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

TL;DR

This paper introduces a novel numerical method combining Fourier and Gegenbauer techniques to accurately compute optimal periodic control policies for chemostats, enhancing performance analysis and control.

Contribution

It develops a new class of integral pseudospectral methods with adaptive features for optimal control of chemostats, including error analysis and convergence proofs.

Findings

Efficient numerical computation of optimal chemostat control policies.

Demonstrated high accuracy and convergence of the proposed methods.

Validated the approach with extensive numerical simulations.

Abstract

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.