Type of Document 
Dissertation 
Author 
Boquet, Grant Michael

Author's Email Address 
gboquet@vt.edu 
URN 
etd03032010180226 
Title 
Geometric Properties of OverDetermined Systems of Linear Partial Difference Equations 
Degree 
PhD 
Department 
Mathematics 
Advisory Committee 
Advisor Name 
Title 
Ball, Joseph A. 
Committee Chair 
Haskell, Peter E. 
Committee Member 
Linnell, Peter A. 
Committee Member 
Renardy, Michael J. 
Committee Member 

Keywords 
 algebraic geometry
 Multidimensional linear systems
 vessels
 autonomous behavioral systems
 Livšic systems

Date of Defense 
20100219 
Availability 
unrestricted 
Abstract
We relate linear constant coefficient systems of partial difference equations (a discretization of a system of linear partial differential equations) satisfying some collection of scalar polynomial equations to systems defined over the coordinate ring of an algebraic variety. This motivates the extension of behavioral systems theory (a generalization of classical systems theory where inputs and outputs are lumped together) to the setting where the ring of operators is an affine domain and the signal space is restricted to signals which satisfy the same scalar polynomial equations. By recognizing the role of the kernel representation’s Gröbner basis in the Cauchy problem, we extend notions of controllability from the classical behavioral setting to accommodate this generalization. We then address the question as to when an autonomous behavior admits a Livšicsystem statespace representation, where the state update equations are overdetermined leading to the requirement that the input and output signals satisfy their own compatibility difference equations. This leads to a frequency domain setting involving input and output holomorphic vector bundles and a transfer function given by a meromorphic bundle map. An analogue of the Hankel realization theorem developed by J. Ball and V. Vinnikov then leads to a Livšicsystem statespace representation for an autonomous behavior satisfying some natural additional conditions.

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