Periodic fractional control in bioprocesses for clean water and ecosystem health

Kareem T. Elgindy; Muneerah AL Nuwairan; Liew Siaw Ching; Kareem T. Elgindy; Muneerah AL Nuwairan; Liew Siaw Ching

doi:10.3934/math.2026072

AIMS Mathematics

2026, Volume 11, Issue 1: 1712-1760. doi: 10.3934/math.2026072

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Periodic fractional control in bioprocesses for clean water and ecosystem health

1.
Department of Mathematics and Sciences, College of Humanities and Sciences, Ajman University, Ajman, P.O. Box 346, United Arab Emirates
2.
Nonlinear Dynamics Research Center (NDRC), Ajman University, Ajman, P.O. Box 346, United Arab Emirates
3.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
4.
Department of Mathematics and Applied Mathematics, School of Mathematics and Physics, Xiamen University Malaysia, Sepang 43900, Malaysia

Received: 02 December 2025 Revised: 03 January 2026 Accepted: 07 January 2026 Published: 19 January 2026
MSC : 34A08, 37N25, 49J15, 92D25

This paper develops a fractional-order chemostat model for biological water treatment using a Caputo fractional derivative with sliding memory (CFDS) to represent history-dependent microbial dynamics. We pose an optimal control problem that minimizes average pollutant concentration through periodic dilution-rate modulation subject to operational constraints. The analysis reduces the dynamics to a one-dimensional fractional differential equation, establishes existence and uniqueness of an optimal periodic solution, and derives the corresponding bang-bang control via the fractional Pontryagin maximum principle combined with a Fourier–Gegenbauer pseudospectral scheme. Sensitivity results show that the fractional order $ \alpha $, scaling parameter $ \vartheta $, and memory length $ L $ significantly influence treatment performance. Numerical simulations demonstrate substantial reductions in substrate levels compared with steady-state operation, underscoring the potential of fractional modeling for improving water treatment efficiency.
- bang-bang control,
- Caputo fractional derivative,
- chemostat model,
- fractional-order control,
- memory effects,
- optimal periodic control,
- water treatment
Citation: Kareem T. Elgindy, Muneerah AL Nuwairan, Liew Siaw Ching. Periodic fractional control in bioprocesses for clean water and ecosystem health[J]. AIMS Mathematics, 2026, 11(1): 1712-1760. doi: 10.3934/math.2026072

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Abstract

This paper develops a fractional-order chemostat model for biological water treatment using a Caputo fractional derivative with sliding memory (CFDS) to represent history-dependent microbial dynamics. We pose an optimal control problem that minimizes average pollutant concentration through periodic dilution-rate modulation subject to operational constraints. The analysis reduces the dynamics to a one-dimensional fractional differential equation, establishes existence and uniqueness of an optimal periodic solution, and derives the corresponding bang-bang control via the fractional Pontryagin maximum principle combined with a Fourier–Gegenbauer pseudospectral scheme. Sensitivity results show that the fractional order $ \alpha $, scaling parameter $ \vartheta $, and memory length $ L $ significantly influence treatment performance. Numerical simulations demonstrate substantial reductions in substrate levels compared with steady-state operation, underscoring the potential of fractional modeling for improving water treatment efficiency.

