International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 860326, 15 pages
http://dx.doi.org/10.1155/2011/860326
Research Article

The Stability Cone for a Matrix Delay Difference Equation

M. M. Kipnis1 and V. V. Malygina2

1Department of Mathematics, South Ural State University, Chelyabinsk 454080, Russia
2Department of Applied Mathematics and Mechanics, Perm State Technical University, Perm 614990, Russia

Received 22 December 2010; Accepted 20 March 2011

Academic Editor: Frank Werner

Copyright © 2011 M. M. Kipnis and V. V. Malygina. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We construct a stability cone, which allows us to analyze the stability of the matrix delay difference equation 𝑥 𝑛 = 𝐴 𝑥 𝑛 − 1 + 𝐵 𝑥 𝑛 − 𝑘 . We assume that 𝐴 and 𝐵 are 𝑚 × 𝑚 simultaneously triangularizable matrices. We construct 𝑚 points in ℝ 3 which are functions of eigenvalues of matrices 𝐴 ,   𝐵 such that the equation is asymptotically stable if and only if all the points lie inside the stability cone.