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## Archive of Issues

Russia Izhevsk; Yekaterinburg
Year
2016
Volume
26
Issue
1
Pages
15-26
 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 Academya, Udmurt State Universityb, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesc Abstract 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. Keywords discrete time-varying 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. 15-26. References Demidovich V.B. On a criterion of stability for difference equations, Differ. Uravn., 1969, vol. 5, no. 7, pp. 1247-1255 (in Russian). Gaishun I.V. Sistemy s diskretnym vremenem (Discrete-time systems), Minsk: Institute of Mathematics of the National Academy of Sciences of Belarus, 2001, 400 p. Izobov N.A. Linear systems of ordinary differential equations, Journal of Soviet Mathematics, 1976, vol. 5, issue 1, pp. 46-96. Vinograd R.E. On the central characteristic exponent of a system of differential equations, Mat. Sb. (N. S.), 1957, vol. 42, no. 2, pp. 207-222 (in Russian). Millionshchikov V.M. Proof of attainability of central exponents of linear systems, Sib. Mat. Zh., 1969, vol. 10, no. 1, pp. 99-104 (in Russian). Millionshchikov V.M. Robust properties of linear systems of differential equations, Differ. Uravn., 1969, vol. 5, no. 10, pp. 1775-1784 (in Russian). Bylov B.F., Izobov N.A. Necessary and sufficient conditions for stability of characteristic exponents of linear system, Differ. Uravn., 1969, vol. 5, no. 10, pp. 1794-1803 (in Russian). Bylov B.F., Vinograd R.E., Grobman D.M., Nemytskii V.V. Teoriya pokazatelei Lyapunova (Theory of Lyapunov exponents), Moscow: Nauka, 1966, 576 p. Full text