An Algebraic Principle for the Stability of Difference Operators

✍ Scribed by Zhuoyuan Zhang

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 292 KB
• Volume: 136
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.3177

✦ Synopsis

This paper gives necessary and sufficient algebraic criteria for the stability of the difference system x(t)=Ax(t&r 1 )+Bx(t&r 2 ).

1997 Academic Press

Consider the stability of the zero solution of the difference equation

The problem is to decide whether all the roots * of the corresponding characteristic equation, det \ I& : n k=1 A k e &*rk + =0, have negative real part. Various methods and results were developed for the algebraic criteria for its stability (see [2 5]). This paper gives the general necessary and sufficient algebraic criteria for the stability of any N-dimensional difference system with two delays (three when they are rationally linear dependent).

For simplicity, we will denote det( } ) as | } | in this paper. First, let us give two definitions.

Definition 1. The difference operator D(r, A) is called stable if all the roots * of its corresponding characteristic equation satisfy Re *<&$ for some $>0.

If there exists a neighborhood I(r) of r in R n + , such that D(s, A) is stable when s # I(r), we say that D(r, A) is locally stable.