Introduction

Abstract

The realization problem that we will state here can be divided into the following three problems A, B and C. Where I/O is the set of input/output maps that may be the input/output relation of a given black-box. CD is the category of dynamical systems which may have the same behavior (equivalently, input/output relation) as the black-box.

A. The existence and uniqueness in algebraic sense.

For any input/output map a I/O , find out at least one dynamical system CD such that the behavior of it is a . Also prove that any two dynamical systems that have the same behavior a are isomorphic in the sense of the category CD.

B. The finite dimensionality of the dynamical systems.

Clarify when a dynamical system CD is finite dimensional. Because finite dimensional dynamical systems are actually appearing by linear (or non-linear) circuits or computer programs, it is very important that these conditions become clear.

C. Deriving the dynamical systems from finite data.

Partial realization problems are to find the minimal dynamical system fit to a given finite input/output’s data and to clarify when the minimal dynamical systems are isomorphic.