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Section
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Mathematics
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Title
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On the spectral set of a linear discrete system with stable Lyapunov exponents
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Author(-s)
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Banshchikova I.N.ab,
Popova S.N.bc
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Affiliations
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Izhevsk State Agricultural Academya,
Udmurt State Universityb,
Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesc
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Abstract
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Let us fix a certain class of perturbations of the coefficient matrix $A(\cdot)$ for a discrete time-varying 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.
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Keywords
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discrete time-varying linear system, Lyapunov exponents, perturbations of coefficients
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UDC
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517.929.2
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MSC
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39A06, 39A30
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DOI
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10.20537/vm160102
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Received
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1 February 2016
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Language
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Russian
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Citation
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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. 15-26.
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References
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