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Stability of a delay difference system

Abstract

We consider the stability problem for the difference system x n = Axn-1 + Bxn-k, where A, B are real matrixes and the delay k is a positive integer. In the case A = -I, the equation is asymptotically stable if and only if all eigenvalues of the matrix B lie inside a special stability oval in the complex plane. If k is odd, then the oval is in the right half-plane, otherwise, in the left half-plane. If ||A|| + ||B|| < 1, then the equation is asymptotically stable. We derive explicit sufficient stability conditions for A I and A -I.

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Correspondence to Mikhail Kipnis.

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Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Kipnis, M., Komissarova, D. Stability of a delay difference system. Adv Differ Equ 2006, 031409 (2006). https://doi.org/10.1155/ADE/2006/31409

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Keywords

  • Differential Equation
  • Positive Integer
  • Partial Differential Equation
  • Stability Condition
  • Ordinary Differential Equation