Journal of Mathematical Sciences

, Volume 230, Issue 5, pp 752–756 | Cite as

Spectral Set of a Linear System with Discrete Time

  • S. N. Popova
  • I. N. Banshchikova


Fix a certain class of perturbations of the coefficient matrix A(·) of a discrete linear homogeneous system of the form
$$ x\left(m+1\right)=A(m)x(m),\kern1em m\in \kern0.5em \mathrm{N},\kern1em x\in {\mathrm{R}}^n, $$
where the matrix A(·) is completely bounded on ℕ. The spectral set of this system corresponding to a given class of perturbations is the collection of complete spectra of the Lyapunov exponents of perturbed systems when perturbations runs over the whole class considered. In this paper, we examine the class R of multiplicative perturbations of the form
$$ y\left(m+1\right)=A(m)R(m)x(m),\kern1em m\in \mathrm{N},\kern1em y\in {\mathrm{R}}^n, $$

where the matrix R(·) is completely bounded on ℕ. We obtain conditions that guarantee the coincidence of the spectral set λ(R) corresponding to the class R with the set of all nondecreasing n-tuples of n numbers.

Keywords and phrases

linear system with discrete time Lyapunov exponent perturbation of coefficients 

AMS Subject Classification

39A06 39A30 


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Udmurt State UniversityIzhevskRussia
  2. 2.Institute of Mathematics and MechanicsUral Branch of the Russian Academy of SciencesYekaterinburgRussia
  3. 3.Izhevsk State Agricultural AcademyIzhevskRussia

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