Impulsive control for stationary oscillation of nonlinear delay systems and applications

Shipeng Li; Shipeng Li

doi:10.3934/mmc.2023023

Mathematical Modelling and Control

2023, Volume 3, Issue 4: 267-277. doi: 10.3934/mmc.2023023

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Research article

Impulsive control for stationary oscillation of nonlinear delay systems and applications

Shipeng Li ^,

Shandong Energy Group CO. LTD, Ji'nan 250000, China

Received: 10 June 2023 Accepted: 10 November 2023 Published: 15 November 2023

In this paper, the problem of existence-uniqueness and global exponential stability of periodic solution (i.e., stationary oscillation) for a class of nonlinear delay systems with impulses was studied. Some new sufficient conditions ensuring the existence of stationary oscillation for the addressed equations were derived by using the inequality technique that has been reported in recent publications. Our proposed method, which is different with the existing results in the literature, shows that nonlinear delay systems may admit a stationary oscillation using proper impulsive control strategies even if it was originally unstable or divergent. As an application, we considered the single species logarithmic population model and established a new criterion to guarantee the existence of positive stationary oscillation. Some numerical examples and their computer simulations were also given at the end of this paper to show the effectiveness of our development control method.
- impulsive control,
- nonlinear delay systems,
- stationary oscillation,
- logarithmic population model,
- inequality technique
Citation: Shipeng Li. Impulsive control for stationary oscillation of nonlinear delay systems and applications[J]. Mathematical Modelling and Control, 2023, 3(4): 267-277. doi: 10.3934/mmc.2023023

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Abstract

In this paper, the problem of existence-uniqueness and global exponential stability of periodic solution (i.e., stationary oscillation) for a class of nonlinear delay systems with impulses was studied. Some new sufficient conditions ensuring the existence of stationary oscillation for the addressed equations were derived by using the inequality technique that has been reported in recent publications. Our proposed method, which is different with the existing results in the literature, shows that nonlinear delay systems may admit a stationary oscillation using proper impulsive control strategies even if it was originally unstable or divergent. As an application, we considered the single species logarithmic population model and established a new criterion to guarantee the existence of positive stationary oscillation. Some numerical examples and their computer simulations were also given at the end of this paper to show the effectiveness of our development control method.

