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## Archive of Issues

Russia Izhevsk; Yekaterinburg
Year
2017
Volume
27
Issue
4
Pages
481-498
 Section Mathematics Title On the property of integral separation of discrete-time systems Author(-s) Banshchikova I.N.a, Popova S.N.ab Affiliations Udmurt State Universitya, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesb Abstract 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. Keywords discrete time-varying 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 discrete-time systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 4, pp. 481-498. References Bylov B.F. On reduction of a system of linear equations to a diagonal form, Mat. sb. (N.S.), 1965, vol. 67 (109), no. 3, pp. 338-344 (in Russian). Millionshchikov V.M. Structurally stable 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 the stability of the characteristic exponents of a 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 i ee prilozheniya k voprosam ustoichivosti (Theory of Lyapunov exponents and its application to problems of stability), Moscow: Nauka, 1966, 576 p. Czornik A. Perturbation theory for Lyapunov exponents of discrete linear systems, Kraków: AGH University of Science and Technology Press, 2012, 110 p. Babiarz A., Czornik A., Makarov E., Niezabitowski M., Popova S. Pole placement theorem for discrete time-varying linear systems, SIAM Journal on Control and Optimization, 2017, vol. 55, no. 2, pp. 671-692. DOI: 10.1137/15M1033666 Babiarz A., Banshchikova I., Czornik A., Makarov E., Niezabitowski M., Popova S. On assignability of Lyapunov spectrum of discrete linear time-varying system with control, 21st International Conference on Methods and Models in Automation and Robotics (MMAR), 2016, pp. 697-701. DOI: 10.1109/MMAR.2016.7575221 Horn R.A., Johnson C.R. Matrix analysis, Cambridge: Cambridge University Press, 1986. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989, 655 p. Demidovich V.B. A certain criterion for the stability of 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. Adrianova L.Ya. Vvedenie v teoriyu lineinykh sistem differentsial'nykh uravnenii (Introduction to the theory of linear systems of differential equations), Saint Petersburg: Saint Petersburg State University, 1992, 240 p. Bylov B.F. Reduction of a linear system to block-triangular form, Differ. Uravn., 1987, vol. 23, no. 12, pp. 2027-2031 (in Russian). 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 (in Russian). DOI: 10.20537/vm160102 Full text