Details:
Title  Canonical State Representations and Hilbert Functions of Multidimensional Systems  Author(s)  Ulrich Oberst  Text  AMS subject classification: 93B,93C,13P  Type  Technical Report, Misc  Abstract  A basic and substantial theorem of onedimensional systems theory,
due to R. Kalman, says that an arbitrary input/output behavior with
proper transfer matrix admits an observable state representation which,
in particular, is a realization of the transfer matrix. The state equations
have the characteristic property that any local, better temporal, state at
time zero and any input give rise to a unique global state or trajectory of
the system or, in other terms, that the global state is the unique solution
of a suitable Cauchy problem. With an adaption of this state property
to the multidimensional situation or rather its algebraic counterpart we
prove that any behavior governed by a linear system of partial differen
tial or difference equations with constant coe±cients is isomorphic to a
canonical state behavior which is constructed by means of Groebner bases.
In contrast to the onedimensional situation and to J.C. Willems' multi
dimensional state space models the canonical state behavior is not nec
essarily a first order system. Further Kalman representations and first
order models are due to J.F Pommaret and E. Zerz. As a byproduct of
the state space construction we derive a new algorithm for the computa
tion of the Hilbert function of any finitely generated polynomial module
or behavior. J. Wood, P. Rocha et al. recognized the systems theoretic
significance of this Hilbert function in context with complexity and struc
ture indices. The theorems of this paper are constructive and have been
implemented in MAPLE in the twodimensional case and demonstrated
in a simple, but instructive example. For the standard onedimensional
systems the algorithms of this paper compare well with those from the
literature.  Keywords  state, Hilbert function, behavior, multidimensional system, partial differential equation, partial difference equation, polynomial module  Length  48 
Language  English  Year  2005  Month  June  Translation 
No  Refereed 
No  Organization 
University of Innsbruck  Institution 
Institut für Mathematik (Universität Innsbruck) 
