# Causal and Stable Input/Output Structures on Multidimensional Behaviours

Wood, J, Sule, V R and Rogers, E (2005) Causal and Stable Input/Output Structures on Multidimensional Behaviours. SIAM Journal on Control and Optimization, 43, (4), 1493-1520.

## Description/Abstract

In this work we study multidimensional (nD) linear differential behaviours with a distinguished independent variable called time''. We define in a natural way causality and stability on input/output structures with respect to this distinguished direction. We make an extension of some results in the theory of partial differential equations, demonstrating that causality is equivalent to a property of the transfer matrix which is essentially hyperbolicity of the $P^{c}$ operator defining the behaviour $({\cal{B}})_{0,y}.$ We also quote results which in effect characterise time autonomy for the general systems case. Stability is likewise characterized by a property of the transfer matrix. We prove this result for the 2D case and for the case of a single equation; for the general case it requires solution of an open problem concerning the geometry of a particular set in ${\mathbb{C}}^{n}.$ In order to characterize input/output stability we also develop new results on inclusions of kernels, freeness of variables, and closure with respect to ${\cal{S}}, {\cal{S}}^{\prime}$ and associated spaces, which are of independent interest. We also discuss stability of autonomous behaviours, which we beleive to be governed by a corresponding condition.

Item Type: Article
Divisions : Faculty of Physical Sciences and Engineering > Electronics and Computer Science > Southampton Wireless Group
ePrint ID: 259276
Accepted Date and Publication Date:
Status
2005Published
Date Deposited: 08 Mar 2005