标题： 生物过程中的周期性分数控制用于清洁水和生态系统健康 显示英文标题

标题： Periodic Fractional Control in Bioprocesses for Clean Water and Ecosystem Health

摘要： 本文提出了一种新颖的分数阶化学计量模型（FOCM），用于优化生物水处理过程，通过使用具有滑动记忆的Caputo分数阶导数来引入记忆效应。传统整数阶模型无法捕捉微生物系统中固有的时变行为和记忆效应。我们的工作通过开发一个分数阶框架来解决这一局限性，从而更准确地表示微生物生长和底物降解动力学。主要目标是通过稀释率的最优周期控制（OPC）最小化处理水中平均污染物浓度，同时满足处理能力和周期边界条件的约束。主要贡献包括：(1) 将二维分数阶系统简化为计算上可行的一维分数阶微分方程，同时保留关键动力学；(2) 通过Schauder不动点定理和凸性论证严格证明了最优周期解的存在性和唯一性；(3) 通过分数阶Pontryagin最大原理（PMP）推导出bang-bang最优控制（OC）策略；(4) 全面的数值模拟表明与稳态操作相比性能显著提升。我们的结果表明，与稳态操作相比，周期性分数控制可将平均底物浓度降低多达40%，其中分数阶{\alpha }、动态缩放参数$\vartheta$和滑动记忆长度 L 是决定记忆效应、控制响应性和切换频率的关键因素。所提出的框架将分数阶微积分与环境工程相结合，为设计具有改进污染物去除效率的可持续水处理系统提供了新的见解。 显示英文摘要

摘要： This paper presents a novel fractional-order chemostat model (FOCM) for optimizing biological water treatment processes, incorporating memory effects through the use of Caputo fractional derivative with sliding memory (CFDS). Traditional integer-order models fail to capture the time-dependent behaviors and memory effects inherent in microbial systems. Our work addresses this limitation by developing a fractional-order framework that represents microbial growth and substrate degradation dynamics more accurately. The primary objective is to minimize the average pollutant concentration in treated water through optimal periodic control (OPC) of the dilution rate, subject to constraints on treatment capacity and periodic boundary conditions. Key contributions include: (1) reduction of the 2D fractional-order system to a computationally tractable 1D fractional differential equation while preserving essential dynamics; (2) rigorous proof of the existence and uniqueness of optimal periodic solutions using Schauder's fixed-point theorem and convexity arguments; (3) derivation of bang-bang optimal control (OC) strategies by using a fractional Pontryagin maximum principle (PMP); and (4) comprehensive numerical simulations demonstrating significant performance improvements over steady-state operation. Our results show that periodic fractional control can reduce average substrate concentrations by up to 40% compared to steady-state operation, with the fractional order {\alpha}, the dynamic scaling parameter $\vartheta$, and the sliding memory length L serving as critical factors that govern memory effects, control responsiveness, and switching frequency. The proposed framework bridges fractional calculus with environmental engineering, offering new insights for designing sustainable water treatment systems with improved pollutant removal efficiency.