# Efficient and More Accurate Representation of Solution Trajectories in Numerical Optimal Control

## TL;DR

This paper introduces a novel trajectory representation method for optimal control that minimizes residual errors across the entire trajectory, resulting in higher accuracy and computational efficiency compared to traditional interpolation schemes.

## Contribution

It proposes a new residual minimization approach that improves solution accuracy and reduces mesh resolution requirements in numerical optimal control.

## Key findings

- Higher accuracy in trajectory solutions with fewer collocation points
- Reduced computational complexity due to smaller problem dimensions
- Enhanced robustness of solutions in optimal control applications

## Abstract

We show via examples that, when solving optimal control problems, representing the optimal state and input trajectory directly using interpolation schemes may not be the best choice. Due to the lack of considerations for solution trajectories in-between collocation points, large errors may occur, posing risks if this solution is to be applied. A novel solution representation method is proposed, capable of yielding a solution of much higher accuracy for the same discretization mesh. This is achieved by minimizing the integral of the residual error for the overall trajectory, instead of forcing the errors to be zero only at collocation points. In this way, the requirement for mesh resolution can be significantly reduced, leaving the problem dimensions relatively small. This particular formulation also avoids some of the drawbacks found in the earlier work of integrated residual minimization, leading to more efficient computations.