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

### 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.

### References

[1]

Daleckii Yu. L., Krein M. G., Stability of Solutions to Differential Equations in Banach Space, Amer. Math. Soc., Providence, RI (1974).

[2]

Godunov S. K., “Problem of the dichotomy of the spectrum of a matrix,” Sib. Math. J., 27, No. 5, 649–660 (1986).

[3]

Bulgakov A. Ya. and Godunov S. K., “Circular dichotomy of the spectrum of a matrix,” Sib. Math. J., 29, No. 5, 734–744 (1988).

[4]

Bulgakov A. Ya., “The basis of guaranteed accuracy in the problem of separation of invariant subspaces for non-self-adjoint matrices,” Sib. Adv. Math., 1, No. 1, 64–108; No. 2, 1–56 (1991).

[5]

Godunov S. K., Modern Aspects of Linear Algebra, Amer. Math. Soc., Providence, RI (1998).

[6]

Abramov A. A., “On the boundary conditions at a singular point for linear ordinary differential equations,” USSR Comput. Math. Math. Phys., 11, No. 1, 363–367 (1971).

[7]

Roberts J. D., “Linear model reduction and solution of the algebraic Riccati equation by use of the sign function,” Int. J. Control, 32, No. 4, 677–687 (1980).

[8]

Balzer L. A., “Accelerated convergence of the matrix sign function method of solving Lyapunov, Riccati and other matrix equations,” Int. J. Control, 32, No. 6, 1057–1078 (1980).

[9]

Demidenko G. V., “On a functional approach to constructing projections onto invariant subspaces of matrices,” Sib. Math. J., 39, No. 4, 683–699 (1998).

[10]

Demidenko G. V., “On constructing approximate projections onto invariant subspaces of linear operators,” Int. J. Diff. Equ. Appl., 3, No. 2, 135–146 (2001).

[11]

Demidenko G. V., “A method of constructing the projections onto the invariant subspaces of matrices [in Russian],” Sib. Zh. Ind. Mat., 1, No. 1, 104–113 (1998).

[12]

Demidenko G., “On a functional approach to spectral problems of linear algebra,” Selcuk J. Appl. Math., 2, No. 2, 39–52 (2001).

[13]

Godunov S. K., Kiriluk O. P., and Kostin V. I., Spectral portraits of matrices [in Russian], preprint, Russ. Acad. Sci., Sib. Branch, Inst. Math., No. 3, Novosibirsk (1990).

[14]

Bulgak H., “Pseudoeigenvalues, spectral portrait of a matrix and their connections with different criteria of stability,” in: Error Control and Adaptivity in Scientific Computing (H. Bulgak and C. Zenger, eds.), pp. 95–124, Kluwer Acad. Publ., Amsterdam (1999) (NATO Sci. Ser. II; V. 536).

[15]

Demidenko G. V., Matrix Equations [in Russian], Novosib. Gos. Univ., Novosibirsk (2009).

[16]

Demidenko G. V., “Systems of differential equations with periodic coefficients,” J. Appl. Ind. Math., 8, No. 1, 20–27 (2014).

[17]

Demidenko G. V. and Bondar A. A., “Exponential dichotomy of systems of linear difference equations with periodic coefficients,” Sib. Math. J., 57, No. 6, 969–980 (2016).

*Mathematical notes of NEFU*, 26(4), pp. 3-13. doi: https://doi.org/10.25587/SVFU.2019.84.91.001.

This work is licensed under a Creative Commons Attribution 4.0 International License.