Mean-Square Stability and Instability Criteria for the Gikhman–Ito Stochastic Diffusion Functional Differential Systems Subject to External Disturbances of the Type of Random Variables

The authors investigate the asymptotic stability in quadratic mean of the trivial solution of the Gikhman–Ito stochastic diffusion functional differential equations in terms of the eigenvalues of the matrix constructed from the coefficients of these equations.

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Published in	Cybernetics and systems analysis Vol. 59; no. 2; pp. 283 - 295
Main Authors	Yasynskyy, V. K., Yurchenko, I. V.
Format	Journal Article
Language	English
Published	New York Springer US 01.03.2023 Springer Springer Nature B.V
Subjects	Analysis Artificial Intelligence Control Differential equations Diffusion Eigenvalues Mathematical analysis Mathematics Mathematics and Statistics Processor Architectures Random variables Software Engineering/Programming and Operating Systems Stability criteria Systems Theory criterion external disturbances stability of the solution Gikhman–Ito stochastic functional differential equations
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Abstract	The authors investigate the asymptotic stability in quadratic mean of the trivial solution of the Gikhman–Ito stochastic diffusion functional differential equations in terms of the eigenvalues of the matrix constructed from the coefficients of these equations.
AbstractList	The authors investigate the asymptotic stability in quadratic mean of the trivial solution of the Gikhman-Ito stochastic diffusion functional differential equations in terms of the eigenvalues of the matrix constructed from the coefficients of these equations.
Audience	Academic
Author	Yurchenko, I. V. Yasynskyy, V. K.
Author_xml	– sequence: 1 givenname: V. K. surname: Yasynskyy fullname: Yasynskyy, V. K. email: v.yasynskyy@chnu.edu.ua organization: Yuriy Fedkovych Chernivtsi National University – sequence: 2 givenname: I. V. surname: Yurchenko fullname: Yurchenko, I. V. organization: Yuriy Fedkovych Chernivtsi National University
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References	TsarkovEFRandom Perturbations of Functional Differential Equations1989RigaZinatne[in Russian] V. S. Korolyuk, E. F. Tsarkov, and V. K. Yasynskyy, Probability, Statistics, and Random Processes. Theory and Computer Practice, Vol. 3, Random Processes: Theory and Computer Practice [in Ukrainian], Zoloti Lytavry, Chernivtsi (2009). M. L. Sverdan, E. F. Tsarkov, and V. K. Yasynskyy, Stability in Stochastic Modeling of Complex Dynamic Systems [in Ukrainian], Nad Prutom, Snyatyn (1996). E. F. Tsarkov and V. K. Yasynskyy, Quasilinear Stochastic Functional Differential Equations [in Russian], Orientir, Riga (1990). AndreevaEAKolmanovskiiVBShaikhetLEControl of Systems with Aftereffect1992MoscowNauka[in Russian] V. K. Yasynskyy and A. Ya. Dovgun’, Stabilization of Stochastic Dynamic Systems of Automatic Control [in Ukrainian], ChNU, Chernivtsi (2015). GantmakherFRMatrix Theory1967MoscowNauka[in Russian] KushnerHJStochastic Stability and Control1969AcadPress V. K. Yasynskyy, I. V. Yurchenko, and U. M. Kysiluk, Existence and Uniqueness of a Strong Solution to a Stochastic Functional Differential Equation with External Random Disturbances, Nauk. Visnyk Uzhhorod. Univ., Ser. Matem. Inform., Issue No. 1 (30), 143–153 (2017). V. K. Yasynskyy and E. V. Yasynskyy, The Problems of Stability and Stabilization of Dynamic Systems with Finite Aftereffect [in Ukrainian], TViMS, Kyiv (2005). V. K. Yasynskyy and T. O. Lukashiv, Stabilization of Stochastic Diffusion Dynamic Systems of Random Structure [in Ukrainian], ChNU, Chernivtsi (2013). YurchenkoIVYasynskyyVKExistence of Lyapunov-Krasovskii functionals for stochastic differential-functional Ito-Skorokhod equations under the condition of solutions’ stability on probability with finite aftereffectCybern. Syst. Analysis201854695797010.1007/s10559-018-0099-81411.34111 J. K. Hale, Theory of Functional Differential Equations, Ser. Applied Mathematical Sciences, Springer, New York (1977). GikhmanIISkorokhodAVStochastic Differential Equations and their Applications1982KyivNaukova Dumka0557.60041[in Russian] KorolyukVSMusurivskiiVIYurchenkoIVStability of dynamic systems with aftereffect with due regard for Markov perturbationsCybern. Syst. Analysis200743687688510.1007/s10559-007-0112-01153.34352 562_CR5 562_CR13 562_CR12 562_CR8 562_CR10 562_CR1 562_CR2 EF Tsarkov (562_CR11) 1989 VS Korolyuk (562_CR15) 2007; 43 562_CR3 FR Gantmakher (562_CR14) 1967 IV Yurchenko (562_CR4) 2018; 54 EA Andreeva (562_CR9) 1992 II Gikhman (562_CR7) 1982 HJ Kushner (562_CR6) 1969
