some familiarity with the undergraduate topics of polynomials, systems of linear differential equations, and some linear algebra. In particular, everything described below will be explained from first principles.

Suppose we have a driven physical system whose evolution is governed by a system of linear differential equations. If we use some measured outputs of the system to drive the system (via a static linear feedback law) then the elementary theory of linear differential equations tells us that the behaviour of the resulting closed system is encapsulated by its characteristic polynomial.

The pole placement problem asks, given such a driven linear physical system, for which feedback laws will the resulting closed system have a given characteristic polynomial? Interestingly, this (apparently linear) problem turns out to be highly non-linear, and its analysis uses some ideas from classical algebraic geometry, including Grassmannians and the Schubert calculus. We describe the pole placement problem, its relation to geometry, and also its resolution using the Schubert calculus.