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Proceedings of the London Mathematical Society 2004 88(1):185-203; doi:10.1112/S0024611503014308
© 2004 by London Mathematical Society
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© London Mathematical Society

Formulas for Liapunov Functions for Systems of Linear Difference Equations

John R. Graef, Chuanxi Qian and Bo Zhang

Department of Mathematics, University of Tennessee at Chattanooga Chattanooga, TN 37403, USA. E-mail: john-graef{at}utc.edu
Department of Mathematics and Statistics, Mississippi State University Mississippi State, MS 39762, USA. E-mail: qian{at}math.msstate.edu
Department of Mathematics and Computer Science, Fayetteville State University Fayetteville, NC 28301, USA. E-mail: bzhang{at}uncfsu.edu

Received 17 June 2002.

Explicit quadratic Liapunov functions that provide necessary and sufficient conditions for the asymptotic stability of the system of linear difference equations x (t + 1) = Ax(t) are constructed by transforming the original systems to y (t + 1) = Gy(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. 2000 Mathematicals Subject Classification 39A11, 93D05.

Key Words: linear systems of difference equations • Liapunov functions • asymptotic stability • quadratic functions


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