Stabilization by delay feedback control for highly nonlinear HSDDEs driven by Lévy noise

Document Type : Research Article

Author

School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo 454003, China

Abstract

This research aims to investigate the stabilization of highly nonlinear hybrid stochastic differential delay equations (HSDDEs) with L'evy noise by delay feedback control. The coefficients of these systems satisfy a more general polynomial growth condition instead of classical linear growth condition. Precisely, an appropriate Lyapunov functional is constructed to analyze the stabilization of such systems in the sense of $H_{\infty}$-stability and asymptotic stability. The theoretical analysis indicates that the delay can affect the stability of highly nonlinear hybrid stochastic systems.  

Keywords

Main Subjects


[1] D. Applebaum, L´evy Processes and Stochastic Calculus, 2nd Edition, Cambridge University Press,
2009.
[2] D. Applebaum, M. Siakalli, Stochastic stabilization of dynamical systems using L´evy noise, Stoch.
Dyn. 10 (2010) 509–527.
[3] A. Bahar, X. Mao, Stochastic delay population dynamics, Int. J. Pure Appl. Math. 11 (2004) 377–
400.
[4] M. Fatehi Nia, E. Mirzavand, Stochastic dynamics of Izhikevich-Fitzhugh neuron model, J. Math.
Model. 12 (2024) 199–214.
[5] C. Fei, W. Fei, X. Mao, M. Shen, L. Yan, Stability analysis of highly nonlinear hybrid multiple-
delay stochastic differential equations, J. Appl. Anal. Comput. 9 (2019) 1053–1070.
[6] D. He, L. Xu, Boundedness analysis of stochastic delay differential equations with L´evy noise,
Appl. Math. Comput. 421 (2022) 126902.
[7] L. Hu, X. Mao, Y. Shen, Stability and boundedness of nonlinear hybrid stochastic differential delay
equations, Syst. Control Lett. 62 (2013) 178–187.
[8] L. Huang, X. Mao, Delay-dependent exponential stability of neutral stochastic delay systems, IEEE
Trans. Autom. Control. 54 (2009) 147–152.
[9] M. Loeve, Probability theory, D. Van Nostrand Company, Inc, 1955.
[10] G. Li, Stabilization of stochastic regime-switching Poisson jump equations by delay feedback con-
trol, J. Inequal. Appl. 1 (2022) 20.
[11] M. Li, F. Deng, Almost sure stability with general decay rate of neutral stochastic delayed hybrid
systems with L´evy noise, Nonlinear Anal. Hybrid Syst. 24 (2017) 171–185.
[12] G. Li, Z. Hu, F. Deng, H. Zhang, Stabilization via delay feedback for highly nonlinear stochastic
time-varying delay systems with Markovian switching and Poisson jump, Electron. J. Qual. Theory
Differ. Equ. 49 (2022) 1–20.
[13] W. Li, C. Fei, C. Mei, W. Fei, X. Mao, Delay tolerance for stable hybrid stochastic differential
equations with L´evy noise based on Razumikhin technique, Syst. Control Lett. 176 (2023) 105530.
[14] X. Li, X. Mao, Stabilisation of highly nonlinear hybrid stochastic differential delay equations by
delay feedback control, Automatica 112 (2020) 108657.
[15] G. Li, Q. Yang, Stability analysis between the hybrid stochastic delay differential equations with
jumps and the Euler-Maruyama method, J. Appl. Anal. Comput. 11 (2021) 1259–1272.
[16] G. Li, Q. Yang, Stabilization of hybrid stochastic systems with L´evy noise by discrete-time feedback
control, Int. J. Control. 95 (2020) 197–205.
[17] D. Liu, W. Wang, J. Menaldi, Almost sure asymptotic stabilization of differential equations with
time-varying delay by L´evy noise, Nonlinear Dyn. 79 (2015) 163–172.
[18] M. Liu, Y. Zhu, Stationary distribution and ergodicity of a stochastic hybrid competition model
with L´evy jumps, Nonlinear Anal. Hybrid Syst. 30 (2018) 225–239.
[19] Z. Lu, J. Hu, X. Mao, Stabilisation by delay feedback control for highly nonlinear hybrid stochastic
differential equations, Discrete Contin. Dyn. Syst. Ser. B. 24 (2019) 4099–4116.
[20] X. Mao, Stability of stochastic differential equations with Markovian switching, Stoch. Proc. Appl.
79 (1999) 45–67.
[21] W. Mao, L. Hu, X. Mao, The asymptotic stability of hybrid stochastic systems with pantograph
delay and non-Gaussian L´evy noise, J. Franklin Inst. 357 (2020) 1174–1198.
[22] X. Mao, A. Matasov, A. Piunovskiy, Stochastic differential delay equations with Markovian switch-
ing, Bernoulli. 6 (2000) 73–90.
[23] X. Mao, L. Shaikhet, Delay-dependent stability criteria for stochastic differential delay equations
with Markovian switching, Stab. Control Theory Appl. 3 (2000) 88–102.
[24] X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College
Press, London, 2006.
[25] M. Rhaima, L. Mchiri, A. Makhlouf, H∞ and Asymptotic Stability via delay feedback for hybrid
neutral stochastic delay differential equations with L´evy noise, IMA J. Math. Control Inform. 40
(2023) 106–132.
[26] T. Tian, K. Burrage, P. Burrage, M. Carletti, Stochastic delay differential equations for genetic
regulatory networks, J. Comput. Appl. Math. 205 (2007) 696–707.
[27] F. Wan, P. Hu, H. Chen, Stability analysis of neutral stochastic dfferential delay equations driven
by L´evy noises, Appl. Math. Comput. 375 (2020) 125080.
[28] L. Wu, X. Su, P. Shi, Sliding mode control with bounded L2 gain performance of Markovian jump
singular time-delay systems, Automatica. 48 (2012) 1929–1933.
[29] A. Wu, H. Yu, Z. Zeng, Variable-delay feedback control for stabilisation of highly nonlinear hybrid
stochastic neural networks with time-varying delays, Int. J. Control. 97 (2023) 744–755.
[30] C. Yuan, X. Mao, Stability of stochastic delay hybrid systems with jumps, Eur. J. Control. 6 (2010)
595–608.
[31] Q. Zhu, Asymptotic stability in the pth moment for stochastic differential equations with L´evy noise,
J. Math. Anal. Appl. 416 (2014) 126-142.