An adaptive mesh refinement method considering control errors for pseudospectral discretization

Abstract

This paper presents an adaptive mesh refinement method that considers control errors for solving pseudospectral optimal control problems. Firstly, a method for estimating errors in both states and controls is presented. Based on the estimation results, an adaptive mesh refinement method is subsequently devised. This method increases and reduces the number of collocation points in accordance with a theoretical convergence rate that incorporates both state and control errors. Furthermore, in addition to dividing intervals resulting from a large number of collocation points, new intervals are also generated when control errors exceed tolerance. As a result, the mesh density near the point with the largest control error is effectively increased, thereby improving the discretization accuracy. The effectiveness of the method is illustrated through three numerical examples, and its performance is evaluated in comparison to other adaptive mesh refinement methods. The numerical results demonstrate that the proposed method exhibits superior performance in terms of capturing the nonsmooth and discontinuous changes and achieving an accurate solution, while requiring fewer iterations.