TY - JOUR

T1 - On conditions for exponential dichotomy for systems of difference equations under perturbation of coefficients

AU - Bondar, Anna A.

PY - 2019/10/1

Y1 - 2019/10/1

N2 - 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.

AB - 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.

KW - Discrete Lyapunov equations

KW - Exponential dichotomy

KW - Systems of difference equations

KW - Theorem on continuous dependence

UR - http://www.scopus.com/inward/record.url?scp=85079187439&partnerID=8YFLogxK

U2 - 10.25587/SVFU.2019.84.91.001

DO - 10.25587/SVFU.2019.84.91.001

M3 - Article

AN - SCOPUS:85079187439

VL - 26

SP - 3

EP - 13

JO - Mathematical Notes of NEFU

JF - Mathematical Notes of NEFU

SN - 2411-9326

IS - 4

ER -