Feedback control of linear systems and the real Schubert calculus University of Washington Mathematics Colloquium Frank Sottile MSRI 17 November 1998

In 1981, Brockett and Byrnes showed how the static feedback laws which control a given linear system are determined through the Schubert calculus of enumerative geometry. This talk will describe that connection and propose numerical homotopy methods to solve the resulting systems of polynomials.

The related questions of finding real feedback laws and of trying to do Schubert's calculus over the real numbers are intertwined with a precise conjecture of Shapiro and Shapiro. We discuss this connection and present some impressive computational evidence in support of this conjecture. We close with a new result, proving a version of this conjecture and showing it is possible to do Schubert's calculus over the real numbers.