High-Order Adaptive Gegenbauer Integral Spectral Element Method for Solving Nonlinear Optimal Control Problems

TL;DR

This paper introduces an adaptive spectral element method using Gegenbauer polynomials for efficiently solving nonlinear optimal control problems, especially handling solution nonsmoothness through local adaptivity.

Contribution

It develops a novel adaptive algorithm combining Gegenbauer collocation and local mesh refinement to improve accuracy and efficiency in solving complex optimal control problems.

Findings

Achieves high accuracy with fewer discretization points.

Effectively brackets discontinuities and nonsmooth points.

Demonstrates superior performance on benchmark problems.

Abstract

In this work, we propose an adaptive spectral element algorithm for solving nonlinear optimal control problems. The method employs orthogonal collocation at the shifted Gegenbauer-Gauss points combined with very accurate and stable numerical quadratures to fully discretize the multiple-phase integral form of the optimal control problem. The proposed algorithm relies on exploiting the underlying smoothness properties of the solutions for computing approximate solutions efficiently. In particular, the method brackets discontinuities and "points of nonsmoothness" through a novel local adaptive algorithm, which achieves a desired accuracy on the discrete dynamical system equations by adjusting both the mesh size and the degree of the approximating polynomials. A rigorous error analysis of the developed numerical quadratures is presented. Finally, the efficiency of the proposed method is demonstrated on three test examples from the open literature.