Global robust stability of switched nonlinear positive time-varying delay systems with interval uncertainties and all unstable subsystems

Suriyon Yimnet; Kanyuta Poochinapan; Suriyon Yimnet; Kanyuta Poochinapan

doi:10.3934/math.2026026

AIMS Mathematics

2026, Volume 11, Issue 1: 594-617. doi: 10.3934/math.2026026

Previous Article Next Article

Research article Special Issues

Global robust stability of switched nonlinear positive time-varying delay systems with interval uncertainties and all unstable subsystems

Suriyon Yimnet ¹,
Kanyuta Poochinapan ^{2,3,4,5
,
,}

1.
Department of Applied Mathematics and Statistics, Faculty of Science and Technology, Phetchabun Rajabhat University, Phetchabun 67000, Thailand
2.
Advanced Research Center for Computational Simulation, Chiang Mai University, Chiang Mai 50200, Thailand
3.
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
4.
Data Science Research Center, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand
5.
Centre of Excellence in Mathematics, MHESI, Bangkok 10400, Thailand

Received: 29 September 2025 Revised: 01 December 2025 Accepted: 04 December 2025 Published: 08 January 2026
MSC : 34D05, 34D23, 93D05, 93D20

This paper reports on results concerning the global robust stability and global asymptotic stability of switched nonlinear positive time-varying delay systems when all subsystems are unstable. Based on the assumptions of admissible sector nonlinearity and interval uncertainty, a new nonlinear multiple discretized copositive Lyapunov-Krasovskii functional was constructed, and a mode-dependent dwell time switching technique was utilized to derive sufficient conditions for global robust stability of the system under consideration. Furthermore, when the considered systems did not have without interval uncertainty, adequacy criteria were also proposed to ensure global asymptotic stability. By comparing with previous results, it was verified that our results were less conservative and more general than those of older studies. The theoretical results obtained were presented in illustrative examples applicable to delayed neural networks.
- nonlinear dynamical systems,
- global robust stability,
- global asymptotic stability,
- switched positive systems,
- all unstable subsystems,
- time-varying delay systems,
- interval uncertainty analysis
Citation: Suriyon Yimnet, Kanyuta Poochinapan. Global robust stability of switched nonlinear positive time-varying delay systems with interval uncertainties and all unstable subsystems[J]. AIMS Mathematics, 2026, 11(1): 594-617. doi: 10.3934/math.2026026

Related Papers:

Abstract

This paper reports on results concerning the global robust stability and global asymptotic stability of switched nonlinear positive time-varying delay systems when all subsystems are unstable. Based on the assumptions of admissible sector nonlinearity and interval uncertainty, a new nonlinear multiple discretized copositive Lyapunov-Krasovskii functional was constructed, and a mode-dependent dwell time switching technique was utilized to derive sufficient conditions for global robust stability of the system under consideration. Furthermore, when the considered systems did not have without interval uncertainty, adequacy criteria were also proposed to ensure global asymptotic stability. By comparing with previous results, it was verified that our results were less conservative and more general than those of older studies. The theoretical results obtained were presented in illustrative examples applicable to delayed neural networks.

