December 13, 2004
Victor Vinnikov
Title: Linear Matrix Inequalities, Matrix Convexity, And Realization Theory Of Noncommutative Rational Functions
In view of widespread applications of Linear Matrix Inequalities (LMIs) in system and control theory, it is a natural and important question which convex sets with an algebraic boundary can be represented by LMIs, and how can one systematically find such a representation if it exists. If the variables are scalars then there are certain very strong restrictions which are necessary (and in some cases also known to be sufficient) for the existence of an LMI representation. However, if the variables are matrices of an arbitrary size appearing as noncommuting variables in noncommutative algebraic expressions, then there are good reasons to conjecture that convexity implies an LMI representations (which can furthermore be obtained explicitly by a ``good'' algorithm). In this talk I will sketch a proof of this conjecture for the special case of ``sublevel sets'' of a convex noncommutative rational function. The main ingredient in the proof is realization theory for noncommutative rational functions, due originally to Schutzenberger and revisited recently by several researchers from different perspectives. One of the main issues is whether the singularities of a noncommutative rational function coincide with the singularities of the resolvent in a minimal realization. This is a joint work with Bill Helton and Scott McCullough.