Feedback Control of Linear Systems and 
    the Real Schubert calculus

Frank Sottile 
University of Wisconsin


   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 a new result which
shows it is possible to do Schubert's calculus over the reals and proves a
version of this conjecture.  In a similar vein, problems of enumerating
rational curves in the Grassmannian (also known as quantum enumerative 
geometry) may also be solved over the real numbers.