Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems

Pages: 559 - 575, Volume 29, Issue 2, February 2011      doi:10.3934/dcds.2011.29.559

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Sofia O. Lopes - CMAT e Departamento de Matemática e Aplicações, Universidade do Minho, 4800-058 Guimarães, Portugal (email)
Fernando A. C. C. Fontes - ISR-Porto, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal (email)
Maria do Rosário de Pinho - ISR-Porto, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal (email)

Abstract: We address necessary conditions of optimality (NCO), in the form of a maximum principle, for optimal control problems with state constraints. In particular, we are interested in the NCO that are strengthened to avoid the degeneracy phenomenon that occurs when the trajectory hits the boundary of the state constraint. In the literature on this subject, we can distinguish two types of constraint qualifications (CQ) under which the strengthened NCO can be applied: CQ involving the optimal control and CQ not involving it. Each one of these types of CQ has its own merits. The CQs involving the optimal control are not so easy to verify, but, are typically applicable to problems with less regularity on the data. In this article, we provide conditions under which the type of CQ involving the optimal control can be reduced to the other type. In this way, we also provide nondegenerate NCO that are valid under a different set of hypotheses.

Keywords:  Optimal Control, State Constraints, Maximum Principle, Degeneracy Phenomenon, Constraint Qualifications.
Mathematics Subject Classification:  49K15, 49K30.

Received: September 2009;      Revised: April 2010;      Available Online: October 2010.