» Jirstrand M., Foorsman K. - Some finiteness issues in differential algebraic systems theory
Jirstrand M., Foorsman K. - Some finiteness issues in differential algebraic systems theory
||Some finiteness issues in differential algebraic systems theory
||Jirstrand M., Foorsman K.
||In this paper a few algorithmic problems relating to differential algebraic systems theory are addressed. We will point out some problems arising as one treats systems of polynomial differential equations using the language of differential algebra. In particular we will see what happens when we search for the input-output equations, which are possible to derive from the original system. Most of the problems are related to computational difficulties, e.g. the existence of a finite set of differential equations describing the system.
Differential algebra made its way into systems theory via the discoveries of Michel Fliess in the mid 80's and has now grown quite extensive. A very good survey is given in . Several authors have studied how the algorithms of differential algebra, mostly elimination theory, can be used in systems theory. Let us here mention [1, 7, 8, 14, 19].
The paper is organized as follows: section 2 presents an example which serves as a background for later sections. Sections 3 and 4 display some properties of this example, which are probably rather surprising and counter-intuitive to the non-expert. Section 5, 6 and 7 explain how most of the problems that arose can be resolved. Section 8 describes some problems that are still open, and section 9 summarizes the results of the paper.
1.1 Basic Definitions and Notation
We suppose that the reader is familiar with some basic concepts from commutative and differential algebra, such as ring, ideal, prime ideal and characteristic sets. Some references are [11, 12, 21, 22]. Here follow a few definitions and theorems we use.
к denotes an arbitrary field of characteristic zero.
The ring of polynomials in the variables x\, ..., xn with coefficients from к is written
k[Xl, . . . , Xn\.
The ideal generated by a set P of polynomials is denoted (P).
The ring of differential polynomials in the differential indeterminates x\, ..., xn is denoted k\x±, ..., xn\.