References

[1]	T. Bayen, A. Rapaport, F. Z. Tani, Improvement of performances of the chemostat used for continuous biological water treatment with periodic controls, Automatica, 121 (2020), 109199. https://doi.org/10.1016/j.automatica.2020.109199 doi: 10.1016/j.automatica.2020.109199
[2]	K. T. Elgindy, Sustainable water treatment through fractional-order chemostat modeling with sliding memory and periodic boundary conditions: A mathematical framework for clean water and sanitation, Fractal Fract., 10 (2026), 1–25. https://doi.org/10.3390/fractalfract10010004 doi: 10.3390/fractalfract10010004
[3]	K. T. Elgindy, New optimal periodic control policy for the optimal periodic performance of a chemostat using a Fourier–Gegenbauer-based predictor-corrector method, J. Process Contr., 127 (2023), 102995. https://doi.org/10.1016/j.jprocont.2023.102995 doi: 10.1016/j.jprocont.2023.102995
[4]	V. E. Tarasov, Fractional dynamics: Applications of fractional calculus to dynamics of particles, fields and media, New York: Springer Science & Business Media, 2011. https://doi.org/10.1007/978-3-642-14003-7
[5]	R. Magin, Fractional calculus in bioengineering, part 1, Crit. Rev. Biomed. Eng., 32 (2004), 1–104. https://doi.org/10.1615/CritRevBiomedEng.v32.i1.10 doi: 10.1615/CritRevBiomedEng.v32.i1.10
[6]	R. Caponetto, G. Dongola, L. Fortuna, I. Petras, Fractional order systems: Modeling and control applications, New Jersey: World Scientific, 72 (2010). https://doi.org/10.1142/7709
[7]	V. V. Kulish, J. L. Lage, Application of fractional calculus to fluid mechanics, J. Fluid. Eng., 124 (2002), 803–806. https://doi.org/10.1115/1.1478062 doi: 10.1115/1.1478062
[8]	B. Ghanbari, H. Günerhan, H. Srivastava, An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model, Chaos Soliton. Fract., 138 (2020), 109910. https://doi.org/10.1016/j.chaos.2020.109910 doi: 10.1016/j.chaos.2020.109910
[9]	A. Charkaoui, A. B. Loghfyry, A novel multi-frame image super-resolution model based on regularized nonlinear diffusion with Caputo time fractional derivative, Commun. Nonlinear Sci., 139 (2024), 108280. https://doi.org/10.1016/j.cnsns.2024.108280 doi: 10.1016/j.cnsns.2024.108280
[10]	L. Loudahi, A. Ali, J. Yuan, J. Ahmad, L. G. Amin, Y. Wei, Fractal–fractional analysis of a water pollution model using fractional derivatives, Fractal Fract., 9 (2025), 321. https://doi.org/10.3390/fractalfract9050321 doi: 10.3390/fractalfract9050321
[11]	M. Du, Z. Wang, H. Hu, Measuring memory with the order of fractional derivative, Sci. Rep., 3 (2013), 3431. https://doi.org/10.1038/srep03431 doi: 10.1038/srep03431
[12]	C. Ionescu, A. Lopes, D. Copot, J. T. Machado, J. H. Bates, The role of fractional calculus in modeling biological phenomena: A review, Commun. Nonlinear Sci., 51 (2017), 141–159. https://doi.org/10.1016/j.cnsns.2017.04.001 doi: 10.1016/j.cnsns.2017.04.001
[13]	K. T. Elgindy, Fourier–Gegenbauer pseudospectral method for solving periodic fractional optimal control problems, Math. Comput. Simulat., 225 (2024), 148–164. https://doi.org/10.1016/j.matcom.2024.05.003 doi: 10.1016/j.matcom.2024.05.003
[14]	K. T. Elgindy, Fourier–Gegenbauer pseudospectral method for solving periodic higher-order fractional optimal control problems, Int. J. Comput. Meth., 21 (2024), 2450015. https://doi.org/10.1142/S0219876224500154 doi: 10.1142/S0219876224500154
[15]	K. T. Elgindy, Fourier-Gegenbauer pseudospectral method for solving time-dependent one-dimensional fractional partial differential equations with variable coefficients and periodic solutions, Math. Comput. Simulat., 218 (2024), 544–555. https://doi.org/10.1016/j.matcom.2023.11.034 doi: 10.1016/j.matcom.2023.11.034
[16]	I. Podlubny, Fractional differential equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, San Diego: Academic Press, 198 (1998).
[17]	A. Kilbas, Theory and applications of fractional differential equations, North-Holland Math. Stud., 204 (2006).
[18]	S. Bourafa, M. S. Abdelouahab, R. Lozi, On periodic solutions of fractional-order differential systems with a fixed length of sliding memory, J. Innov. Appl. Math. Comput. Sci., 1 (2021), 64–78. 10.58205/jiamcs.v1i1.6 doi: 10.58205/jiamcs.v1i1.6
