  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.
