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Section
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Mathematics
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Title
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On the property of integral separation of discrete-time systems
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Author(-s)
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Banshchikova I.N.a,
Popova S.N.ab
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Affiliations
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Udmurt State Universitya,
Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesb
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Abstract
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This paper is devoted to the study of the property of an integral separation of discrete time-varying linear systems. By definition, the system $x(m+1)=A(m)x(m),$ $m\in\mathbb N,$ $x\in\mathbb R^n,$ is called a system with integral separation if it has a basis of solutions $x^1(\cdot),\ldots,x^n(\cdot)$ such that for some $\gamma>0$, $a>1$ and all natural $m>s$, $i\leqslant n-1$ the inequalities
$$
\dfrac{\|x^{i+1}(m)\|}{\|x^{i+1}(s)\|}\geqslant\gamma a^{m-s}\dfrac{\|x^{i}(m)\|}{\|x^{i}(s)\|}.
$$
are satisfied. The concept of integral separation of systems with continuous time was introduced by B.F. Bylov in 1965. The criteria for the integral separation of systems with discrete time are proved: reducibility to diagonal form with an integrally separated diagonal; stability and nonmultiplicity of Lyapunov exponents. The property of diagonalizability of discrete-time systems is also studied in detail. The evidence takes into account the specifics of these systems.
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Keywords
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discrete time-varying linear system, Lyapunov exponents, integral separability, diagonalizability
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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/vm170401
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Received
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1 September 2017
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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 property of integral separation of discrete-time systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 4, pp. 481-498.
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References
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