In this paper, the fixed/prescribed-time stability issues were considered for stochastic systems with time delay. First, some new fixed-time stability and prescribed-time stability criteria for stochastic systems with delay and multi-delay were established. Second, based on the new fixed/prescribed stability criteria, the fixed-time stabilization of the stochastic system with time-delay and the prescribed-time stabilization of the stochastic reaction-diffusion system with multi-delay were investigated, respectively. Third, two new fixed/prescribed-time delay-independent control mechanisms were designed. The primary advantage of the innovative fixed/prescribed-time controller lies in its independence from delayed states. This makes the controller applicable to systems with unknown delays. Finally, three numerical examples were provided to illustrate the feasibility of the stated theoretical results.
Citation: Yabo Zhao, Huaiqin Wu. Fixed/Prescribed stability criterions of stochastic system with time-delay[J]. AIMS Mathematics, 2024, 9(6): 14425-14453. doi: 10.3934/math.2024701
In this paper, the fixed/prescribed-time stability issues were considered for stochastic systems with time delay. First, some new fixed-time stability and prescribed-time stability criteria for stochastic systems with delay and multi-delay were established. Second, based on the new fixed/prescribed stability criteria, the fixed-time stabilization of the stochastic system with time-delay and the prescribed-time stabilization of the stochastic reaction-diffusion system with multi-delay were investigated, respectively. Third, two new fixed/prescribed-time delay-independent control mechanisms were designed. The primary advantage of the innovative fixed/prescribed-time controller lies in its independence from delayed states. This makes the controller applicable to systems with unknown delays. Finally, three numerical examples were provided to illustrate the feasibility of the stated theoretical results.
[1] |
H. Deng, M. Krstic, Output-feedback stochastic nonlinear stabilization, IEEE Trans. Automat. Contr., 44 (1999), 328–333. http://doi.org/10.1109/9.746260 doi: 10.1109/9.746260
![]() |
[2] |
Z. Liu, L. Xue, W. Zhang, Universal adaptive control strategies for stochastic nonlinear time-delay systems with odd rational powers, Automatica, 125 (2021), 109419. https://doi.org/10.1016/j.automatica.2020.109419 doi: 10.1016/j.automatica.2020.109419
![]() |
[3] |
W. Li, M. Krstic, Stochastic nonlinear prescribed-time stabilization and inverse optimality, IEEE Trans. Automat. Contr., 67 (2022), 1179–1193. https://doi.org/10.1109/TAC.2021.3061646 doi: 10.1109/TAC.2021.3061646
![]() |
[4] |
Z. Cao, C. Li, Z. He, X. Zhang, L. You, Synchronization of coupled stochastic reaction-diffusion neural networks with multiple weights and delays via pinning impulsive control, IEEE Trans. Netw. Sci. Eng., 9 (2022), 820–833. https://doi.org/10.1109/TNSE.2021.3137255 doi: 10.1109/TNSE.2021.3137255
![]() |
[5] |
X. Liu, D. W. C. Ho, Q. Song, W. Xu, Finite/Fixed-time pinning synchronization of complex networks with stochastic disturbances, IEEE Trans. Cybernetics, 49 (2019), 2398–2403. https://doi.org/10.1109/TCYB.2018.2821119 doi: 10.1109/TCYB.2018.2821119
![]() |
[6] | K. Gu, V. L. Kharitonov, J. Chen, Stability of time-delay systems, MA: Birkhäuser Boston, 2003. https://doi.org/10.1007/978-1-4612-0039-0 |
[7] |
Z. Wang, H. Qiao, K. J. Burnham, On stabilization of bilinear uncertain time-delay stochastic systems with Markovian jumping parameters, IEEE Trans. Automat. Contr., 47 (2002), 640–646. https://doi.org/10.1109/9.995042 doi: 10.1109/9.995042
![]() |
[8] |
Q. Fu, J. Cai, S. Zhong, Y. Yu, Pinning impulsive synchronization of stochastic memristor-based neural networks with time-varying delays, Int. J. Control Autom. Syst., 17 (2019), 243–252. https://doi.org/10.1007/s12555-018-0295-3 doi: 10.1007/s12555-018-0295-3
![]() |
[9] |
B. J. Karaki, M. S. Mahmoud, Consensus of time-delay stochastic multiagent systems with impulsive behavior and exogenous disturbances, Neurocomputing, 439 (2021), 86–95. https://doi.org/10.1016/j.neucom.2020.12.077 doi: 10.1016/j.neucom.2020.12.077
![]() |
