Skip to main content
Erschienen in: Journal of Dynamical and Control Systems 1/2023

24.05.2022

Practical Stability with Respect to a Part of the Variables of Stochastic Differential Equations Driven by G-Brownian Motion

verfasst von: Tomás Caraballo, Faten Ezzine, Mohamed Ali Hammami

Erschienen in: Journal of Dynamical and Control Systems | Ausgabe 1/2023

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

In this paper, practical stability with respect to a part of the variables of stochastic differential equations driven by G-Brownian motion (G-SDEs) is studied. The analysis of the global practical uniform p th moment exponential stability, as well as the global practical uniform exponential stability with respect to a part of the variables of G-SDEs, is investigated by means of the G-Lyapunov functions. An illustrative example to show the usefulness of the practical stability with respect to a part of the variables notion is also provided.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Literatur
1.
Zurück zum Zitat Bai X, Lin Y. On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients. Acta Math Appl Sin Engl Ser 2010;30:589–610.MathSciNetCrossRefMATH Bai X, Lin Y. On the existence and uniqueness of solutions to stochastic differential equations driven by G-Brownian motion with integral-Lipschitz coefficients. Acta Math Appl Sin Engl Ser 2010;30:589–610.MathSciNetCrossRefMATH
2.
Zurück zum Zitat BenAbdallah A, Ellouze I, Hammami M A. Practical stability of nonlinear time-varying cascade systems. J Dyn Control Syst 2009;15:45–62.MathSciNetCrossRefMATH BenAbdallah A, Ellouze I, Hammami M A. Practical stability of nonlinear time-varying cascade systems. J Dyn Control Syst 2009;15:45–62.MathSciNetCrossRefMATH
3.
Zurück zum Zitat Ben Hamed B, Ellouze I, Hammami M A. Practical uniform stability of nonlinear differential delay equations. Mediterr J Math 2011;8:603–16.MathSciNetCrossRefMATH Ben Hamed B, Ellouze I, Hammami M A. Practical uniform stability of nonlinear differential delay equations. Mediterr J Math 2011;8:603–16.MathSciNetCrossRefMATH
4.
Zurück zum Zitat Caraballo T, Garrido-Atienza M J, Real J. Asymptotic stability of nonlinear stochastic evolution equations. Stoch Anal Appl 2003;21:301–27.MathSciNetCrossRefMATH Caraballo T, Garrido-Atienza M J, Real J. Asymptotic stability of nonlinear stochastic evolution equations. Stoch Anal Appl 2003;21:301–27.MathSciNetCrossRefMATH
5.
Zurück zum Zitat Caraballo T, Garrido-Atienza M J, Real J. Stochastic stabilization of differential systems with general decay rate. Syst Control Lett 2003;48: 397–406.MathSciNetCrossRefMATH Caraballo T, Garrido-Atienza M J, Real J. Stochastic stabilization of differential systems with general decay rate. Syst Control Lett 2003;48: 397–406.MathSciNetCrossRefMATH
6.
Zurück zum Zitat Caraballo T, Hammami M A, Mchiri L. On the practical global uniform asymptotic stability of stochastic differential equations. Stochastics–An International Journal of Probability and Stochastic Processes 2016;88:45–56.MathSciNetCrossRefMATH Caraballo T, Hammami M A, Mchiri L. On the practical global uniform asymptotic stability of stochastic differential equations. Stochastics–An International Journal of Probability and Stochastic Processes 2016;88:45–56.MathSciNetCrossRefMATH
7.
Zurück zum Zitat Caraballo T, Ezzine F, Hammami M, Mchiri L. Practical stability with respect to a part of variables of stochastic differential equations. Stochastics—An International Journal of Probability and Stochastic Processes 2020;6:1–18.MATH Caraballo T, Ezzine F, Hammami M, Mchiri L. Practical stability with respect to a part of variables of stochastic differential equations. Stochastics—An International Journal of Probability and Stochastic Processes 2020;6:1–18.MATH
