Rapid exponential stabilization of Lotka-McKendrick's equation via event-triggered impulsive control

Mohsen Dlala; Sharifah Obaid Alrashidi; Mohsen Dlala; Sharifah Obaid Alrashidi

doi:10.3934/mbe.2021449

Mathematical Biosciences and Engineering

2021, Volume 18, Issue 6: 9121-9131. doi: 10.3934/mbe.2021449

Previous Article Next Article

Research article

Rapid exponential stabilization of Lotka-McKendrick's equation via event-triggered impulsive control

Department of Mathematics, College of Sciences, Qassim University, Buraydah, Saudi Arabia

Academic Editor: Tonghua Zhang

Received: 13 July 2021 Accepted: 11 October 2021 Published: 25 October 2021

This paper investigates the problem of rapid exponential stabilization for linear Lotka-McKendrick's equation. Based on a new event-triggered impulsive control (ETIC) method, an impulsive control is designed to solve the rapid exponential stabilization of the dynamic population Lotka-McKendrick's equation. The effectiveness of our control is verified through a numerical example.

Keywords:

Citation: Mohsen Dlala, Sharifah Obaid Alrashidi. Rapid exponential stabilization of Lotka-McKendrick's equation via event-triggered impulsive control[J]. Mathematical Biosciences and Engineering, 2021, 18(6): 9121-9131. doi: 10.3934/mbe.2021449

Related Papers:

[1]	Patrick A. Blamo Jr, Hong Ngoc Thuy Pham, The Han Nguyen . Maximising phenolic compounds and antioxidant capacity from Laurencia intermedia using ultrasound-assisted extraction. AIMS Agriculture and Food, 2021, 6(1): 32-48. doi: 10.3934/agrfood.2021003
[2]	Naima Belguedj, Ghayth Rigane, Ridha Ben Salem, Khodir Madani . Conventional and eco-friendly aqueous extraction methods of date palm fruit compounds: Optimization, comparison, characterization of the date pulp extract and value-added potential. AIMS Agriculture and Food, 2025, 10(1): 218-246. doi: 10.3934/agrfood.2025012
[3]	Budi Suarti, Sukarno, Ardiansyah, Slamet Budijanto . Bio-active compounds, their antioxidant activities, and the physicochemical and pasting properties of both pigmented and non-pigmented fermented de-husked rice flour. AIMS Agriculture and Food, 2021, 6(1): 49-64. doi: 10.3934/agrfood.2021004
[4]	Evi Mintowati Kuntorini, Laurentius Hartanto Nugroho, Maryani, Tri Rini Nuringtyas . Maturity effect on the antioxidant activity of leaves and fruits of Rhodomyrtus tomentosa (Aiton.) Hassk.. AIMS Agriculture and Food, 2022, 7(2): 282-296. doi: 10.3934/agrfood.2022018
[5]	Itxaso Filgueira-Garro, Carolina González-Ferrero, Diego Mendiola, María R. Marín-Arroyo . Effect of cultivar and drying methods on phenolic compounds and antioxidant capacity in olive (Olea europaea L.) leaves. AIMS Agriculture and Food, 2022, 7(2): 250-264. doi: 10.3934/agrfood.2022016
[6]	Nicolas Nagahama, Bruno Gastaldi, Michael N. Clifford, María M. Manifesto, Renée H. Fortunato . The influence of environmental variations on the phenolic compound profiles and antioxidant activity of two medicinal Patagonian valerians (Valeriana carnosa Sm. and V. clarionifolia Phil.). AIMS Agriculture and Food, 2021, 6(1): 106-124. doi: 10.3934/agrfood.2021007
[7]	Marlin Marlin, Marulak Simarmata, Umi Salamah, Waras Nurcholis . Effect of nitrogen and potassium application on growth, total phenolic, flavonoid contents, and antioxidant activity of Eleutherine palmifolia. AIMS Agriculture and Food, 2022, 7(3): 580-593. doi: 10.3934/agrfood.2022036
[8]	Wan Abd Al Qadr Imad Wan-Mohtar, Norfaizah Mahmud, Sugenendran Supramani, Rahayu Ahmad, Nurul Amalina Mohd Zain, Nurshazleen A. M. Hassan, Jayanthi Peryasamy, Sarina Abdul Halim-Lim . Fruiting-body-base flour from an oyster mushroom—a waste source of antioxidative flour for developing potential functional cookies and steamed-bun. AIMS Agriculture and Food, 2018, 3(4): 481-492. doi: 10.3934/agrfood.2018.4.481
[9]	Namfon Samsalee, Rungsinee Sothornvit . Physicochemical, functional properties and antioxidant activity of protein extract from spent coffee grounds using ultrasonic-assisted extraction. AIMS Agriculture and Food, 2021, 6(3): 864-878. doi: 10.3934/agrfood.2021052
[10]	Rossaporn Jiamjariyatam, Orachorn Mekkerdchoo, Pakkapong Phucharoenrak, Lu Zheng . Effects of freeze-drying and vacuum-drying on the quality, total phenolic contents, and antioxidant activities of bee honey in northern Thailand. AIMS Agriculture and Food, 2024, 9(2): 430-444. doi: 10.3934/agrfood.2024025

