### Abstract

In this paper we define a class of optimal control problems which we denote "embedded optimal control problems". These are not true optimal control problems since the control system is not locally controllable on the manifold on which it is defined. Despite this, they allow for a well defined associated optimal control problem which does not admit abnormal extremals. We apply Pontryagin's maximum principle to the embedded optimal control problem to derive the generating differential equations for the normal and abnormal extremals. We show that the normal extremal generating equations in a sense contain the extremal generating equations for the associated optimal control problem. We show that this is not the case for the abnormal extremal generating equations. This has applications to the study of the optimal control of systems constrained to a given submanifold of a configuration space, for example the sphere or hypersphere. We apply the theory to three examples in order to illustrate its applicability and to show how it relates to well known results.

Original language | English (US) |
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Title of host publication | 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011 |

Pages | 7311-7316 |

Number of pages | 6 |

DOIs | |

State | Published - Dec 1 2011 |

Externally published | Yes |

Event | 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011 - Orlando, FL, United States Duration: Dec 12 2011 → Dec 15 2011 |

### Publication series

Name | Proceedings of the IEEE Conference on Decision and Control |
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ISSN (Print) | 0191-2216 |

### Other

Other | 2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011 |
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Country | United States |

City | Orlando, FL |

Period | 12/12/11 → 12/15/11 |

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modeling and Simulation
- Control and Optimization

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## Cite this

*2011 50th IEEE Conference on Decision and Control and European Control Conference, CDC-ECC 2011*(pp. 7311-7316). [6160296] (Proceedings of the IEEE Conference on Decision and Control). https://doi.org/10.1109/CDC.2011.6160296