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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.
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