References

[1]	W. M. Haddad, V. Chellaboina, S. G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, New Jersey: Princeton University Press, 2006. https://doi.org/10.1515/9781400865246
[2]	X. Li, S. Song, Impulsive Systems With Delays: Stability and Control, Singapore: Springer, 2022. https://doi.org/10.1007/978-981-16-4687-4
[3]	J. Cao, G. Stamov, I. Stamova, S. Simeonov, Almost periodicity in impulsive fractional-order reaction-diffusion neural networks with time-varying delays, IEEE Trans. Cybern., 51 (2021), 151–161. https://doi.org/10.1109/TCYB.2020.2967625 doi: 10.1109/TCYB.2020.2967625
[4]	X. Yang, Y. Liu, J. Cao, L. Rutkowski, Synchronization of coupled time-delay neural networks with mode-dependent average dwell time switching, IEEE Trans. Neural Netw. Lear. Syst., 31 (2020), 5483–5496. https://doi.org/10.1109/TNNLS.2020.2968342 doi: 10.1109/TNNLS.2020.2968342
[5]	S. Novo, R. Obaya, V. M. Villarragut, Asymptotic behavior of solutions of nonautonomous neutral dynamical systems, Nonlinear Anal., 199 (2020), 111918. https://doi.org/10.1016/j.na.2020.111918 doi: 10.1016/j.na.2020.111918
[6]	L. Qiang, B.-G. Wang, X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental models with time delay, J. Differ. Equ., 269 (2020), 4440–4476. https://doi.org/10.1016/j.jde.2020.03.027 doi: 10.1016/j.jde.2020.03.027
[7]	S. Niculescu, Delay Effects on Stability: A Robust Control Approach, New York: Springer-Verlag, 2001. https://doi.org/10.1007/1-84628-553-4
[8]	K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Dordrecht: Kluwer Academic Publishers, 1992. https://doi.org/10.1007/978-94-015-7920-9
[9]	I. M. Stamova, G. T. Stamov, On the practical stability with respect to $h$-manifolds of hybrid Kolmogorov systems with variable impulsive perturbations, Nonlinear Anal., 201 (2020), 111775. https://doi.org/10.1016/j.na.2020.111775 doi: 10.1016/j.na.2020.111775
[10]	L. Chen, J. Sun, F. Chen, L. Zhao, Extinction in a Lotkaolterra competitive system with impulse and the effect of toxic substances, Appl. Math. Model., 40 (2016), 2015–2024. https://doi.org/10.1016/j.apm.2015.09.057 doi: 10.1016/j.apm.2015.09.057
[11]	M. Niedzwiecki, M. Ciolek, New semicausal and noncausal techniques for detection of impulsive disturbances in multivariate signals with audio applications, IEEE Trans. Signal Process., 65 (2017), 3881–3892. https://doi.org/10.1109/TSP.2017.2692740 doi: 10.1109/TSP.2017.2692740
[12]	T. Wei, X. Xie, X. Li, Persistence and periodicity of survival red blood cells model with time-varying delays and impulses, Math. Model. Control, 1 (2021), 12–25. https://doi.org/10.3934/mmc.2021002 doi: 10.3934/mmc.2021002
[13]	T. Nagashio, T. Kida, Y. Hamada, T. Ohtani, Robust two-degrees-of-freedom attitude controller design and flight test result for engineering test satellite-Ⅷ spacecraft, IEEE Trans. Control Syst. Techn., 22 (2014), 157–168. https://doi.org/10.1109/10.1109/TCST.2013.2248009 doi: 10.1109/10.1109/TCST.2013.2248009
[14]	I. Stankovic, I. Orovic, M. Dakovic, S. Stankovic, Denoising of sparse images in impulsive disturbance environment, Multimed. Tools Appl., 77 (2018), 5885–5905. https://doi.org/10.1007/s11042-017-4502-7 doi: 10.1007/s11042-017-4502-7
[15]	T. Wei, X. Li, Fixed-time and predefined-time stability of impulsive systems, IEEE/CAA J. Automa. Sin., 10 (2023), 1086–1089. https://doi.org/10.1109/JAS.2023.123147 doi: 10.1109/JAS.2023.123147
[16]	X. Tan, C. Xiang, J. Cao, W. Xu, G. Wen, L. Rutkowski, Synchronization of neural networks via periodic self-triggered impulsive control and its application in image encryption, IEEE Trans. Cybern., 52 (2022), 8246–8257. https://doi.org/10.1109/TCYB.2021.3049858 doi: 10.1109/TCYB.2021.3049858
[17]	T. Yang, Impulsive Control Theory, Berlin: Springer, 2001. https://doi.org/10.1007/3-540-47710-1
[18]	X. Li, S. Song, J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans Autom. Control, 64 (2019), 4024–4034. https://doi.org/10.1109/TAC.2019.2905271 doi: 10.1109/TAC.2019.2905271
[19]	X. Li, X. Yang, J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica, 117 (2020), 108981. https://doi.org/10.1016/j.automatica.2020.108981 doi: 10.1016/j.automatica.2020.108981
[20]	G. Ballinger, X. Liu, Existence, uniqueness and boundedness results for impulsive delay differential equations, Appl. Anal., 74 (2000), 71–93. https://doi.org/10.1080/00036810008840804 doi: 10.1080/00036810008840804
[21]	S. Arora, M. T. Mohan, J. Dabas, Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces, J. Differ. Equ., 307 (2022), 83–113. https://doi.org/10.1016/j.jde.2021.10.049 doi: 10.1016/j.jde.2021.10.049
[22]	H. Leiva, Controllability of the impulsive functional BBM equation with nonlinear term involving spatial derivative, Syst. Control Lett., 109 (2017), 12–16. https://doi.org/10.1016/j.sysconle.2017.09.001 doi: 10.1016/j.sysconle.2017.09.001
[23]	J. Yan, B. Hu, Z.-H. Guan, Controllability criteria on discrete-time impulsive hybrid systems with input delay, IEEE Trans. Syst. Man Cybern.: Syst., 53 (2023), 2304–2316. https://doi.org/10.1109/TSMC.2022.3212533 doi: 10.1109/TSMC.2022.3212533
[24]	T. Zhang, J. Zhou, Y. Liao, Exponentially stable periodic oscillation and Mittag effler stabilization for fractional-order impulsive control neural networks with piecewise Caputo derivatives, IEEE Trans. Cybern., 52 (2022), 9670–9683. https://doi.org/10.1109/TCYB.2021.3054946 doi: 10.1109/TCYB.2021.3054946
[25]	J. Alzabut, T. Abdeljawad, On existence of a globally attractive periodic solution of impulsive delay logarithmic population model, Appl. Math. Comput., 198 (2008), 463–469. https://doi.org/10.1016/j.amc.2007.08.024 doi: 10.1016/j.amc.2007.08.024
[26]	J. Alzabut, G. Stamov, E. Sermutlu, On almost periodic solutions for an impulsive delay logarithmic population model, Math. Comput. Model., 51 (2010), 625–631. https://doi.org/10.1016/j.mcm.2009.11.001 doi: 10.1016/j.mcm.2009.11.001
[27]	Z. Yang, D. Xu, Existence and exponential stability of periodic solution for impulsive delay differential equations and applications, Nonlinear Anal.: Theo. Meth. Appl., 64 (2006), 130–145. https://doi.org/10.1016/j.na.2005.06.014 doi: 10.1016/j.na.2005.06.014
[28]	A. Weng, J. Sun, Globally exponential stability of periodic solutions for nonlinear impulsive delay systems, Nonlinear Anal. Theo. Meth. Appl., 67 (2007), 1938–1946. https://doi.org/10.1016/j.na.2006.08.019 doi: 10.1016/j.na.2006.08.019
[29]	X. Li, M. Bohner, An impulsive delay differential inequality and applications, Comput. Math. Appl., 64 (2012), 1875–1881. https://doi.org/10.1016/j.camwa.2012.03.013 doi: 10.1016/j.camwa.2012.03.013
[30]	D. Yue, S. Xu, Y. Liu, Differential inequality with delay and impulse and its applications to design of robust control, Control Theo. Appl., 16 (1999), 519–524.

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