Abstract
Discrete feedback control was designed to stabilize an unstable hybrid neutral stochastic differential delay system (HNSDDS) under a highly nonlinear constraint in the H∞ and exponential forms. Nevertheless, the existing work just adapted to autonomous cases, and the obtained results were mainly on exponential stabilization. In comparison with autonomous cases, non-autonomous systems are of great interest and represent an important challenge. Accordingly, discrete feedback control has here been adjusted with a time factor to stabilize an unstable non-autonomous HNSDDS, in which new Lyapunov-Krasovskii functionals and some novel technologies are adopted. It should be noted, in particular, that the stabilization can be achieved not only in the routine H∞ and exponential forms, but also the polynomial form and even a general form.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Karimi H R. Robust delay-dependent H∞ control of uncertain time-delay systems with mixed neutral, discrete, and distributed time-delays and Markovian switching parameters. IEEE Transactions on Circuits and Systems I: Regular Papers, 2011, 58(8): 1910–1923
Lee T H, Lakshmanan S, Park J H, et al. State estimation for genetic regulatory networks with mode-dependent leakage delays, time-varying delays, and Markovian jumping parameters. IEEE Transactions on Nanobioscience, 2013, 12(4): 363–375
Revathi V M, Balasubramaniam P, Park J H, et al. H∞ filtering for sample data systems with stochastic sampling and Markovian jumping parameters. Nonlinear Dynamics, 2014, 78(2): 813–830
Feng L, Cao J, Liu L. Stability analysis in a class of Markov switched stochastic Hopfield neural networks. Neural Processing Letters, 2019, 50(1): 413–430
Choi J, Lim C C. A Cholesky factorization based approach for blind FIR channel identification. IEEE Transactions on Signal Processing, 2008, 56(4): 1730–1735
Boukas E K. Stochastic Switching Systems: Analysis and Design. Boston: Birkhauser, 2006
Kolmanovskii V, Koroleva N, Maizenberg T, et al. Neutral stochastic differential delay equations with Markovian switching. Stochastic Analysis and Applications, 2003, 21(4): 819–847
Xu S, Chu Y, Lu J, et al. Exponential dynamic output feedback controller design for stochastic neutral systems with distributed delays. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 2006, 36(3): 540–548
Mao X, Shen Y, Yuan C. Almost surely asymptotic stability of neutral stochastic differential delay equations with Markovian switching. Stochastic Processes and Their Applications, 2008, 118(8): 1385–1406
Bao J, Hou Z, Yuan C. Stability in distribution of neutral stochastic differential delay equations with Markovian switching. Statistics & Probability Letters, 2009, 79(15): 1663–1673
Chen W, Zheng W, Shen Y. Delay-dependent stochastic stability and H∞-control of uncertain neutral stochastic systems with time delay. IEEE Transactions on Automatic Control, 2009, 54(7): 1660–1667
Wu F, Hu S, Huang C. Robustness of general decay stability of nonlinear neutral stochastic functional differential equations with infinite delay. Systems & Control Letters, 2010, 59(3/4): 195–202
Chen Y, Zheng W, Xue A. A new result on stability analysis for stochastic neutral systems. Automatica, 2010, 46(12): 2100–2104
Zhu Q, Cao J. Stability analysis for stochastic neural networks of neutral type with both Markovian jump parameters and mixed time delays. Neurocomputing, 2010, 73: 2671–2680
Pavlovic G, Jankovic S. The Razumikhin approach on general decay stability for neutral stochastic functional differential equations. Journal of the Franklin Institute, 2013, 350(8): 2124–2145
Zhou W, Zhu Q, Shi P, et al. Adaptive synchronization for neutral-type neural networks with stochastic perturbation and Markovian switching parameters. IEEE Transactions on Cybernetics, 2014, 44(12): 2848–2860
Chen H, Shi P, Lim C C, et al. Exponential stability for neutral stochastic Markov systems with time-varying delay and its applications. IEEE Transactions on Cybernetics, 2015, 46(6): 1350–1362
Obradović M, Milošević M. Stability of a class of neutral stochastic differential equations with unbounded delay and Markovian switching and the Euler-Maruyama method. Journal of Computational and Applied Mathematics, 2017, 309: 244–266
Chen H, Yuan C. On the asymptotic behavior for neutral stochastic differential delay equations. IEEE Transactions on Automatic Control, 2018, 64(4): 1671–1678
Li M, Deng F. Almost sure stability with general decay rate of neutral stochastic delayed hybrid systems with lévy noise. Nonlinear Analysis: Hybrid Systems, 2017, 24: 171–185
Shen M, Fei W, Mao X, et al. Stability of highly nonlinear neutral stochastic differential delay equations. Systems & Control Letters, 2018, 115: 1–8
