The Hankel Pencil Conjecture
 
 
Description:  In 1968 Heyman proved a lemma that turned out to be a corner stone for the theory of linear multivariable state space systems, a branch of control theory: he showed that fields have the feedback cyclization- or FC-property, that is, every reachable pair $(A,B)$ of matrices over the field is also a cyclizable pair. The FC-property was shown to hold for a number of further rings (apart from fields) and to not hold for others. In 1981 Bumby, Sontag, Sussman and Vasconcelos (BSSV) conjectured that the polynomial ring $\CC[y]$ is FC. In a series of papers Wiland Schmale shows that apart from possibly two exceptional families, all reachable pairs of matrices over $\CC[y]$ are indeed cyclizable; and in a recent paper with Pramod Sharma he shows that cyclizability of one of these families would follow from the truth of a deceptively simply looking conjecture on a certain family of Toeplitz (or Hankel) pencils. This conjecture has been posted also in IMAGE (the newsletter of the Linear Algebra Society) in April 2003. A `solution' by a prolific linear algrebraist has recently appeared; but an example by Harald Wimmer shows it to be false. However, some true progress, found together with M. Celeste Gouveia can be reported.
Area(s):
Date:  2007-11-28
Start Time:   14:30
Speaker:  Alexander Kovacec (CMUC/Mat. FCTUC)
Place:  5.5
Research Groups: -Algebra and Combinatorics
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