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Stabilization of Stochastic Dynamical Systems of a Random Structure with Markov Switches and Poisson Perturbations

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  • Taras Lukashiv

    (Multiomics Data Science Research Group, Department of Cancer Research, Luxembourg Institute of Health, L-1445 Strassen, Luxembourg
    NORLUX Neuro-Oncology Laboratory, Department of Cancer Research, Luxembourg Institute of Health, L-1210 Luxembourg, Luxembourg
    Department of Mathematical Problems of Control and Cybernetics, Yuriy Fedkovych Chernivtsi National University, 58000 Chernivtsi, Ukraine
    These authors contributed equally to this work.)

  • Yuliia Litvinchuk

    (Department of Mathematical Problems of Control and Cybernetics, Yuriy Fedkovych Chernivtsi National University, 58000 Chernivtsi, Ukraine
    These authors contributed equally to this work.)

  • Igor V. Malyk

    (Department of Mathematical Problems of Control and Cybernetics, Yuriy Fedkovych Chernivtsi National University, 58000 Chernivtsi, Ukraine
    These authors contributed equally to this work.)

  • Anna Golebiewska

    (NORLUX Neuro-Oncology Laboratory, Department of Cancer Research, Luxembourg Institute of Health, L-1210 Luxembourg, Luxembourg)

  • Petr V. Nazarov

    (Multiomics Data Science Research Group, Department of Cancer Research, Luxembourg Institute of Health, L-1445 Strassen, Luxembourg)

Abstract

An optimal control for a dynamical system optimizes a certain objective function. Here, we consider the construction of an optimal control for a stochastic dynamical system with a random structure, Poisson perturbations and random jumps, which makes the system stable in probability. Sufficient conditions of the stability in probability are obtained, using the second Lyapunov method, in which the construction of the corresponding functions plays an important role. Here, we provide a solution to the problem of optimal stabilization in a general case. For a linear system with a quadratic quality function, we give a method of synthesis of optimal control based on the solution of Riccati equations. Finally, in an autonomous case, a system of differential equations was constructed to obtain unknown matrices that are used for the construction of an optimal control. The method using a small parameter is justified for the algorithmic search of an optimal control. This approach brings a novel solution to the problem of optimal stabilization for a stochastic dynamical system with a random structure, Markov switches and Poisson perturbations.

Suggested Citation

  • Taras Lukashiv & Yuliia Litvinchuk & Igor V. Malyk & Anna Golebiewska & Petr V. Nazarov, 2023. "Stabilization of Stochastic Dynamical Systems of a Random Structure with Markov Switches and Poisson Perturbations," Mathematics, MDPI, vol. 11(3), pages 1-22, January.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:3:p:582-:d:1043881
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    References listed on IDEAS

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    1. Taras Lukashiv, 2016. "One Form of Lyapunov Operator for Stochastic Dynamic System with Markov Parameters," Journal of Mathematics, Hindawi, vol. 2016, pages 1-5, September.
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    Cited by:

    1. Juan Carlos Seck-Tuoh-Mora & Joselito Medina-Marin & Norberto Hernández-Romero & Genaro J. Martínez, 2023. "Mean-Field Analysis with Random Perturbations to Detect Gliders in Cellular Automata," Mathematics, MDPI, vol. 11(20), pages 1-13, October.

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