In its simplest form, the chemostat consists of microorganisms or cells which grow continually in a specific phase of growth while competing for a single limiting nutrient. Under certain conditions on the cells' growth rate, substrate concentration, and dilution rate, the theory predicts and numerical experiments confirm that a periodically operated chemostat exhibits an "over-yielding" state in which the performance becomes higher than that at the steady-state operation. In this paper we show that an optimal control policy for maximizing the chemostat performance can be accurately and efficiently derived numerically using a novel class of integral-pseudospectral methods and adaptive h-integral-pseudospectral methods composed through a predictor-corrector algorithm. Some new formulas for the construction of Fourier pseudospectral integration matrices and barycentric shifted Gegenbauer quadratures are derived. A rigorous study of the errors and convergence rates of shifted Gegenbauer quadratures as well as the truncated Fourier series, interpolation operators, and integration operators for nonsmooth and generally T-periodic functions is presented. We introduce also a novel adaptive scheme for detecting jump discontinuities and reconstructing a discontinuous function from the pseudospectral data. An extensive set of numerical simulations is presented to support the derived theoretical foundations.

翻译：以其最简单的形式,化学元件由微生物或细胞组成,这些微生物或细胞在特定增长阶段持续增长,同时竞争单一限制营养素。在细胞增长率、基底浓度和稀释率等某些条件下,理论预测和数字实验证实,定期运行的化学元件显示“超产”状态,其性能高于稳定状态运行的“超产”状态。在本文中,我们表明,利用新型的集成光谱方法和由预测者-校正算法构成的适应性超模光谱方法,可以准确和有效地从数字上准确和有效地得出最大限度提高化学元件性能的最佳控制政策。我们从一个预测者-校正算法中引入了一些用于建造四面形伪光谱集集成矩阵的新的公式,以及一些以粗鲁为中心的变化的变异形方形模型。我们从一个模拟模型模型模型模型中引入了一个用于对非模拟和整个周期功能的整合操作者。我们还介绍了一个用于对不移动的四面系列、内部操作者和集成操作者进行精确的模拟分析的模型分析功能。