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ISSN Печать: 1064-2315
ISSN Онлайн: 2163-9337
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Asymptotics of Control Problem for the Diffusion Process in Markov Environment
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Краткое описание
A stochastic optimization procedure and a limit generator of the original problem are constructed for a system of stochastic differential equations with Markov switching and diffusion perturbation with control, which is determined by the condition for the extremum of the quality function. The complexity of the studied evolutionary model lies primarily in the fact that the system is in conditions of external random influence, which is modeled using a switching process. The main assumption is the condition for uniform ergodicity of the Markov switching process, that is, the existence of a stationary distribution for the switching process over large time intervals. This allows one to construct explicit algorithms for the analysis of the asymptotic behavior of a controlled process. An important property of the generator of the Markov switching process is that the space in which it is defined splits into the direct sum of its zero-subspace and a subspace of values, followed by the introduction of a projector that acts on the subspace of zeros. Another complexity of the model under study is the presence of an approximation scheme determined by normalization. The question of how the behavior of the limit process depends on the prelimit normalization by a small parameter of the stochastic system in an ergodic Markovian environment is studied. A stochastic differential equation is written for determining the limiting processes of transfer and control. For the first time, a model of the control problem for the diffusion transfer process using the stochastic optimization procedure for control is proposed. A singular expansion in the small parameter of the generator of the three-component Markov process is obtained, and the problem of a singular perturbation with the representation of the limiting generator of this process is solved.
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Korolyuk V.S., Limnios N., Stochastic systems in merging phase space, World Scientific, Singapore. 2005.
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Samoilenko I.V., Chabanyuk Y.M., NikitinA.V., KhimkaU.T., Differential equations with small stochastic additions under Poisson approximation conditions, Cybernetics and Systems Analysis, 2017, 53, No. 3, 410-416, DOI: 10.1007/s10559-017-9941-7.
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ChabanyukY.M., NikitinA.V., KhimkaU.T., Asymptotic properties of the impulse perturbation process with control function under Levy approximation conditions, Mat. Stud., 2019, 52, 96-104, DOI: 10.30970/ms.52.1.96-104.
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NikitinA.V., KhimkaU.T.. Asymptotics of normalized control with Markov switchings, Ukrainian Mathematical Journal, 2017, 8, No. 68, 1252-1262, DOI: 10.1007/s11253-017-1291-0.
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Nikitin A.V., Asymptotic dissipativity of stochastic processes with impulsive perturbation in the Levy approximation scheme, Journal of Automation and Information Sciences, 2018, 50, No. 4, 205-211, DOI: 10.1615/JAutomatInfScien.v50.i4.50.
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KhimkaU.T., Chabanyuk Ya.M., A difference stochastic optimization procedure with impulse perturbation, Cybernetics and Systems Analysis, 2013, 49, No. 5, 768-773, DOI: 10.1007/s10559- 013-9564-6.
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Samoilenko I.V., Chabanyuk Y.M., Nikitin A.V., Khimka U.T., Differential equations with small stochastic additions under Poisson approximation conditions, Cybernetics and Systems Analysis, 2017, 53, No. 3, 410-416, DOI: 10.1007/s10559-017-9941-7.
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Nikitin A.V., Asymptotic properties of a stochastic diffusion transfer process with an equilibrium point of a quality criterion, Cybernetics and Systems Analysis, 2015, 51, No. 4, 650-656. DOI: 10.1007/s10559-015-9756-3.