[10] | A. Halanay, Differential equations: stability, oscillations, time lags, Amsterdam: Elsevier, 1966. |
[11] |
H. Li, B. Zhou, M. Hou, G. Duan, On the time-varying Halanay inequality with applications to stability analysis of time-delay systems, J. Franklin I., 358 (2021), 5488–5512. https://doi.org/10.1016/j.jfranklin.2021.04.046 doi: 10.1016/j.jfranklin.2021.04.046
![]() |
[12] |
D. Ruan, X. Guo, Y. Shi, Generalized Halanay inequalities for stability and dissipativity of stochastic functional differential equations, Syst. Control Lett., 173 (2023), 105469. https://doi.org/10.1016/j.sysconle.2023.105469 doi: 10.1016/j.sysconle.2023.105469
![]() |
[13] |
F. Du, J. Lu, Q, Zhang, Delay-dependent finite-time synchronization criterion of fractional-order delayed complex networks, Commun. Nonlinear Sci. Numer. Simulat., 119 (2023), 107072. https://doi.org/10.1016/j.cnsns.2022.107072 doi: 10.1016/j.cnsns.2022.107072
![]() |
[14] |
M. Zhang, Q. Zhu, Finite-time input-to-state stability of switched stochastic time-varying nonlinear systems with time delays, Chaos Soliton. Fract., 162 (2022), 112391. https://doi.org/10.1016/j.chaos.2022.112391 doi: 10.1016/j.chaos.2022.112391
![]() |
[15] |
G. Chen, J. Yang, X. Zhou, Finite-time dissipative control for discrete-time stochastic delayed systems with Markovian switching and interval parameters, Commun. Nonlinear Sci. Numer. Simulat., 110 (2022), 106352. https://doi.org/10.1016/j.cnsns.2022.106352 doi: 10.1016/j.cnsns.2022.106352
![]() |
[16] | P. Cheng, C. Lu, T. Cai, Finite-time stability of impulsive stochastic delayed systems, Math. Method. Appl. Sci., in press. https://doi.org/10.1002/mma.8096 |
[17] |
Q. Fu, W. Jiang, S. Zhong, K. Shi, Novel adaptive synchronization in finite-time and fixed-time for impulsive complex networks with semi-Markovian switching, ISA Trans., 143 (2023), 360–369. https://doi.org/10.1016/j.isatra.2023.09.010 doi: 10.1016/j.isatra.2023.09.010
![]() |
[18] |
W. Zhang, X. Yang, C. Li, Fixed-time stochastic synchronization of complex networks via continuous control, IEEE Trans. Cybernetics, 49 (2019), 3099–3104. https://doi.org/10.1109/TCYB.2018.2839109 doi: 10.1109/TCYB.2018.2839109
![]() |
[19] |
L. Zhao, H. Wu, J. Cao, Finite/Fixed-time bipartite consensus for networks of diffusion PDEs via event-triggered control, Inform. Sciences, 609 (2022), 1435–1450. https://doi.org/10.1016/j.ins.2022.07.151 doi: 10.1016/j.ins.2022.07.151
![]() |
[20] |
H. Du, G. Wen, D. Wu, Y. Cheng, J. Lü, Distributed fixed-time consensus for nonlinear heterogeneous multi-agent systems, Automatica, 113 (2020), 108797. https://doi.org/10.1016/j.automatica.2019.108797 doi: 10.1016/j.automatica.2019.108797
![]() |
[21] |
T. Peng, J. Lu, J. Xiong, Z. Tu, Y. Liu, J. Lou, Fixed-time synchronization of quaternion-valued neural networks with impulsive effects: a non-decomposition method, Commun. Nonlinear Sci. Numer. Simulat., 132 (2024), 107865. https://doi.org/10.1016/j.cnsns.2024.107865 doi: 10.1016/j.cnsns.2024.107865
![]() |
[22] |
T. Zhou, H. Wu, J. Cao, Distributed optimization in predefinded-time for multi-agent systems over a directed network, Inform. Sciences, 615 (2022), 743–757. https://doi.org/10.1016/j.ins.2022.10.034 doi: 10.1016/j.ins.2022.10.034
![]() |
[23] |
L. Mi, C. Chen, B. Qiu, L. Xu, L. Zhang, Fixed-time synchronization analysis for complex-valued neural networks via a new fixed-time stability theorem, IEEE Access, 8 (2020), 172799–172807. https://doi.org/10.1109/ACCESS.2020.3025373 doi: 10.1109/ACCESS.2020.3025373
![]() |
[24] |
C. Hu, H. He, H. Jiang, Fixed/Preassigned-time synchronization of complex networks via improving fixed-time stability, IEEE Trans. Cybernetics, 51 (2020), 2882–2892. https://doi.org/10.1109/TCYB.2020.2977934 doi: 10.1109/TCYB.2020.2977934
![]() |
[25] |
A. Abdurahman, H. Jiang, C. Hu, Improved fixed-time stability results and application to synchronization of discontinuous neural networks with state-dependent switching, Int. J. Robust Nonlinear Control, 31 (2021), 5725–5744. https://doi.org/10.1002/rnc.5566 doi: 10.1002/rnc.5566
![]() |
[26] |
