Numerical Solution of Nonlinear Periodic Optimal Control Problems Using a Fourier Integral Pseudospectral Method

Abstract

This paper presents a Fourier integral pseudospectral (FIPS) method for a general class of nonlinear, periodic optimal control (OC) problems with equality and/or inequality constraints and sufficiently smooth solutions. In this scheme, the integral form of the problem is collocated at an equispaced set of nodes, and all necessary integrals are approximated using highly accurate Fourier integration matrices (FIMs). The proposed method leads to a nonlinear programming problem (NLP) with algebraic constraints, which is solved using a direct numerical optimization method. Sharp convergence and error estimates are derived for Fourier series, interpolants, and quadratures used for smooth and continuous periodic functions. Two nonlinear examples are considered to show the accuracy and efficiency of the numerical scheme.