[19]	C. S. Gokhale, S. Giaimo, P. Remigi, Memory shapes microbial populations, PLoS Comput. Biol., 17 (2021), e1009431. https://doi.org/10.1371/journal.pcbi.1009431 doi: 10.1371/journal.pcbi.1009431
[20]	R. F. Romm, N. Yedidi, O. Gefen, N. K. Nagar, L. Aroeti, I. Ronin, et al., Uncovering phenotypic inheritance from single cells with Microcolony-seq, Cell, 188 (2025), 5313–5331. https://doi.org/10.1016/j.cell.2025.08.001 doi: 10.1016/j.cell.2025.08.001
[21]	M. Khalighi, G. S. Klein, D. Gonze, K. Faust, L. Lahti, Quantifying the impact of ecological memory on the dynamics of interacting communities, PLoS Comput. Biol., 18 (2022), e1009396. https://doi.org/10.1371/journal.pcbi.1009396 doi: 10.1371/journal.pcbi.1009396
[22]	M. M. Amirian, A. J. Irwin, Z. V. Finkel, Extending the monod model of microbal growth with memory, Front. Mar. Sci., 9 (2022), 963734. https://doi.org/10.3389/fmars.2022.963734 doi: 10.3389/fmars.2022.963734
[23]	K. P. Rumbaugh, K. Sauer, Biofilm dispersion, Nat. Rev. Microbiol., 18 (2020), 571–586. https://doi.org/10.1038/s41579-020-0385-0 doi: 10.1038/s41579-020-0385-0
[24]	M. Raboni, V. Torretta, G. Urbini, Influence of strong diurnal variations in sewage quality on the performance of biological denitrification in small community wastewater treatment plants (WWTPs), Sustainability, 5 (2013), 3679–3689. https://doi.org/10.3390/su5093679 doi: 10.3390/su5093679
[25]	A. Novick, M. Weiner, Enzyme induction as an all-or-none phenomenon, P. Natl. Acad. Sci. USA, 43 (1957), 553–566. https://doi.org/10.1073/pnas.43.7.553 doi: 10.1073/pnas.43.7.553
[26]	G. Lambert, E. Kussell, Memory and fitness optimization of bacteria under fluctuating environments, PLoS Genet., 10 (2014), e1004556. https://doi.org/10.1371/journal.pgen.1004556 doi: 10.1371/journal.pgen.1004556
[27]	A. Mitchell, G. H. Romano, B. Groisman, A. Yona, E. Dekel, M. Kupiec, et al., Adaptive prediction of environmental changes by microorganisms, Nature, 460 (2009), 220–224. https://doi.org/10.1038/nature08112 doi: 10.1038/nature08112
[28]	M. B. Miller, B. L. Bassler, Quorum sensing in bacteria, Annu. Rev. Microbiol., 55 (2001), 165–199. https://doi.org/10.1146/annurev.micro.55.1.165 doi: 10.1146/annurev.micro.55.1.165
[29]	S. Renes, Shaped by the past: Resilience and ecological memory in microbial communities, Acta Univ. Agric. Sueciae, 2021.
[30]	M. Vass, S. Langenheder, The legacy of the past: Effects of historical processes on microbial metacommunities, Aquat. Microb. Ecol., 79 (2017), 13–19. https://doi.org/10.3354/ame01816 doi: 10.3354/ame01816
[31]	G. H. Chen, M. C. van Loosdrecht, G. A. Ekama, D. Brdjanovic, Biological wastewater treatment: Principles, modeling and design, London: IWA publishing, 2020. https://doi.org/10.2166/9781789060362
[32]	F. Clarke, Functional analysis, calculus of variations and optimal control, London: Springer, 264 (2013). https://doi.org/10.1007/978-1-4471-4820-3
[33]	N. Ahmed, S. Wang, Optimal control: Existence theory, In: Optimal Control of Dynamic Systems Driven by Vector Measures: Theory and Applications, Cham: Springer, 2021,109–149. doi.org/10.1007/978-3-030-82139-5_4
[34]	D. Liberzon, Calculus of variations and optimal control theory: A concise introduction, Princeton: Princeton University Press, 2011.
[35]	O. P. Agrawal, A general formulation and solution scheme for fractional optimal control problems, Nonlinear Dynam., 38 (2004), 323–337. https://doi.org/10.1007/s11071-004-3764-6 doi: 10.1007/s11071-004-3764-6
[36]	R. Kamocki, Pontryagin maximum principle for fractional ordinary optimal control problems, Math. Method. Appl. Sci., 37 (2014), 1668–1686. https://doi.org/10.1002/mma.2928 doi: 10.1002/mma.2928
[37]	M. Sahabi, A. Y. Cherati, Fractional pseudospectral schemes with applications to fractional optimal control problems, J. Math., 2024 (2024), 9917116. https://doi.org/10.1155/jom/9917116 doi: 10.1155/jom/9917116
