A minima tracking variant of semi-infinite programming for the treatment of path constraints within direct solution of optimal control problems

Abstract

We present an extension of the direct multiple shooting method for the solution of optimal control problems with path constraints. The extension allows for fully automatic and accurate treatment of path constraints in case of relatively coarse discretizations of the control while maintaining a prescribed resolution of the system dynamics. It is based on a reduction principle from classical theory of semi-infinite programming. On the basis of these theoretical foundations the minima tracking algorithm is designed. We theoretically and numerically compare this method with a sampling technique. The numerical test case is a well-known, difficult space shuttle reentry benchmark problem.

Keywords:

• optimal control
• path constraints
• semi-infinite
• local reduction
• direct multiple shooting

AMS Subject Classification :

• 49J15
• 65K05
• 78M50
• 90C34

Acknowledgements

The first author wishes to thank Matthias Heinkenschloss from Rice University, Houston, TX, USA, for his hospitality, and inspiring scientific discussions when research on this topic started. Valuable support by Moritz Diehl, who is now at OPTEC, K.U. Leuven, Belgium, is gratefully acknowledged. The authors thank the unknown reviewer for his/her comments that helped to improve this article.

The first author also gladly acknowledges support by Germany's Federal Ministry of Education and Research (BMBF) within the project ‘Nichtlineare prädiktive Regelung für kontinuierlich betriebene Prozesse der chemischen Verfahrenstechnik’ (FKZ: 03BONCHD) and by the German Research Foundation (DFG) within the International Research Training Group IGK 710 ‘Complex processes: Modelling, Simulation and Optimization’.