Abstract
This paper aims to analyze the behavior of the solutions of a stochastic perturbed system with respect to the solutions of the stochastic unperturbed system. To prove our stability results, we have derived a new Gronwall-type inequality instead of the Lyapunov techniques, which makes it easy to apply in practice and it can be considered as a more general tool in some situations. On the one hand, we present sufficient conditions ensuring the global practical uniform exponential stability of SDEs based on Gronwall’s inequalities. On the other hand, we investigate the global practical uniform exponential stability with respect to a part of the variables of the stochastic perturbed system by using generalized Gronwall’s inequalities. It turns out that, the proposed approach gives a better result comparing with the use of a Lyapunov function. A numerical example is presented to illustrate the applicability of our results.
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References
Bellman, R.: The stability of solutions of linear differential equations. Duke Math. J. 10, 643–647 (1943)
Ben Hamed, B., Ellouze, I., Hammami, M.A.: Practical uniform stability of nonlinear differential delay equations. Mediterr. J. Math. 8, 603–616 (2011)
Caraballo, T., Han, X.: Applied Nonautonomous and Random Dynamical Systems. Springer, New York (2016)
Caraballo, T., Hammami, M.A., Mchiri, L.: On the practical global uniform asymptotic stability of stochastic differential equations. Stochastics 88, 45–56 (2016)
Caraballo, T., Hammami, M.A., Mchiri, L.: Practical exponential stability of impulsive stochastic functional differential equations. Syst. Control Lett. 109, 43–48 (2017)
Caraballo, T., Hammami, M.A., Mchiri, L.: Practical stability of stochastic delay evolution equations. Acta Appl. Math. 142, 91–105 (2016)
Caraballo, T., Ezzine, F., Hammami, M., Mchiri, L.: Practical stability with respect to a part of variables of stochastic differential equations. Stochastics 93, 1–18 (2020)
Caraballo, T., Ezzine, F., Hammami, M.: On the exponential stability of stochastic perturbed singular systems in mean square. Appl. Math. Optim. 84, 1–23 (2021)
Caraballo, T., Ezzine, F., Hammami, M.: New stability criteria for stochastic perturbed singular systems in mean square. Nonlinear Dyn. 105, 241–256 (2021)
Caraballo, T., Ezzine, F., Hammami, M.: Partial stability analysis of stochastic differential equations with a general decay rate. J. Eng. Math. 130, 1–17 (2021)
Dammak, H., Hammami, M.A.: Stabilization and practical asymptotic stability of abstract differential equations. Numer. Funct. Anal. Optim. 37, 1235–1247 (2016)
Dlala, M., Hammami, M.A.: Uniform exponential practical stability of impulsive perturbed systems. J. Dyn. Control Syst. 13, 373–386 (2007)
Dragomir, S.S.: Some Gronwall Type Inequalities and Applications. Nova Science Publishers, New York (2003)
Gordon, S.P.: A stability theory for perturbed differential equations. Int. J. Math. Math. Sci. 2, 283–297 (1979)
Gronwall, T.H.: Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. 20, 293–296 (1919)
Ignatyev, O.: Partial asymptotic stability in probability of stochastic differential equations. Statist. Probab. Lett. 79, 597–601 (2009)
Ignatyev, O.: New criterion of partial asymptotic stability in probability of stochastic differential equations. Appl. Math. Comput. 219, 10961–10966 (2013)
Khalil, H.K.: Nonlinear Systems, 2nd edn. MacMillan, New York (1996)
Mao, X.: Exponential Stability of Stochastic Differential Equations. Marcel Dekker Inc, New York (1994)
Mao, X.: Stochastic Differential Equations and Applications. Ellis Horwood, Chichester (1997)
Ma, W., Luo, X., Zhu, Q.: Practical exponential stability of stochastic age-dependent capital system with Lévy noise. Syst. Control Lett. 144, 104–759 (2020)
Oksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, New York (2003)
Peiffer, K., Rouche, N.: Liapunov’s second method applied to partial stability. J. Mécanique 2, 20–29 (1969)
Rymanstev, V.V.: On the stability of motions with respect to part of variables, Vestnik Moscow University. Ser. Math. Mech. 4, 9–16 (1957)
Rumyantsev, V.V., Oziraner, A.S.: Partial Stability and Stabilization of Motion. Nauka, Moscow (1987). (in Russian)
Shen, G., Wu, X., Yin, X.: Stabilization of stochastic differential equations driven by G-Lévy process with discrete-time feedback control. Discret. Contin. Dyn. Syst. B 26, 755–774 (2021)
Socha, V., Zhu, Q.: Exponential stability with respect to part of the variables for a class of nonlinear stochastic systems with Markovian switchings. Math. Comput. Simul. 155, 2–14 (2018)
Vrabel, R.: Local null controllability of the control-affine nonlinear systems with time-varying disturbances. Direct calculation of the null controllable region. Eur. J. Control 40, 80–86 (2018)
Wang, B., Gao, H.: Exponential stability of solutions to stochastic differential equations driven by G-Levy process. Appl. Math. Optim. 83, 1191–1218 (2021)
Zhu, D.: Practical exponential stability of stochastic delayed systems with G-Brownian motion via vector G-Lyapunov function. Math. Comput. Simul. 199, 307–316 (2022)
Zhu, Q., Wang, H.: Output feedback stabilization of stochastic feedforward systems with unknown control coefficients and unknown output function. Automatica 87, 166–175 (2018)
Zhu, Q.: Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. IEEE Trans. Autom. Control 64, 3764–3771 (2019)
Zhu, Q., Huang, T.: Stability analysis for a class of stochastic delay nonlinear systems driven by G-Brownian motion. Syst. Control Lett. 140, 104–699 (2020)
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The research of Tomás Caraballo has been partially supported by the Spanish Ministerio de Ciencia e Innovación (MCI), Agencia Estatal de Investigación (AEI) and Fondo Europeo de Desarrollo Regional (FEDER) under the project PID2021-122991NB-C21.
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Caraballo, T., Ezzine, F. & Hammami, M.A. Estimates of Exponential Convergence for Solutions of Stochastic Nonlinear Systems. Appl Math Optim 88, 62 (2023). https://doi.org/10.1007/s00245-023-10040-2
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DOI: https://doi.org/10.1007/s00245-023-10040-2
Keywords
- Stochastic differential equations
- Gronwall’s inequalities
- Practical uniform exponential stability
- Practical uniform exponential stability with respect to a part of the variables