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In its simplest form, a chemostat consists of microorganisms or cells thatgrow continually in a specific phase of growth while competing for a singlelimiting nutrient. Under certain conditions of the cell growth rate, substrateconcentration, and dilution rate, the theory predicts and numerical experimentsconfirm that a periodically operated chemostat exhibits an "overyielding" statein which the performance becomes higher than that at steady-state operation. Inthis paper, we show that an optimal periodic control policy for maximizingchemostat performance can be accurately and efficiently derived numericallyusing a novel class of integral pseudospectral (IPS) methods and adaptive h-IPSmethods composed through a predictor-corrector algorithm. New formulas for theconstruction of Fourier pseudospectral (PS) integration matrices andbarycentric-shifted Gegenbauer (SG) quadratures are derived. A rigorous studyof the errors and convergence rates of SG quadratures, as well as the truncatedFourier series, interpolation operators, and integration operators fornonsmooth and generally T-periodic functions, is presented. We also introduce anovel adaptive scheme for detecting jump discontinuities and reconstructing apiecewise analytic function from PS data. An extensive set of numericalsimulations is presented to support the derived theoretical foundations.

New Optimal Periodic Control Policy for the Optimal Periodic Performance of a Chemostat Using a Fourier-Gegenbauer-Based Predictor-Corrector Method