Section

Mathematics

Title

On the spectral set of a linear discrete system with stable Lyapunov exponents

Author(s)

Banshchikova I.N.^{ab},
Popova S.N.^{bc}

Affiliations

Izhevsk State Agricultural Academy^{a},
Udmurt State University^{b},
Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences^{c}

Abstract

Let us fix a certain class of perturbations of the coefficient matrix $A(\cdot)$ for a discrete timevarying linear system
$$x(m+1)=A(m)x(m),\quad m\in\mathbb Z,\quad x\in\mathbb R^n,$$
where $A(\cdot)$ is completely bounded on $\mathbb Z$, i.e., $\sup_{m\in\mathbb Z}\bigl(\A(m)\+\A^{1}(m)\\bigr)<\infty$. The spectral set of this system, corresponding to a given class of perturbations, is a collection of all Lyapunov spectra (with multiplicities) for perturbed systems, when the perturbations range over this class all. The main attention is paid to the class ${\cal R}$ of perturbed systems
$$y(m+1)=A(m)R(m)y(m),\quad m\in\mathbb Z,\quad y\in\mathbb R^n,$$
where $R(\cdot)$ is completely bounded on $\mathbb Z$, as well as its subclasses ${\cal R}_{\delta}$, where $\sup_{m\in\mathbb Z}\R(m)E\<\delta$, $\delta>0$. For an original system with stable Lyapunov exponents, we prove that the spectral set $\lambda({\cal R})$ of class ${\cal R}$ coincides with the set of all ordered ascending sets of $n$ numbers. Moreover, for any $\Delta> 0$ there exists an $\ell =\ell(\Delta)> 0 $ such that for any $\delta<\Delta$ the spectral set $\lambda({\cal R}_{\ell\delta})$ contains the $\delta$neighborhood of the Lyapunov spectrum of the unperturbed system.

Keywords

discrete timevarying linear system, Lyapunov exponents, perturbations of coefficients

UDC

517.929.2

MSC

39A06, 39A30

DOI

10.20537/vm160102

Received

1 February 2016

Language

Russian

Citation

Banshchikova I.N., Popova S.N. On the spectral set of a linear discrete system with stable Lyapunov exponents, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2016, vol. 26, issue 1, pp. 1526.

References

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