In its simplest form, a chemostat consists of microorganisms or cells that grow continually in a specific phase of growth while competing for a single limiting nutrient. Under certain conditions of the cell growth rate, substrate concentration, and dilution rate, the theory predicts and numerical experiments confirm that a periodically operated chemostat exhibits an "overyielding" state in which the performance becomes higher than that at steady-state operation. In this paper, we show that an optimal periodic control policy for maximizing chemostat performance can be accurately and efficiently derived numerically using a novel class of integral pseudospectral (IPS) methods and adaptive h-IPS methods composed through a predictor-corrector algorithm. New formulas for the construction of Fourier pseudospectral (PS) integration matrices and barycentric-shifted Gegenbauer (SG) quadratures are derived. A rigorous study of the errors and convergence rates of SG quadratures, as well as the truncated Fourier series, interpolation operators, and integration operators for nonsmooth and generally T-periodic functions, is presented. We also introduce a novel adaptive scheme for detecting jump discontinuities and reconstructing a piecewise analytic function from PS data. An extensive set of numerical simulations is presented to support the derived theoretical foundations.

翻译：在最简形式下，恒化器由微生物或细胞构成，这些微生物或细胞在特定生长阶段持续生长，并竞争单一限制性营养物。在细胞生长速率、底物浓度和稀释率满足特定条件时，理论预测及数值实验均证实：周期性运行的恒化器会出现"超产"状态，其性能优于稳态运行。本文证明，通过采用新型积分伪谱方法及基于预测-校正算法构建的自适应h-IPS方法，可精确高效地数值推导出最大化恒化器性能的最优周期控制策略。本文推导了傅里叶伪谱积分矩阵与重心位移盖根鲍尔求积公式的新构造方法，并对非光滑及一般T周期函数的SG求积误差与收敛速率、截断傅里叶级数、插值算子及积分算子进行了严格研究。同时，我们提出了一种新型自适应方案，用于检测跳跃间断点并从伪谱数据重构分段解析函数。通过大量数值仿真验证了所建立的理论基础。