Discrete Dynamics in Nature and Society
Volume 2012 (2012), Article ID 320302, 19 pages
Research Article

About the Stabilization of a Nonlinear Perturbed Difference Equation

Institute of Research and Development of Processes, Campus of Leioa, Bizkaia, 48080 Bilbao, Spain

Received 18 October 2011; Accepted 15 January 2012

Academic Editor: Beatrice Paternoster

Copyright © 2012 M. De la Sen. 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.


This paper investigates the local asymptotic stabilization of a very general class of instable autonomous nonlinear difference equations which are subject to perturbed dynamics which can have a different order than that of the nominal difference equation. In the general case, the controller consists of two combined parts, namely, the feedback nominal controller which stabilizes the nominal (i.e., perturbation-free) difference equation plus an incremental controller which completes the stabilization in the presence of perturbed or unmodeled dynamics in the uncontrolled difference equation. A stabilization variant consists of using a single controller to stabilize both the nominal difference equation and also the perturbed one under a small-type characterization of the perturbed dynamics. The study is based on Banach fixed point principle, and it is also valid with slight modification for the stabilization of unstable oscillatory solutions.