Abstract

References

[1]	C. Prieur, E. Trélat, Robust optimal stabilization of the brockett integrator via a hybrid feedback, Math. Control Signals Syst., 17 (2005), 201–216. doi: 10.1007/s00498-005-0152-9
[2]	E. D. Sontag, Stability and stabilization: discontinuities and the effect of disturbances, in Nonlinear Analysis, Differential Equations and Control, Springer, (1999), 551–598.
[3]	P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Autom. Control, 52 (2007), 1680–1685. doi: 10.1109/TAC.2007.904277
[4]	M. Heemels, J. Donkers, A. R. Teel, Periodic event-triggered control for linear systems, IEEE Trans. Autom. Control, 54 (2013), 847–861.
[5]	A. Girard, Dynamic triggering mechanisms for event-triggered controls, IEEE Trans. Autom. Control, 60 (2015), 1992–1997. doi: 10.1109/TAC.2014.2366855
[6]	B. Liu, D. N. Liu, C. X. Dou, Exponential stability via event-triggered impulsive control for continuous-time dynamical systems, in Proceedings of the 33rd Chinese Control Conference, (2014), 4056–4060.
[7]	M. Cao, Z. Ai, L. Peng, Input-to-state stabilization of nonlinear systems via event-triggered impulsive control, IEEE Access, 7 (2019), 118581–118585. doi: 10.1109/ACCESS.2019.2936586
[8]	B. Liu, D. J. Hill, Z. Sun, J. Huang, Event-triggered control via impulses for exponential stabilization of discrete-time delayed systems and networks, Int. J. Robust Nonlin. Control, 29 (2019), 1613–1638. doi: 10.1002/rnc.4450
[9]	R. Postoyan, P. Tabuada, D. Nesic, A. Anta, A framework for the event-triggered stabilization of nonlinear systems, IEEE Trans. Autom. Control, 60 (2015), 982–996. doi: 10.1109/TAC.2014.2363603
[10]	Y. Q. Xia, Y. L. Gao, L. P. Yan, M. Y. Fu, Recent progress in networked control systems —- A survey, Int. J. Autom. Comput., 12 (2015), 343–367. doi: 10.1007/s11633-015-0894-x
[11]	M. S. Mahmoud, Y. Xia, Networked Control Systems, Elsevier, New York, 2019.
[12]	J. Qin, Q. Ma, Y. Shi, L. Wang, Recent advances in consensus of multi-agent systems: a brief survey, IEEE Trans. Indust. Electron., 64 (2017), 4972–4983. doi: 10.1109/TIE.2016.2636810
[13]	C. Nowzari, E. Garcia, J. Cortes, Event-triggered communication and control of networked systems for multi-agent consensus, Automatica, 105 (2019), 1–27. doi: 10.1016/j.automatica.2019.03.009
[14]	C. Penga, F. Li, A survey on recent advances in event-triggered communication and control, IEEE Trans. Indust. Electron., 52 (2018), 58–63.
[15]	X. Ge, Q. Han, L. Ding, Y. Wang, X. Zhang, Dynamic event-triggered distributed coordination control and its applications: a survey of trends and techniques, IEEE Trans. Syst. Man CY-S., 50 (2020), 3112–3125. doi: 10.1109/TSMC.2020.3010825
[16]	Z. Yao, N. H. El-Farra, Resource-aware model predictive control of spatially distributed processes using event-triggered communication, in 52nd IEEE Conference on Decision and Control, 8 (2013), 3726–3731.
[17]	N. Espitia, A. Girard, N. Marchand, C. Prieur, Event-based control of linear hyperbolic systems of conservation laws, Automatica, 70 (2016), 275–287. doi: 10.1016/j.automatica.2016.04.009
[18]	N. Espitia, A. Tanwani, S. Tarbouriech, Stabilization of boundary controlled hyperbolic pdes via lyapunov-based event triggered sampling and quantization, in 56th IEEE Conference on Decision and Control, (2017), 1266–1271.
[19]	N. Espitia, I. Karafyllis, M. Krstic, Event-triggered boundary control of constant-parameter reaction-diffusion pdes: a small-gain approach, in 2020 American Control Conference (ACC), (2020), 3437–344.
[20]	L. Baudouin, S. Marx, S. Tarbouriech, Event-triggered damping of a linear wave equation, IFAC PapersOnLine, 52 (2019), 58–63.
[21]	V. Barbu, M. Iannelli, M. Martcheva, On the controllability of the Lotka-Mckendrick model of population dynamics, J. Math. Anal. Appl., 253 (2001), 142–165. doi: 10.1006/jmaa.2000.7075
