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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
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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

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