Fast and accurate method for computing non-smooth solutions to constrained control problems

TL;DR

This paper presents a flexible mesh approach in an integrated residual method for solving constrained control problems with non-smooth solutions, achieving superlinear convergence and faster computation compared to fixed mesh schemes.

Contribution

Introducing a decision-variable mesh node placement in an integrated residual method to improve convergence and efficiency in non-smooth control problem solutions.

Findings

Flexible mesh locates discontinuities automatically

Achieves superlinear convergence

Reduces computational time

Abstract

Introducing flexibility in the time-discretisation mesh can improve convergence and computational time when solving differential equations numerically, particularly when the solutions are discontinuous, as commonly found in control problems with constraints. State-of-the-art methods use fixed mesh schemes, which cannot achieve superlinear convergence in the presence of non-smooth solutions. In this paper, we propose using a flexible mesh in an integrated residual method. The locations of the mesh nodes are introduced as decision variables, and constraints are added to set upper and lower bounds on the size of the mesh intervals. We compare our approach to a uniform fixed mesh on a real-world satellite reorientation example. This example demonstrates that the flexible mesh enables the solver to automatically locate the discontinuities in the solution, has superlinear convergence and faster solve times, while achieving the same accuracy as a fixed mesh.