Stabilization of Stochastic Dynamic Systems with Markov Parameters and Concentration Point

Taras Lukashiv (), Igor V. Malyk, Venkata P. Satagopam and Petr V. Nazarov ()
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Taras Lukashiv: Multiomics Data Science Research Group, Department of Cancer Research, Luxembourg Institute of Health, L-1445 Strassen, 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

Mathematics, 2025, vol. 13, issue 14, 1-14

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.

Keywords: optimal control; Lyapunov function; system of stochastic differential equations; Markov switches; concentration point (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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