Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T13:59:18.815Z Has data issue: false hasContentIssue false
  • English
  • Français

Formulas for Liapunov functions for systems of linear difference equations

Published online by Cambridge University Press:  13 January 2004

John R. Graef ,
Chuanxi Qian  and
Bo Zhang
John R. Graef
Affiliation:
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, TN 37403, USA. E-mail: john-graef@utc.edu
Chuanxi Qian
Affiliation:
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 39762, USA. E-mail: qian@math.msstate.edu
Bo Zhang
Affiliation:
Department of Mathematics and Computer Science, Fayetteville State University, Fayetteville, NC 28301, USA. E-mail: bzhang@uncfsu.edu
Get access

Abstract

Explicit quadratic Liapunov functions that provide necessary and sufficient conditions for the asymptotic stability of the system of linear difference equations $x (t + 1) = A x(t)$ are constructed by transforming the original systems to $y (t + 1) = G y(t)$ , where $G$ is a companion matrix associated with the characteristic polynomial of $A$. A necessary and sufficient condition for all roots of the characteristic polynomial to lie in the unit circle $|z| < 1$ on the complex plane is also derived.

Keywords

linear systems of difference equationsLiapunov functionsasymptotic stabilityquadratic functions39A1193D05
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 88 , Issue 1 , January 2004 , pp. 185 - 203
Copyright
2004 London Mathematical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)