Feng L, Wu Z, Cao J, et al. Exponential stability for nonlinear hybrid stochastic systems with time varying delays of neutral type. Applied Mathematics Letters, 2020, 107: 106468
Feng L, Liu L, Cao J, et al. General decay stability for non-autonomous neutral stochastic systems with time-varying delays and Markovian switching. IEEE Transactions on Cybernetics, 2022, 52(6): 5441–5453
Zhao Y, Zhu Q. Stability of highly nonlinear neutral stochastic delay systems with non-random switching signals. Systems & Control Letters, 2022, 165: 105261
Hu L, Mao X. Almost sure exponential stabilisation of stochastic systems by state-feedback control. Automatica, 2008, 44: 465–471
Ji Y, Chizeck H J. Controllability, stabilizability and continuous-time Markovian jump linear quadratic control. IEEE Transactions on Automatic Control, 1990, 35: 777–788
Mao X, Yin G, Yuan C. Stabilization and destabilization of hybrid systems of stochastic differential equations. Automatica, 2007, 43: 264–273
Wu L, Su X, Shi P. Sliding mode control with bounded L2 gain performance of Markovian jump singular time-delay systems. Automatica, 2012, 48(8): 1929–1933
Mao X. Stabilization of continuous-time hybrid stochastic differential equations by discrete-time feedback control. Automatica, 2013, 49(12): 3677–3681
Mao X, Liu W, Hu L, et al. Stabilization of hybrid stochastic differential equations by feedback control based on discretetime state observations. Systems & Control Letters, 2014, 73: 88–95
You S, Liu W, Lu J, et al. Stabilization of hybrid systems by feedback control based on discrete-time state observations. SIAM Journal on Control and Optimization, 2015, 53(2): 905–925
Song G, Zheng B, Luo Q, et al. Stabilisation of hybrid stochastic differential equations by feedback control based on discrete-time observations of state and mode. IET Control Theory & Applications, 2017, 11(3): 301–307
Lewis A L. Option Valuation Under Stochastic Volatility: With Mathematica Code. Newport Beach: Finance Press, 2000
Luo Q, Mao X. Stochastic population dynamics under regime switching. Journal of Mathematical Analysis and Applications, 2007, 334: 69–84
Zhu Y, Wang K, Ren Y. Dynamics of a mean-reverting stochastic volatility equation with regime switching. Communications in Nonlinear Science and Numerical Simulation, 2020, 83: 105110
Fei C, Fei W, Mao X, et al. Stabilization of highly nonlinear hybrid systems by feedback control based on discrete-time state observations. IEEE Transactions on Automatic Control, 2020, 65(7): 2899–2912
Feng L, Liu Q, Cao J, et al. Stabilization in general decay rate of discrete feedback control for non-autonomous Markov jump stochastic systems. Applied Mathematics and Computation, 2022, 417: 126771
Mei C, Fei C, Shen M, et al. Discrete feedback control for highly nonlinear neutral stochastic delay differential equations with Markovian switching. Information Sciences, 2022, 592: 123–136
Zhao Y, Zhu Q. Stabilization of stochastic highly nonlinear delay systems with neutral term. IEEE Transactions on Automatic Control, 2023, 68(4): 2544–2551
Zhu Q. Stabilization of stochastic nonlinear delay systems with exogenous disturbances and the event-triggered feedback control. IEEE Transactions on Automatic Control, 2019, 64(9): 3764–3771
Zhao Y, Zhu Q. Stabilization by delay feedback control for highly nonlinear switched stochastic systems with time delays. International Journal of Robust & Nonlinear Control, 2021, 31(8): 3070–3089
Feng L, Liu L, Cao J, et al. General stabilization of non-autonomous hybrid systems with delays and random noises via delayed feedback control. Communications in Nonlinear Science and Numerical Simulation, 2023, 117: 106939
Feng L, Li S, Song R, et al. Suppression of explosion by polynomial noise for nonlinear differential systems. Science China Information Sciences, 2018, 61(7): Art 70215
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
Conflict of Interest The authors declare that they have no conflict of interest.
Additional information
This research was supported by the National Natural Science Foundation of China (61833005), the Humanities and Social Science Fund of Ministry of Education of China (23YJAZH031), the Natural Science Foundation of Hebei Province of China (A2023209002, A2019209005) and the Tangshan Science and Technology Bureau Program of Hebei Province of China (19130222g).
Rights and permissions
About this article
Cite this article
Feng, L., Zhang, C., Cao, J. et al. A note on the general stabilization of discrete feedback control for non-autonomous hybrid neutral stochastic systems with a delay. Acta Math Sci 44, 1145–1164 (2024). https://doi.org/10.1007/s10473-024-0320-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10473-024-0320-y
Key words
- hybrid neutral stochastic differential delay system
- discrete feedback control
- general stabilization
- polynomial stabilization