Advantages of nonlinear-programming-based methodologies for inequality path-constrained optimal control problems - A numerical study

摘要

We address some advantages of nonlinear programming (NLP)-based methods for inequality path-constrained optimal control problems. The analysis is carried out on the differential algebraic equation (DAE)-constrained optimization problem presented in [J. T. Betts and S. L. Campbell, Discretize Then Optimize, Technical report M&CT-TECH-03-01, The Boeing Company, Chicago, IL, 2003], which relates to a one-dimensional transient heat conduction problem. This optimal control problem possesses the essential characteristics of a typical inequality path-constrained optimal control problem. In a DAE-constrained optimization setting, the index of the path constraint can be arbitrarily increased, and this facilitates the study of the effect of the index of a path constraint on the solution obtained by using an NLP-based methodology. We show that the direct transcription approach leads to a problem whose limiting behavior does not satisfy the linear independence constraint qualification but satisfies the weaker Mangasarian-Fromowitz constraint qualification. The direct transcription approach leads to a convex quadratic programming problem with the result that the control obtained is the unique global minimizer. We also contrast this with the classical indirect approach where we show that it is difficult to numerically integrate the ODEs over the constrained arc because of stability and error control reasons. These observations explain some of the results of Betts and Campbell. This supports the fact that NLP-based methodologies have additional flexibility with respect to constraint qualifications, and this can be put to use in the case of inequality path-constrained optimal control problems to obtain well-defined solutions.