8.
Zurück zum Zitat Caraballo T, Ezzine F, Hammami M. Partial stability analysis of stochastic differential equations with a general decay rate. J Eng Math 2021;130(4):1–17.MathSciNetMATH Caraballo T, Ezzine F, Hammami M. Partial stability analysis of stochastic differential equations with a general decay rate. J Eng Math 2021;130(4):1–17.MathSciNetMATH
10.
Zurück zum Zitat Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion path. Potential Anal 2010; 34:139–61.MathSciNetCrossRefMATH Denis L, Hu M, Peng S. Function spaces and capacity related to a sublinear expectation: application to G-Brownian motion path. Potential Anal 2010; 34:139–61.MathSciNetCrossRefMATH
11.
Zurück zum Zitat Gao F. Pathwise properties and homomorphic flows for stochastic differential equations driven by G-Brownian motion. Stoch Process Appl 2009;119: 3356–82.CrossRefMATH Gao F. Pathwise properties and homomorphic flows for stochastic differential equations driven by G-Brownian motion. Stoch Process Appl 2009;119: 3356–82.CrossRefMATH
12.
Zurück zum Zitat Ignatyev O. Partial asymptotic stability in probability of stochastic differential equations. Stat Probab Lett 2009;79:597–601.MathSciNetCrossRefMATH Ignatyev O. Partial asymptotic stability in probability of stochastic differential equations. Stat Probab Lett 2009;79:597–601.MathSciNetCrossRefMATH
13.
Zurück zum Zitat Ignatyev O. New criterion of partial asymptotic stability in probability of stochastic differential equations. Appl Math Comput 2013;219:10961–6.MathSciNetMATH Ignatyev O. New criterion of partial asymptotic stability in probability of stochastic differential equations. Appl Math Comput 2013;219:10961–6.MathSciNetMATH
14.
Zurück zum Zitat Li X, Peng S. Stopping times and related g-itô calculus with G-Brownian motion. Stoch Process Appl 2009;121:1492–508.CrossRefMATH Li X, Peng S. Stopping times and related g-itô calculus with G-Brownian motion. Stoch Process Appl 2009;121:1492–508.CrossRefMATH
15.
Zurück zum Zitat Li X, Lin X, Lin Y. Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion. Math Anal Appl 2016;439:235–55.MathSciNetCrossRefMATH Li X, Lin X, Lin Y. Lyapunov-type conditions and stochastic differential equations driven by G-Brownian motion. Math Anal Appl 2016;439:235–55.MathSciNetCrossRefMATH
16.
17.
Zurück zum Zitat Lin Y. Stochastic differential eqations driven by G-Brownian motion with reflecting boundary. Electron J Probab 2013;18:1–23.MathSciNetCrossRef Lin Y. Stochastic differential eqations driven by G-Brownian motion with reflecting boundary. Electron J Probab 2013;18:1–23.MathSciNetCrossRef
18.
Zurück zum Zitat Luo P, Wang F. Stochastic differential equations driven by G-Brownian motion and ordinary differential equations. Stoch Process Appl 2014;124:3869–85.MathSciNetCrossRefMATH Luo P, Wang F. Stochastic differential equations driven by G-Brownian motion and ordinary differential equations. Stoch Process Appl 2014;124:3869–85.MathSciNetCrossRefMATH
19.
Zurück zum Zitat Mao X. Stochastic differential equations and applications. Chichester: Ellis Horwood; 1997.MATH Mao X. Stochastic differential equations and applications. Chichester: Ellis Horwood; 1997.MATH
21.
Zurück zum Zitat Sontag E, Wang Y. On characterizations of input-to-state stability with respect to compact sets. IFAC Nonlinear Control Systems Design, Tahoe City; 1995. Sontag E, Wang Y. On characterizations of input-to-state stability with respect to compact sets. IFAC Nonlinear Control Systems Design, Tahoe City; 1995.
22.
