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Russia Izhevsk; Yekaterinburg
Section  Mathematics 
Title  On the property of integral separation of discretetime systems 
Author(s)  Banshchikova I.N.^{a}, Popova S.N.^{ab} 
Affiliations  Udmurt State University^{a}, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences^{b} 
Abstract  This paper is devoted to the study of the property of an integral separation of discrete timevarying 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 $m>s$, $i\leqslant n1$ the inequalities $$ \dfrac{\x^{i+1}(m)\}{\x^{i+1}(s)\}\geqslant\gamma a^{ms}\dfrac{\x^{i}(m)\}{\x^{i}(s)\}. $$ 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 discretetime systems is also studied in detail. The evidence takes into account the specifics of these systems. 
Keywords  discrete timevarying linear system, Lyapunov exponents, integral separability, diagonalizability 
UDC  517.929.2 
MSC  39A06, 39A30 
DOI  10.20537/vm170401 
Received  1 September 2017 
Language  Russian 
Citation  Banshchikova I.N., Popova S.N. On the property of integral separation of discretetime systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 4, pp. 481498. 
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