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Stabilization of Stochastic Dynamic Systems with Markov Parameters and Concentration Point

Author

Listed:
  • 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
    Luxembourg Centre for Systems Biomedicine, University of Luxembourg, L-4370 Belvaux, Luxembourg)

  • Igor V. Malyk

    (Department of Mathematical Problems of Control and Cybernetics, Yuriy Fedkovych Chernivtsi National University, 58000 Chernivtsi, Ukraine)

  • Venkata P. Satagopam

    (Luxembourg Centre for Systems Biomedicine, University of Luxembourg, L-4370 Belvaux, Luxembourg)

  • Petr V. Nazarov

    (Multiomics Data Science Research Group, Department of Cancer Research, Luxembourg Institute of Health, L-1445 Strassen, Luxembourg
    Bioinformatics and AI Unit, Department of Medical Informatics, Luxembourg Institute of Health, L-1445 Strassen, Luxembourg)

Abstract

This paper addresses the problem of optimal stabilization for stochastic dynamical systems characterized by Markov switches and concentration points of jumps, which is a scenario not adequately covered by classical stability conditions. Unlike traditional approaches requiring a strictly positive minimal interval between jumps, we allow jump moments to accumulate at a finite point. Utilizing Lyapunov function methods, we derive sufficient conditions for exponential stability in the mean square and asymptotic stability in probability. We provide explicit constructions of Lyapunov functions adapted to scenarios with jump concentration points and develop conditions under which these functions ensure system stability. For linear stochastic differential equations, the stabilization problem is further simplified to solving a system of Riccati-type matrix equations. This work provides essential theoretical foundations and practical methodologies for stabilizing complex stochastic systems that feature concentration points, expanding the applicability of optimal control theory.

Suggested Citation

  • Taras Lukashiv & Igor V. Malyk & Venkata P. Satagopam & Petr V. Nazarov, 2025. "Stabilization of Stochastic Dynamic Systems with Markov Parameters and Concentration Point," Mathematics, MDPI, vol. 13(14), pages 1-14, July.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:14:p:2307-:d:1705236
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    References listed on IDEAS

    as
    1. 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.
    2. 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.
    3. E. Savku & G.-W Weber, 2022. "Stochastic differential games for optimal investment problems in a Markov regime-switching jump-diffusion market," Annals of Operations Research, Springer, vol. 312(2), pages 1171-1196, May.
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