References

[1]	H. K. Khalil, J. W. Grizzle, Nonlinear systems, 3 Eds, Upper Saddle River, NJ: Prentice hall, 2002.
[2]	L. Faybusovich, T. Mouktonglang, T. Tsuchiya, Implementation of infinite-dimensional interior-point method for solving multi-criteria linear-quadratic control problem, Optim. Method. Softw., 21 (2006), 315–341. https://doi.org/10.1080/10556780500079086 doi: 10.1080/10556780500079086
[3]	T. C. Lee, Z. P. Jiang, Uniform asymptotic stability of nonlinear switched systems with an application to mobile robots, IEEE T. Automat. Contr., 53 (2008), 1235–1252. https://doi.org/10.1109/TAC.2008.923688 doi: 10.1109/TAC.2008.923688
[4]	B. Wongsaijai, K. Poochinapan, Optimal decay rates of the dissipative shallow water waves modeled by coupling the Rosenau-RLW equation and the Rosenau-Burgers equation with power of nonlinearity, Appl. Math. Comput., 405 (2021), 126202. https://doi.org/10.1016/j.amc.2021.126202 doi: 10.1016/j.amc.2021.126202
[5]	S. Nanta, S. Yimnet, K. Poochinapan, B. Wongsaijai, On the identification of nonlinear terms in the generalized Camassa-Holm equation involving dual-power law nonlinearities, Appl. Numer. Math., 160 (2021), 386–421. https://doi.org/10.1016/j.apnum.2020.10.006 doi: 10.1016/j.apnum.2020.10.006
[6]	J. Zhao, H. Zhao, Y. Song, Z. Y. Sun, D. Yu, Fast finite-time consensus protocol for high-order nonlinear multi-agent systems based on event-triggered communication scheme, Appl. Math. Comput., 508 (2026), 129631. https://doi.org/10.1016/j.amc.2025.129631 doi: 10.1016/j.amc.2025.129631
[7]	D. Liberzon, Switching in systems and control, Boston: Birkhauser, 2003. https://doi.org/10.1007/978-1-4612-0017-8
[8]	V. N. Phat, T. Botmart, P. Niamsup, Switching design for exponential stability of a class of nonlinear hybrid time-delay systems, Nonlinear Anal.-Hybri., 3 (2009), 1–10. https://doi.org/10.1016/j.nahs.2008.10.001 doi: 10.1016/j.nahs.2008.10.001
[9]	A. Y. Aleksandrov, Y. Chen, A. V. Platonov, L. Zhang, Stability analysis for a class of switched nonlinear systems, Automatica, 47 (2011), 2286–2291. https://doi.org/10.1016/j.automatica.2011.08.016 doi: 10.1016/j.automatica.2011.08.016
[10]	Y. Sun, L. Wang, On stability of a class of switched nonlinear systems, Automatica, 49 (2013), 305–307. https://doi.org/10.1016/j.automatica.2012.10.011 doi: 10.1016/j.automatica.2012.10.011
[11]	Y. Tian, Y. Cai, Y. Sun, Stability of switched nonlinear time-delay systems with stable and unstable subsystems, Nonlinear Anal.-Hybri., 24 (2017), 58–68. https://doi.org/10.1016/j.nahs.2016.11.003 doi: 10.1016/j.nahs.2016.11.003
[12]	L. Li, L. Liu, Y. Yin, Stability analysis for discrete-time switched nonlinear system under MDADT switching, IEEE Access, 5 (2017), 18646–18653. https://doi.org/10.1109/ACCESS.2017.2751584 doi: 10.1109/ACCESS.2017.2751584
[13]	Y. Deng, H. Zhang, Y. Dai, Y. Li, Interval stability/stabilization for linear stochastic switched systems with time-varying delay, Appl. Math. Comput., 428 (2022), 127201. https://doi.org/10.1016/j.amc.2022.127201 doi: 10.1016/j.amc.2022.127201
[14]	H. Zhang, Y. Dai, C. Zhu, Region stability analysis and precise tracking control of linear stochastic systems, Appl. Math. Comput., 465 (2024), 128402. https://doi.org/10.1016/j.amc.2023.128402 doi: 10.1016/j.amc.2023.128402
[15]	Y. Huang, J. E. Zhang, Asymptotic stability of impulsive stochastic switched system with double state-dependent delays and application to neural networks and neural network-based lecture skills assessment of normal students, AIMS Math., 9 (2024), 178–204. https://doi.org/10.3934/math.2024011 doi: 10.3934/math.2024011
