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Kolloquium Technische Kybernetik im Wintersemester 2016/17

--- Titel: Linear Matrix Inequalities in Matrix Variables

Veranstaltungsdatum:  28. März 2017 16:00 Uhr

Prof. Bill Helton
Dept. of Mathematics
University of California San Diego, U.S.A.

 

Tuesday, 2017-03-28 16:00
IST-Seminar-Room V9.22 - Pfaffenwaldring 9 - Campus Stuttgart-Vaihingen

 

Abstract

 

One of the main developments in optimization over the last 20 years is Semi-Definite Programming. It treats problems which can be expressed as a linear matrix inequality (LMI). Any such problem is necessarily convex, so the determining the scope and range of applicability comes down to the question:

How much more restricted are LMIs than Convex Matrix Inequalities?

There are several main branches of this pursuit. First there are two fundamentally different classes of linear systems problems. Ones whose statements do depend on the dimension of the system "explicitly" and ones (called dimension free) whose statements do not. Dimension dependent systems problems lead to traditional semialgebraic geometry problems, while dimensionless systems problems lead directly to a new area which might be called noncommutative semialgebraic geometry. The classic problems of control are dimension free (noncommutative). Indeed, problems which are entirely specified by a signal flow diagram are dimension free.

In this talk after laying out the distinctions above we give conjectures about the LMI vs convexity question their current status and pursuits that go beyond this.

   

  

Biographical Information

 

Bill Helton typically works on functional analysis problems arising from a variety of areas. His earlier articles concerned circuit theory, distributed systems, and aspects of the theory of operators on Hilbert space which come from circuits, systems, differential and integral equations, spectral theory and non-commutative differential geometry. The theoretical studies of amplifier design SISO by Youla's and MIMO by Helton were the first papers in H-infinity engineering, and developed methods used by the originators of H-infinity control. The focus of Helton's recent work is treating the algebra behind matrix inequalities in a systematic way, computer algebra aimed at linear systems research where one has unknowns which are matrices. Helton's group is the main provider to Mathematica of general noncommutative computer algebra capability.

Bill Helton received the bachelor's degree in mathematics from the University of Texas, Austin, the Master's and Ph.D degree in mathematics from Stanford University. He was at SUNY, Stony Brook, as an Assistant and Associate Professor. He visited UCLA for six months and subsequently moved to UC San Diego where he is currently Professor of Mathematics Emeritus and Research Professor. He won a TAC Outstanding Paper Award and he was a Guggenheim Fellow, is an IEEE Fellow and an AMS Fellow.