[38]	Y. Yang, M. N. Skandari, J. Zhang, A pseudospectral method for continuous-time nonlinear fractional programming, Filomat, 38 (2024), 1947–1961. https://doi.org/10.2298/FIL2406947Y doi: 10.2298/FIL2406947Y
[39]	M. S. Ali, M. Shamsi, H. K. Arab, D. F. Torres, F. Bozorgnia, A space–time pseudospectral discretization method for solving diffusion optimal control problems with two-sided fractional derivatives, J. Vib. Control, 25 (2019), 1080–1095. https://doi.org/10.1177/107754631881119 doi: 10.1177/107754631881119
[40]	C. L. Grady Jr, G. T. Daigger, N. G. Love, C. D. Filipe, Biological wastewater treatment, 3 Eds., New York: CRC press, 2011. https://doi.org/10.1201/b13775
[41]	K. T. Elgindy, Optimal periodic control of unmanned aerial vehicles based on Fourier integral pseudospectral and edge-detection methods, Unmanned Syst., 2024, 1–19. https://doi.org/10.1142/S230138502550075X doi: 10.1142/S230138502550075X
[42]	P. W. Smeets, Stochastic modelling of drinking water treatment in quantitative microbial risk assessment, London: IWA Publishing, 2010.
[43]	R. Goebel, R. G. Sanfelice, A. R. Teel, Hybrid dynamical systems: Modeling, stability, and robustness, Princeton University Press, 2012.
[44]	M. Lowe, R. Qin, X. Mao, A review on machine learning, artificial intelligence, and smart technology in water treatment and monitoring, Water, 14 (2022), 1384. https://doi.org/10.3390/w14091384 doi: 10.3390/w14091384
[45]	G. Olsson, B. Newell, Wastewater treatment systems: Modelling, diagnosis and control, London: IWA Publishing, 2005.
[46]	K. V. Gernaey, M. C. van Loosdrecht, M. Henze, M. Lind, S. B. Jørgensen, Activated sludge wastewater treatment plant modelling and simulation: state of the art, Environ. Modell. Softw., 19 (2004), 763–783. https://doi.org/10.1016/j.envsoft.2003.03.005 doi: 10.1016/j.envsoft.2003.03.005
[47]	S. Revollar, R. Vilanova, P. Vega, M. Francisco, M. Meneses, Wastewater treatment plant operation: Simple control schemes with a holistic perspective, Sustainability, 12 (2020), 768. https://doi.org/10.3390/su12030768 doi: 10.3390/su12030768
[48]	L. Åmand, G. Olsson, B. Carlsson, Aeration control–a review, Water Sci. Technol., 67 (2013), 2374–2398. https://doi.org/10.2166/wst.2013.139 doi: 10.2166/wst.2013.139
[49]	D. Vrecko, N. Hvala, A. Stare, O. Burica, M. Strazar, M. Levstek, Improvement of ammonia removal in activated sludge process with feedforward-feedback aeration controllers, Water Sci. Technol., 53 (2006), 125–132. https://doi.org/10.2166/wst.2006.098 doi: 10.2166/wst.2006.098
[50]	U. Jeppsson, M. N. Pons, The cost benchmark simulation model—current state and future perspective, Control Eng. Pract., 12 (2004), 299–304. https://doi.org/10.1016/j.conengprac.2003.07.001 doi: 10.1016/j.conengprac.2003.07.001
[51]	U. Jeppsson, M. N. Pons, I. Nopens, J. Alex, J. Copp, K. V. Gernaey, et al., Benchmark simulation model no. 2: general protocol and exploratory case studies, Water Sci. Technol., 56 (2007), 67–78. https://doi.org/10.2166/wst.2007.604 doi: 10.2166/wst.2007.604
[52]	I. Santín, C. Pedret, R. Vilanova, Applying variable dissolved oxygen set point in a two level hierarchical control structure to a wastewater treatment process, J. Process Control, 28 (2015), 40–55. https://doi.org/10.1016/j.jprocont.2015.02.005 doi: 10.1016/j.jprocont.2015.02.005
[53]	J. Guerrero, A. Guisasola, J. A. Baeza, The nature of the carbon source rules the competition between PAO and denitrifiers in systems for simultaneous biological nitrogen and phosphorus removal, Water Res., 45 (2011), 4793–4802. https://doi.org/10.1016/j.watres.2011.06.019 doi: 10.1016/j.watres.2011.06.019
[54]	H. K. Khalil, J. W. Grizzle, Nonlinear systems, 3 Eds., Upper Saddle River: Prentice Hall, 2002.
[55]	C. Li, F. Zhang, A survey on the stability of fractional differential equations, Eur. Phys. J.-Spec. Top., 193 (2011), 27–47. https://doi.org/10.1140/epjst/e2011-01379-1 doi: 10.1140/epjst/e2011-01379-1
[56]	J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, New York: Springer Science & Business Media, 42 (2013). https://doi.org/10.1007/978-1-4612-1140-2

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