[22]	N. Hegoburu, M. Tucsnak, Null controllability of the Lotka-Mckendrick system with spatial diffusion, Math. Control Relat. Fields, 8 (2018), 707–720. doi: 10.3934/mcrf.2018030
[23]	N. Hegoburu, P. Magal, M. Tucsnak, Controllability with positivity constraints of the Lotka-Mckendrick syste, SIAM J. Control Optim., 56 (2018), 723–750. doi: 10.1137/16M1103087
[24]	B. Ainseba, Exact and approximate controllability of the age and space population dynamics structured model, J. Math. Anal. Appl., 275 (2002), 562–574. doi: 10.1016/S0022-247X(02)00238-X
[25]	O. Kavian, O. Traoré, Approximate controllability by birth control for a nonlinear population dynamics model, ESAIM Contr. Optim. Calc. Var., 17 (2011), 1198–1213. doi: 10.1051/cocv/2010043
[26]	O. Traore, Null controllability of a nonlinear population dynamics problem, Int. J. Math. Math. Sci., 2006 (2006), 1–20.
[27]	D. Maity, M. Tucsnak, E. Zuazua, Controllability and positivity constraints in population dynamics with age structuring and diffusion, J. de Mathématiques Pures et Appliquées, 129 (2019), 153–179. doi: 10.1016/j.matpur.2018.12.006
[28]	N. Hegoburu, S. Anita, Null controllability via comparison results for nonlinear age-structured population dynamics, Math. Control Signal. Syst., 31 (2019).
[29]	S. P. Wang, Z. R. He, Approximate controllability of population dynamics with size dependence and spatial distribution, ANZIAM J., 58 (2017), 474–481.
[30]	M. L. Gurtin, R. C. MacCamy, Nonlinear age dependent population dynamics, Arch. Ration. Mech. Anal., 54 (1974), 281–300. doi: 10.1007/BF00250793
[31]	F. Kappel, K. Zhang, Approximation of linear age-structured population models using Legendre polynomials, J. Math. Anal. Appl., 180 (1993), 518–549. doi: 10.1006/jmaa.1993.1414
[32]	M. Iannelli, F. Milner, The Basic Approach to Age-Structured Population Dynamics, Springer, Dordrecht, 2017.
[33]	A. G. McKendrick, Applications of mathematics to medical problems, Proc. Edinburgh Math. Soc., 44 (1925), 88–130. doi: 10.1017/S0370164600020824
[34]	A. J. Lotka, The stability of the normal age distribution, Proc. Nat. Acad. Sci., 8 (1922), 339–345. doi: 10.1073/pnas.8.11.339
[35]	S. Anita, Analysis and Control of Age-Dependent Population Dynamics, Springer-Verlag, New York, 2000.
[36]	R. Hao, Y. Zhang, Z. Cao, J. Li, Q. Xu, L. Ye, et al., Control strategies and their effects on the covid-19 pandemic in 2020 in representative countries, J. Biosaf. Biosecur., 3 (2021), 76–81. doi: 10.1016/j.jobb.2021.06.003
[37]	V. Nicosia, P. E. Vértes, W. R. Schafer, V. Latora, E. T. Bullmore, Phase transition in the economically modeled growth of a cellular nervous system, Proc. Natl. Acad. Sci., 110 (2013), 7880–7885. doi: 10.1073/pnas.1300753110
[38]	V. Nicosia, M. Valencia, M. Chavez, A. Díaz-Guilera, V. Latora, Remote synchronization reveals network symmetries and functional modules, Phys. Rev. Lett., 110 (2013), 174102. doi: 10.1103/PhysRevLett.110.174102
[39]	M. Dlala, A. S. Almutairi, Rapid exponential stabilization of nonlinear wave equation derived from brain activity via event-triggered impulsive control, Math., 9 (2021), 516. doi: 10.3390/math9050516
[40]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1992.
[41]	M. Rasheed, S. Laverty, B. Bannish, Numerical solutions of a linear age-structured population model, in AIP Conference Proceedings 2096, (2019), 1–5.

This article has been cited by:

Yi Xuan Choo, Widiastuti Setyaningsih, Hui Ling Tan, Lai Kuan Teh, Chin Xuan Tan, Storage stability of fresh, sonicated, and pasteurized noni juices, 2025, 1556-3758, 10.1515/ijfe-2024-0062

Reader Comments

Your name:*

Email:*
© 2021 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)