References_xml	– reference: KorolyukVSMusurivskiiVIYurchenkoIVStability of dynamic systems with aftereffect with due regard for Markov perturbationsCybern. Syst. Analysis200743687688510.1007/s10559-007-0112-01153.34352 – reference: V. K. Yasynskyy and A. Ya. Dovgun’, Stabilization of Stochastic Dynamic Systems of Automatic Control [in Ukrainian], ChNU, Chernivtsi (2015). – reference: GikhmanIISkorokhodAVStochastic Differential Equations and their Applications1982KyivNaukova Dumka0557.60041[in Russian] – reference: V. S. Korolyuk, E. F. Tsarkov, and V. K. Yasynskyy, Probability, Statistics, and Random Processes. Theory and Computer Practice, Vol. 3, Random Processes: Theory and Computer Practice [in Ukrainian], Zoloti Lytavry, Chernivtsi (2009). – reference: GantmakherFRMatrix Theory1967MoscowNauka[in Russian] – reference: M. L. Sverdan, E. F. Tsarkov, and V. K. Yasynskyy, Stability in Stochastic Modeling of Complex Dynamic Systems [in Ukrainian], Nad Prutom, Snyatyn (1996). – reference: V. K. Yasynskyy and E. V. Yasynskyy, The Problems of Stability and Stabilization of Dynamic Systems with Finite Aftereffect [in Ukrainian], TViMS, Kyiv (2005). – reference: AndreevaEAKolmanovskiiVBShaikhetLEControl of Systems with Aftereffect1992MoscowNauka[in Russian] – reference: J. K. Hale, Theory of Functional Differential Equations, Ser. Applied Mathematical Sciences, Springer, New York (1977). – reference: E. F. Tsarkov and V. K. Yasynskyy, Quasilinear Stochastic Functional Differential Equations [in Russian], Orientir, Riga (1990). – reference: V. K. Yasynskyy, I. V. Yurchenko, and U. M. Kysiluk, Existence and Uniqueness of a Strong Solution to a Stochastic Functional Differential Equation with External Random Disturbances, Nauk. Visnyk Uzhhorod. Univ., Ser. Matem. Inform., Issue No. 1 (30), 143–153 (2017). – reference: V. K. Yasynskyy and T. O. Lukashiv, Stabilization of Stochastic Diffusion Dynamic Systems of Random Structure [in Ukrainian], ChNU, Chernivtsi (2013). – reference: KushnerHJStochastic Stability and Control1969AcadPress – reference: TsarkovEFRandom Perturbations of Functional Differential Equations1989RigaZinatne[in Russian] – reference: YurchenkoIVYasynskyyVKExistence of Lyapunov-Krasovskii functionals for stochastic differential-functional Ito-Skorokhod equations under the condition of solutions’ stability on probability with finite aftereffectCybern. Syst. Analysis201854695797010.1007/s10559-018-0099-81411.34111 – volume-title: Stochastic Stability and Control year: 1969 ident: 562_CR6 – ident: 562_CR1 – ident: 562_CR2 – ident: 562_CR12 – volume-title: Matrix Theory year: 1967 ident: 562_CR14 – ident: 562_CR13 – volume-title: Stochastic Differential Equations and their Applications year: 1982 ident: 562_CR7 – ident: 562_CR10 doi: 10.1007/978-1-4612-9892-2 – volume-title: Random Perturbations of Functional Differential Equations year: 1989 ident: 562_CR11 – volume: 54 start-page: 957 issue: 6 year: 2018 ident: 562_CR4 publication-title: Cybern. Syst. Analysis doi: 10.1007/s10559-018-0099-8 – volume-title: Control of Systems with Aftereffect year: 1992 ident: 562_CR9 – ident: 562_CR5 – ident: 562_CR3 – volume: 43 start-page: 876 issue: 6 year: 2007 ident: 562_CR15 publication-title: Cybern. Syst. Analysis doi: 10.1007/s10559-007-0112-0 – ident: 562_CR8
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SubjectTerms	Analysis Artificial Intelligence Control Differential equations Diffusion Eigenvalues Mathematical analysis Mathematics Mathematics and Statistics Processor Architectures Random variables Software Engineering/Programming and Operating Systems Stability criteria Systems Theory
Title	Mean-Square Stability and Instability Criteria for the Gikhman–Ito Stochastic Diffusion Functional Differential Systems Subject to External Disturbances of the Type of Random Variables
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