TY - JOUR

T1 - Stabilization of Stochastic Dynamical Systems of a Random Structure with Markov Switches and Poisson Perturbations

AU - Lukashiv, Taras

AU - Litvinchuk, Yuliia

AU - Malyk, Igor V.

AU - Golebiewska, Anna

AU - Nazarov, Petr V.

N1 - Funding Information:
This work was supported by the Luxembourg National Research Fund C21/BM/15739125/DIOMEDES to T.L., P.V.N. and A.G.
Publisher Copyright:
© 2023 by the authors.

PY - 2023/2

Y1 - 2023/2

N2 - 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.

AB - 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.

KW - Lyapunov function

KW - Markov switches

KW - optimal control

KW - Poisson perturbations

KW - system of stochastic differential equations

UR - http://www.scopus.com/inward/record.url?scp=85147863353&partnerID=8YFLogxK

U2 - 10.3390/math11030582

DO - 10.3390/math11030582

M3 - Article

AN - SCOPUS:85147863353

SN - 2227-7390

VL - 11

JO - Mathematics

JF - Mathematics

IS - 3

M1 - 582

ER -