On conditions for exponential dichotomy for systems of difference equations under perturbation of coefficients

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Abstract

The problem of the exponential dichotomy for systems of linear difference equations with constant coefficients is considered. We investigate the question of admissible perturbations for the coefficient matrix under which the exponential dichotomy is preserved. Assuming the initial system of linear difference equations is exponentially dichotomous, we establish conditions for perturbations under which the perturbed system is also exponentially dichotomous. The conditions are written in the form of estimates on the norm of perturbation matrices and are of constructive character. Any spectral information was not used to obtain them, since the problem of finding the spectrum for non-self-adjoint matrices is ill-conditioned from the perturbation theory point of view. In the present paper, we apply an approach based on the solvability of the discrete Lyapunov matrix equations. Therefore, the established results can be used for the numerical study of the dichotomy problem.

Original languageEnglish
Pages (from-to)3-13
Number of pages11
JournalMathematical Notes of NEFU
Volume26
Issue number4
DOIs
Publication statusPublished - 1 Oct 2019

Keywords

  • Discrete Lyapunov equations
  • Exponential dichotomy
  • Systems of difference equations
  • Theorem on continuous dependence

OECD FOS+WOS

  • 1.02 COMPUTER AND INFORMATION SCIENCES
  • 1.01 MATHEMATICS

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