Zurück zum Zitat Peng S. Nonlinear expectations and stochastic calculus under uncertaintly-with robust central limit theorem and G-Brownian motion. Berlin: Springer; 2010. Peng S. Nonlinear expectations and stochastic calculus under uncertaintly-with robust central limit theorem and G-Brownian motion. Berlin: Springer; 2010.
23.
Zurück zum Zitat Peiffer K, Rouche N. Liapunov’s second method applied to partial stability. J Mec 1969;2:323–34.MathSciNetMATH Peiffer K, Rouche N. Liapunov’s second method applied to partial stability. J Mec 1969;2:323–34.MathSciNetMATH
25.
Zurück zum Zitat Peng S. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stoch Process Appl 2008;118:223–5.MathSciNetCrossRef Peng S. Multi-dimensional G-Brownian motion and related stochastic calculus under G-expectation. Stoch Process Appl 2008;118:223–5.MathSciNetCrossRef
26.
Zurück zum Zitat Pham Q C, Tabareau N, Slotine J E. A contraction theory approach to stochastic incremental stability. IEEE Trans Autom Control 2009;54: 1285–90.MathSciNetMATH Pham Q C, Tabareau N, Slotine J E. A contraction theory approach to stochastic incremental stability. IEEE Trans Autom Control 2009;54: 1285–90.MathSciNetMATH
27.
Zurück zum Zitat Rouche N, Habets P, Lalog M. Stability theory by liapunov’s direct method. New York: Springer; 1977.CrossRefMATH Rouche N, Habets P, Lalog M. Stability theory by liapunov’s direct method. New York: Springer; 1977.CrossRefMATH
28.
Zurück zum Zitat Rymanstev V V. On the stability of motions with respect to part of variables. Mosc Univ Math Bull 1957;4:9–16. Rymanstev V V. On the stability of motions with respect to part of variables. Mosc Univ Math Bull 1957;4:9–16.
29.
Zurück zum Zitat Rumyantsev V V, Oziraner A S. Partial stability and stabilization of motion. Moscow (in Russian): Nauka; 1987.MATH Rumyantsev V V, Oziraner A S. Partial stability and stabilization of motion. Moscow (in Russian): Nauka; 1987.MATH
30.
Zurück zum Zitat Savchenko AYa, Ignatyev O. 1989. Some Problems of Stability Theory. Naukova Dumka Kiev (in Russian). Savchenko AYa, Ignatyev O. 1989. Some Problems of Stability Theory. Naukova Dumka Kiev (in Russian).
31.
Zurück zum Zitat Vorotnikov V I. Partial stability and control. Boston: Birkhäuser; 1998.MATH Vorotnikov V I. Partial stability and control. Boston: Birkhäuser; 1998.MATH
32.
Zurück zum Zitat Vorotnikov V I, Rumyantsev V V. Stability and control with respect to a part of the phase coordinates of dynamic systems: theory, methods, and applications. Moscow (in Russian): Scientific World; 2001.MATH Vorotnikov V I, Rumyantsev V V. Stability and control with respect to a part of the phase coordinates of dynamic systems: theory, methods, and applications. Moscow (in Russian): Scientific World; 2001.MATH
33.
Zurück zum Zitat Zhang D, Chen Z. Exponential stability for stochastic differential equation driven by G-Brownian motion. Appl Math Lett 2012;25:1906–10.MathSciNetCrossRefMATH Zhang D, Chen Z. Exponential stability for stochastic differential equation driven by G-Brownian motion. Appl Math Lett 2012;25:1906–10.MathSciNetCrossRefMATH
Metadaten
Titel
Practical Stability with Respect to a Part of the Variables of Stochastic Differential Equations Driven by G-Brownian Motion
verfasst von
Tomás Caraballo
Faten Ezzine
Mohamed Ali Hammami
Publikationsdatum
24.05.2022
Verlag
Springer US
Erschienen in
Journal of Dynamical and Control Systems / Ausgabe 1/2023
Print ISSN: 1079-2724
Elektronische ISSN: 1573-8698
DOI
https://doi.org/10.1007/s10883-022-09593-2

Weitere Artikel der Ausgabe 1/2023

Journal of Dynamical and Control Systems 1/2023 Zur Ausgabe

    Marktübersichten

    Die im Laufe eines Jahres in der „adhäsion“ veröffentlichten Marktübersichten helfen Anwendern verschiedenster Branchen, sich einen gezielten Überblick über Lieferantenangebote zu verschaffen.