[16]	Y. Liu, H. Wang, Q. Zhu, Fast finite-time stabilization of switched stochastic low-order nonlinear systems with asymmetric time-varying output constraints, J. Franklin I., 362 (2025), 107823. https://doi.org/10.1016/j.jfranklin.2025.107823 doi: 10.1016/j.jfranklin.2025.107823
[17]	D. Angeli, P. D. Leenheer, E. D. Sontag, Persistence results for chemical reaction networks with time-dependent kinetics and no global conservation laws, SIAM J. Appl. Math., 71 (2011), 128–146. https://doi.org/10.1137/090779401 doi: 10.1137/090779401
[18]	O. Mason, R. Shorten, On linear copositive Lyapunov functions and the stability of switched positive linear systems, IEEE T. Automat. Contr., 52 (2007), 1346–1349. https://doi.org/10.1109/TAC.2007.900857 doi: 10.1109/TAC.2007.900857
[19]	M. S. Mahmoud, Switched delay-dependent control policy for water-quality systems, IET Control Theory A., 3 (2009), 1599–1610. https://doi.org/10.1049/iet-cta.2008.0474 doi: 10.1049/iet-cta.2008.0474
[20]	Y. G. Sun, Y. Z. Tian, X. J. Xie, Stabilization of positive switched linear systems and its application in consensus of multi-agent systems, IEEE T. Automat. Contr., 62 (2017), 6608–6613. https://doi.org/10.1109/TAC.2017.2713951 doi: 10.1109/TAC.2017.2713951
[21]	Y. Dong, Y. Tian, Stability and $L_1$-gain analysis of switched positive systems with unstable subsystems and sector nonlinearities, J. Franklin I., 361 (2024), 107255. https://doi.org/10.1016/j.jfranklin.2024.107255 doi: 10.1016/j.jfranklin.2024.107255
[22]	J. Lian, J. Liu, New results on stability of switched positive systems: an average dwell-time approach, IET Control Theory A., 7 (2013), 1651–1658. https://doi.org/10.1049/iet-cta.2013.0280 doi: 10.1049/iet-cta.2013.0280
[23]	M. Xiang, Z. Xiang, X. Lu, Q. Liu, Stability, $L_1$-gain and control synthesis for positive switched systems with time-varying delay, Nonlinear Anal.-Hybri., 9 (2013), 9–17. https://doi.org/10.1016/j.nahs.2013.01.001 doi: 10.1016/j.nahs.2013.01.001
[24]	W. Xiang, J. Lam, J. Shen, Stability analysis and $L_1$-gain characterization for switched positive systems under dwell-time constraint, Automatica, 58 (2017), 1–8. https://doi.org/10.1016/j.automatica.2017.07.016 doi: 10.1016/j.automatica.2017.07.016
[25]	S. Yimnet, P. Niamsup, Finite-time stability and boundedness for linear switched singular positive time-delay systems with finite-time unstable subsystems, Syst. Sci. Control Eng., 8 (2020), 541–568. https://doi.org/10.1080/21642583.2020.1839812 doi: 10.1080/21642583.2020.1839812
[26]	L. J. Liu, Z. Zhang, H. R. Karimi, S. Y. Chen, Stability analysis of discrete‐time switched positive linear systems with unstable subsystem, Asian J. Control, 27 (2025), 1964–1974. https://doi.org/10.1002/asjc.3562 doi: 10.1002/asjc.3562
[27]	J. Ni, Y. Sheng, Q. Xiao, Z. Zeng, T. Huang, Stability of partially unstable discrete-time switched positive nonlinear systems with delays, Appl. Math. Comput., 507 (2025), 129591. https://doi.org/10.1016/j.amc.2025.129591 doi: 10.1016/j.amc.2025.129591
[28]	Q. Yu, Y. Feng, Stability analysis of switching systems with all modes unstable based on a $\Phi$-dependent max-minimum dwell time method, AIMS Math., 9 (2024), 4863–4881. https://doi.org/10.3934/math.2024236 doi: 10.3934/math.2024236
