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Russia Izhevsk
Section  Mathematics 
Title  About invariant sets of control systems with random coefficients 
Author(s)  Rodina L.I.^{a} 
Affiliations  Udmurt State University^{a} 
Abstract  This work is devoted to the investigation of invariant sets of control systems with impulse influences that are parameterized by a metric dynamic system. Such systems describe various stochastic models of population dynamics, economy, quantum electronics and mechanics. We obtain the conditions of existence of invariant sets for the attainability set of system as well as conditions of asymptotic approach of system solutions to a given set. The obtained results are illustrated by examples of population dynamics which is subject to crafts, when the moments of trade preparations and the sizes of these preparations are random variables. For given models we investigate various dynamic modes of development which essentially differ from modes of the deterministic models and better display the processes occurring in real ecological systems. Conditions under which the population size is in the given set and conditions of asymptotic extinction of population with probability equal to one are received; estimations for a mathematical expectation and a dispersion of time of population extinction are also obtained. 
Keywords  сontrol systems with random coefficients, dynamical systems, invariant sets, probabilistic models of population dynamics 
UDC  517.935, 517.938 
MSC  34A60, 37N35, 49J15, 93B03 
DOI  10.20537/vm140409 
Received  20 October 2014 
Language  Russian 
Citation  Rodina L.I. About invariant sets of control systems with random coefficients, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 4, pp. 109121. 
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