Y. Xu, X. Wu, B. Mao, J. Lü, C. Xie, Fixed-time synchronization in the pth moment for time-varying delay stochastic multilayer networks, IEEE Trans. Syst. Man Cybern. B, 52 (2020), 1135–1144. https://doi.org/10.1109/TSMC.2020.3012469 doi: 10.1109/TSMC.2020.3012469
![]() |
[27] |
C. Chen, L. Li, H. Peng, Y. Yang, L. Mi, H. Zhao, A new fixed-time stability theorem and its application to the fixed-time synchronization of neural networks, Neural Networks, 123 (2020), 412–419. https://doi.org/10.1016/j.neunet.2019.12.028 doi: 10.1016/j.neunet.2019.12.028
![]() |
[28] | J. D. S$\acute{a}$nchez-Torres, E. N. Sanchez, A. G. Loukianov, Predefined-time stability of dynamical systems with sliding modes, In: 2015 American control conference, 2015, 5842–5846. https://doi.org/10.1109/ACC.2015.7172255 |
[29] | T. Wang, L. Zhang, N. Xu, K. H. Alharbi, Adaptive critic learning for approximate optimal event-triggered tracking control of nonlinear systems with prescribed performances, Int. J. Control, in press. https://doi.org/10.1080/00207179.2023.2250880 |
[30] |
Z. Gao, N. Zhao, X. Zhao, B. Niu, N. Xu, Event-triggered prescribed performance adaptive secure control for nonlinear cyber physical systems under denial-of-service attacks, Commun. Nonlinear Sci. Numer. Simulat., 131 (2024), 107793. https://doi.org/10.1016/j.cnsns.2023.107793 doi: 10.1016/j.cnsns.2023.107793
![]() |
[31] |
S. Liu, B. Niu, H. Karimi, X. Zhao, Self-triggered fixed-time bipartite fault-tolerant consensus for nonlinear multiagent systems with function constraints on states, Chaos Soliton. Fract., 178 (2024), 114367. https://doi.org/10.1016/j.chaos.2023.114367 doi: 10.1016/j.chaos.2023.114367
![]() |
[32] |
J. D. S$\acute{a}$nchez-Torres, D. G$\acute{o}$mez-Guti$\acute{e}$rrez, E. L$\acute{o}$pez, A. G. Loukianov, A class of fixed-time stable dynamical systems, IMA J. Math. Control Inform., 35 (2018), i1–i29. https://doi.org/10.1093/imamci/dnx004 doi: 10.1093/imamci/dnx004
![]() |
[33] |
X. Wang, J. Cao, J. Wang, J. Qi, Q. Sun, A novel fast fixed-time control strategy and its application to fixed-time synchronization control of delayed neural networks, Neural Process. Lett., 54 (2022), 145–164. https://doi.org/10.1007/s11063-021-10624-5 doi: 10.1007/s11063-021-10624-5
![]() |
[34] |
R. Wei, J. Cao, A. Alsaedi, Finite-time and fixed-time synchronization analysis of inertial memristive neural networks with time-varying delays, Cogn. Neurodyn., 12 (2018), 121–134. https://doi.org/10.1007/s11571-017-9455-z doi: 10.1007/s11571-017-9455-z
![]() |
[35] |
Y. Liu, X. Wang, X. Zhou, J. Cao, A novel control law design for prescribed-time/fixed-time stochastic synchronization control of neural networks, Arab. J. Sci. Eng., 48 (2023), 6659–6671. https://doi.org/10.1007/s13369-022-07499-6 doi: 10.1007/s13369-022-07499-6
![]() |
[36] |
J. Yin, S. Khoo, Z. Man, X. Yu, Finite-time stability and instability of stochastic nonlinear systems, Automatica, 47 (2011), 2671–2677. https://doi.org/10.1016/j.automatica.2011.08.050 doi: 10.1016/j.automatica.2011.08.050
![]() |
[37] |
J. Yu, S. Yu, J. Li, Y. Yan, Fixed-time stability theorem of stochastic nonlinear systems, Int. J. Control, 92 (2019), 2194–2200. https://doi.org/10.1080/00207179.2018.1430900 doi: 10.1080/00207179.2018.1430900
![]() |
[38] |
S. Sui, C. L. P. Chen, S. Tong, A novel adaptive NN prescribed performance control for stochastic nonlinear systems, IEEE Trans. Neur. Net. Lear., 32 (2021), 3196–3205. https://doi.org/10.1109/TNNLS.2020.3010333 doi: 10.1109/TNNLS.2020.3010333
![]() |
[39] |
J. Chen, X. Li, X. Wu, G. Shen, Prescribed-time synchronization of complex dynamical networks with and without time-varying delays, IEEE Trans. Netw. Sci. Eng., 9 (2022), 4017–4027. https://doi.org/10.1109/TNSE.2022.3191348 doi: 10.1109/TNSE.2022.3191348
![]() |
[40] |
Z. Li, X. Liu, K. Wu, Y. Yao, Fixed-time boundary stabilisation for delay reaction-diffusion systems, Int. J. Control, 97 (2024), 272–283. https://doi.org/10.1080/00207179.2022.2140313 doi: 10.1080/00207179.2022.2140313
![]() |
[41] |
M. Liu, H. Zhao, H. Jiang, C. Hu, Z. Yu, Z. Li, Fixed/Preassigned-time synchronization control of complex networks with time varying delay, IEEE Access, 10 (2022), 16819–16829. https://doi.org/10.1109/ACCESS.2022.3149595 doi: 10.1109/ACCESS.2022.3149595
![]() |