[29]	M. Wu, Y. He, J. H. She, Stability analysis and robust control of time-delay systems, Berlin, Heidelberg: Springer, 2010. https://doi.org/10.1007/978-3-642-03037-6
[30]	V. Kharitonov, Time-delay systems: Lyapunov functionals and matrices, New York: Springer Science and Business Media, 2012. https://doi.org/10.1007/978-0-8176-8367-2
[31]	J. Zhang, Z. Han, Robust stabilization of switched positive linear systems with uncertainties, Int. J. Control Autom., 11 (2013), 41–47. https://doi.org/10.1007/s12555-012-0287-7 doi: 10.1007/s12555-012-0287-7
[32]	Y. Li, H. B. Zhang, Q. X. Zheng, Robust stability and $L_1$-gain analysis of interval positive switched T-S fuzzy systems with mode-dependent dwell time, Neurocomputing, 235 (2017), 90–97. https://doi.org/10.1016/j.neucom.2017.01.003 doi: 10.1016/j.neucom.2017.01.003
[33]	T. Mouktonglang, K. Poochinapan, S. Yimnet, Robust finite-time control of discrete-time switched positive time-varying delay systems with exogenous disturbance and their application, Symmetry, 14 (2022), 735. https://doi.org/10.3390/sym14040735 doi: 10.3390/sym14040735
[34]	S. Feng, J. Wang, J. Zhao, Stability and robust stability of switched positive linear systems with all modes unstable, IEEE-CAA J Automatic., 6 (2019), 167–176. https://doi.org/10.1109/JAS.2017.7510718 doi: 10.1109/JAS.2017.7510718
[35]	Z. Liu, X. Zhang, X. Lu, Q. Liu, Stabilization of positive switched delay systems with all modes unstable, Nonlinear Anal.-Hybri., 29 (2018), 110–120. https://doi.org/10.1016/j.nahs.2018.01.004 doi: 10.1016/j.nahs.2018.01.004
[36]	T. Rojsiraphisal, P. Niamsup, S. Yimnet, Global uniform asymptotic stability criteria for linear uncertain switched positive time-varying delay systems with all unstable subsystems, Mathematics, 8 (2020), 2118. https://doi.org/10.3390/math8122118 doi: 10.3390/math8122118
[37]	T. Mouktonglang, K. Poochinapan, S. Yimnet, A Switching strategy for stabilization of discrete-time switched positive time-varying delay systems with all modes being unstable and application to uncertain data, Axioms, 12 (2023), 440. https://doi.org/10.3390/axioms12050440 doi: 10.3390/axioms12050440
[38]	N. Zhang, Y. Sun, Stabilization of switched positive linear delay system with all subsystems unstable, Int. J. Robust Nonlin., 34 (2024), 2441–2456. https://doi.org/10.1002/rnc.7091 doi: 10.1002/rnc.7091
[39]	X. Zhang, Y. Sun, X. Zhu, Analysis of exponential stability and $L_1$-gain performance of positive switched impulsive systems with all unstable subsystems, Nonlinear Anal.-Model., 30 (2025), 157–175. https://doi.org/10.15388/namc.2025.30.38319 doi: 10.15388/namc.2025.30.38319
[40]	B. Yang, Z. Zhao, R. Ma, Y. Liu, F. Wang, Stabilization of sector-bounded switched nonlinear systems with all unstable modes, In: 2017 29th Chinese Control and Decision Conference (CCDC), Chongqing, China, 2017, 2465–2470. https://doi.org/10.1109/CCDC.2017.7978928
[41]	J. Zhang, X. Zhao, X. Cai, Absolute exponential $L_1$-gain analysis and synthesis of switched nonlinear positive systems with time-varying delay, Appl. Math. Comput., 284 (2016), 24–36. https://doi.org/10.1016/j.amc.2016.02.050 doi: 10.1016/j.amc.2016.02.050
[42]	L. Wu, Z. Feng, W. X. Zheng, Exponential stability analysis for delayed neural networks with switching parameters: Average dwell time approach, IEEE T. Neural Networ., 21 (2010), 1396–1407. https://doi.org/10.1109/TNN.2010.2056383 doi: 10.1109/TNN.2010.2056383

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)