@article{Elgindy2026, 
author = {Kareem T. Elgindy and Muneerah AL Nuwairan and Liew Siaw Ching},
title = {Periodic fractional control in bioprocesses for clean water and ecosystem health},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {1},
pages = {1712-1760},
keywords = {bang-bang control, Caputo fractional derivative, chemostat model, fractional-order control, memory effects, optimal periodic control, water treatment},
url = {https://www.sciopen.com/article/10.3934/math.2026072},
doi = {10.3934/math.2026072},
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    α, scaling parameter    ϑ, 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.}
}