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A scheme for study of linear stochastic time-delay dynamical systems under continuous and impulsive fluctuations

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Abstract

The paper deals with linear stochastic systems perturbed by both continuous and jump random noises. A presence of the lasts leads to the fact that at random times the state vector of the system receives random finite increments. But compound jump processes may have different pulse distributions describing time intervals between the impulses and their magnitudes that brings to a more complex time history of the systems than in the case of continuous noises only. We consider a problem of calculation of the first-order moment functions and the second-order central moment functions (hereinafter referred to as the first moment functions) for the state vectors of linear stochastic difference-differential systems excited by random continuous Wiener and jump Poisson processes. The technique combining the classic method of steps and a scheme of state space extension is used to derive a sequence of jump-diffusion stochastic differential equations (DEs) without delays satisfied by extended state vectors and then a chain of ordinary DEs’ systems governing the first moment functions. A usage of vector-matrix notation allows to obtain this chain in a form that is suitable for a practical computer program implementation. Finally an example of analysis of a stochastic SDOF system demonstrates an application of the scheme.

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Acknowledgments

This work is partly supported by Russian Foundation for Basic Research under Grant No. 14-01-96019.

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Authors and Affiliations

  1. Department of Higher Mathematics, Perm State University, Perm, 614990, Russia

    Igor Poloskov

  2. Department of Control and Information Security, Perm State University, Perm, 614990, Russia

    Vladimir Malanin

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  1. Igor Poloskov
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  2. Vladimir Malanin
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Correspondence to Igor Poloskov.

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Poloskov, I., Malanin, V. A scheme for study of linear stochastic time-delay dynamical systems under continuous and impulsive fluctuations. Int. J. Dynam. Control 4, 195–203 (2016). https://doi.org/10.1007/s40435-015-0172-3

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  • DOI: https://doi.org/10.1007/s40435-015-0172-3

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