An Integral Penalty-Barrier Direct Transcription Method for Optimal Control

TL;DR

The paper introduces the penalty-barrier finite element method (PBF), a convergent direct transcription approach for optimal control that overcomes limitations of collocation methods, especially in cases with singular arcs.

Contribution

The paper presents PBF, a novel direct transcription method that guarantees convergence under weak assumptions and avoids collocation-related issues in optimal control problems.

Findings

PBF converges where collocation methods fail.

PBF has similar computational cost and sparsity as collocation methods.

PBF allows flexible trade-offs between cost, optimality, and constraint violations.

Abstract

Some direct transcription methods can fail to converge, e.g. when there are singular arcs. We recently introduced a convergent direct transcription method for optimal control problems, called the penalty-barrier finite element method (PBF). PBF converges under very weak assumptions on the problem instance. PBF avoids the ringing between collocation points, for example, by avoiding collocation entirely. Instead, equality path constraint residuals are forced to zero everywhere by an integral quadratic penalty term. We highlight conceptual differences between collocation- and penalty-type direct transcription methods. Theoretical convergence results for both types of methods are reviewed and compared. Formulas for implementing PBF are presented, with details on the formulation as a nonlinear program (NLP), sparsity and solution. Numerical experiments compare PBF against several collocation methods with regard to robustness, accuracy, sparsity and computational cost. We show that the computational cost, sparsity and construction of the NLP functions are roughly the same as for orthogonal collocation methods of the same degree and mesh. As an advantage, PBF converges in cases where collocation methods fail. PBF also allows one to trade off computational cost, optimality and violation of differential and other equality equations against each other.