Periodic Fractional Control in Bioprocesses for Clean Water and Ecosystem Health

TL;DR

This paper develops a fractional-order chemostat model with memory effects for biological water treatment, optimizing periodic control to reduce pollutants and improve system responsiveness, supported by rigorous analysis and efficient numerical methods.

Contribution

It introduces a novel fractional framework linking microbial memory to control strategies, with proofs of optimal solutions and advanced numerical techniques for practical implementation.

Findings

Up to 40% reduction in substrate concentrations compared to steady-state.

Efficient numerical discretization scales favorably for real-world applications.

Memory parameters significantly influence system performance and control effectiveness.

Abstract

This paper introduces a novel fractional-order chemostat model (FOCM) incorporating Caputo fractional derivative with sliding memory (CFDS) to capture microbial memory effects in biological water treatment, addressing limitations of integer-order models that overlook time-dependent behaviors and fail to capture microbial memory such as delayed growth responses to past nutrient availability, history-dependent adaptation to inflow fluctuations, and persistent historical effects over hours to days, which are biologically critical in wastewater treatment. By optimizing periodic dilution rate control, we minimize the average pollutant output, constrained by treatment capacity and periodic boundaries. Key contributions include: (1) a rigorous fractional framework linking microbial kinetics to memory-driven control; (2) reduction to a 1D fractional-order differential equation (FDE) for computational efficiency; (3) proofs of optimal periodic solution (OPS) existence/uniqueness via Schauder's theorem and convexity; (4) bang-bang control derivation using fractional Pontryagin maximum principle (PMP) and Fourier-Gegenbauer pseudospectral (FG-PS) method with a specialized edge-detection technique to handle control discontinuities, ensuring efficient numerical resolution of switching points and discontinuities inherent in bang-bang strategies; and (5) a comprehensive sensitivity analysis revealing how the fractional order {\alpha}, scaling parameter {\vartheta}, and memory length L critically influence system performance, with simulations showing up to 40% reduction in substrate concentrations versus steady-state and demonstrating computational tractability through efficient discretization that scales favorably for practical implementation. Scientifically, this advances fractional calculus in bioprocesses, revealing memory's role in improving responsiveness and efficiency.