A Pseudospectral Approach to High Index DAE Optimal Control Problems

TL;DR

This paper demonstrates that pseudospectral methods can directly solve high index DAE optimal control problems without index reduction, challenging traditional beliefs about their difficulty and providing a new perspective on DAE solvability.

Contribution

It introduces a novel approach using pseudospectral methods to directly solve high index DAE optimal control problems, bypassing complex index reduction procedures.

Findings

Successfully solved an index-three DAE optimal control problem using PS methods.

Validated PS solutions with industry-standard verification practices.

Showed PS methods can handle high index DAEs directly, contrary to traditional assumptions.

Abstract

Historically, solving optimal control problems with high index differential algebraic equations (DAEs) has been considered extremely hard. Computational experience with Runge-Kutta (RK) methods confirms the difficulties. High index DAE problems occur quite naturally in many practical engineering applications. Over the last two decades, a vast number of real-world problems have been solved routinely using pseudospectral (PS) optimal control techniques. In view of this, we solve a "provably hard," index-three problem using the PS method implemented in DIDO, a state-of-the-art MATLAB optimal control toolbox. In contrast to RK-type solution techniques, no laborious index-reduction process was used to generate the PS solution. The PS solution is independently verified and validated using standard industry practices. It turns out that proper PS methods can indeed be used to "directly" solve high index DAE optimal control problems. In view of this, it is proposed that a new theory of difficulty for DAEs be put forth.