New optimal periodic control policy for the optimal periodic performance of a chemostat using a Fourier–Gegenbauer-based predictor-corrector method

• Main Author: Elgindy, K.T (Author)
• Format: Article
• Language: English
• Published: Elsevier Ltd 2023
• Subjects: Adaptive method; Chemostat model; Fourier interpolation; Gegenbauer polynomials; h-integral pseudospectral; Integration matrix; Optimal control; Predictor-corrector; Pseudospectral method
• Online Access: View Fulltext in Publisher; View in Scopus
• ISBN: 09591524 (ISSN)
• DOI: 10.1016/j.jprocont.2023.102995

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. © 2023 Elsevier Ltd