State for 2-D Systems zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG P. Rocha* and J. C. Willems Systems and Control Group Mathematics Institute University of Groningen P.O. Box 800 9700 AV Groningen, The Netherlands zyxwvutsrqponmlkjihgfedcbaZYXWVUTSR Submitted by Paul A. Fuhrmann ABSTRACT A new definition of state for N-D systems is given in a noncausal context. This definition is based on a deterministic Markoviau-like property. It is shown that, for the particular case of (AR) %D systems, it yields systems that can be described by a special kind of first-order equations. The solutions of these equations can be simulated by means of a local line-by-line computational scheme. 1. INTRODUCTION The main motivation of this paper is to examine the concept of state for N-D systems. However, for simplicity of exposition, we will concentrate mainly on discrete 2-D systems. The theory of dynamical systems has been mainly concerned with 1-D systems, with phenomena evolving in time. 2-D systems have been introduced to describe phenomena depending on two independent variables, often regarded as spatial variables, as in image analysis. Our approach to 2-D systems is inspired by some of the recent work [5] in the area of 1-D dynamical systems. However, an important difference which we will emphasize is the following. Whereas the 1-D systems considered in [5] are defined over time, and have therefore a natural preferred direction (namely forward time, past and future), we will not view 2-D systems as having a preferred direction. In fact, when considering 2-D systems there are *On leave from the Grupo de Matematica Aplicada, Faculdade de Ciencias, Universidade do Porte. Supported by the Calouste Gulbenkian Foundation, Portugal. LINEAR ALGEBRA AND zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF ITS APPLICATIONS 122/123/124:1OQ3-1038 (1989) zyxwvutsrqponmlkjih 0 Elsevier Science Publishing Co., Inc., 1989 1003 655 Avenue of the Americas, New York, NY 10010 0024-3795/89/$3.50