Periodic Fractional Control in Bioprocesses for Clean Water and Ecosystem Health

TL;DR

This work develops a memory-aware, fractional-order chemostat model for biological water treatment by employing the Caputo Fractional Derivative with Sliding Memory to capture microbial memory. The 2D system is reduced to a 1D fractional differential equation, and a periodic-optimal control problem is formulated to minimize the average substrate concentration under realistic bounds and periodicity. Existence and (under certain conditions) uniqueness of the optimal periodic solution are established, and the optimal control is shown to be bang-bang via a fractional Pontryagin maximum principle. Numerical results using Fourier–Gegenbauer pseudospectral discretization demonstrate robust convergence, with memory parameters ($eta$, $L$) and scaling ($ heta$) significantly influencing switching behavior and pollutant removal, achieving up to ~40% improvement over steady-state operation. Collectively, the findings highlight memory effects as a powerful design lever for improving water treatment efficiency and ecosystem health, while offering scalable computational approaches for real-time implementation.

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 α, 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.

Figures (12)

• Figure 1: Time evolution (in hours) of (a) the PSC $s^*(t)$, (b) the OPC $D^*(t)$, and (c) the corresponding biomass concentration $x^*(t)$ of the RFOCP. The symbols show the predicted solution values obtained at $N = 300$ equally-spaced collocation points from the numerical optimization, while the corrected solution (solid lines) is computed using a reconstructed bang-bang control law with $M=400$ interpolation points. Dashed lines indicate the average substrate concentration $\bar{s}$ and average dilution rate $\bar{D}$, respectively.
• Figure 2: Time evolution (in hours) of (a) the PSC $s^*(t)$, (b) the OPC $D^*(t)$, and (c) the biomass concentration $x^*(t)$ of the RFOCP. The symbols show the predicted solution values obtained at $N = 400$ equally-spaced collocation points from the numerical optimization, while the corrected solution (solid lines) is computed using a reconstructed bang-bang control law with $M=500$ interpolation points. Dashed lines indicate the average substrate concentration $\bar{s}$ and average dilution rate $\bar{D}$, respectively.
• Figure 3: Algebraic convergence of the PSC $s^*(t)$ for the RFOCP. The plot shows the $L^2$-error norm in $s^*(t)$ as a function of the number of collocation points $N$. The reference solution is computed at $N=400$.
• Figure 4: Algebraic convergence of the OOFV $J(D^*)$ for the RFOCP. The absolute error in the computed objective value is shown as a function of the number of collocation points $N$, with the reference value taken at $N=400$.
• Figure 5: Switching times $\xi_k$ (in hours) of the optimal bang-bang control as a function of the fractional order $\alpha$ in the FOCM, obtained using $N = 300$ and $M = 400$. All other parameter values were taken from Table \ref{['tab:parameters']}. Each symbol corresponds to a different switching event, illustrating how the control structure changes with the order of the FD.

Theorems & Definitions (22)

• Theorem 1: Existence of OPC
• proof
• Remark 1
• Theorem 2: Possible Existence of Non-Constant OPCs
• proof
• Remark 2
• Corollary 1
• Theorem 3: Uniqueness of OPC
• proof
• Lemma 1