Periodic fractional control in bioprocesses for clean water and ecosystem health

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.