### Abstract

We consider an optimal control problem defined on a Lie group whose associated Hamiltonian function is left-invariant under the action of a subgroup of the Lie group. Necessary conditions for optimality are derived using Lie-Poisson reduction for semidirect products, which allows us to study the Hamiltonian system in a space of lower dimension. Our main contribution is a reduced sufficient condition for optimality that relies on the nonexistence of conjugate points. We derive coordinate formulae for computing conjugate points in the reduced Hamiltonian system, and we relate these conjugate points to local optimality in the original optimal control problem. These conditions are applied to an optimal control problem that can be used to model either a kinematic airplane or a Kirchhoff elastic rod in a gravitational field.

Original language | English (US) |
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Article number | 7040306 |

Pages (from-to) | 5855-5861 |

Number of pages | 7 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 2015-February |

Issue number | February |

DOIs | |

State | Published - Jan 1 2014 |

Event | 2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States Duration: Dec 15 2014 → Dec 17 2014 |

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### ASJC Scopus subject areas

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

### Cite this

*Proceedings of the IEEE Conference on Decision and Control*,

*2015-February*(February), 5855-5861. [7040306]. https://doi.org/10.1109/CDC.2014.7040306

**Geometric optimal control for symmetry breaking cost functions.** / Borum, Andy D.; Bretl, Timothy.

Research output: Contribution to journal › Conference article

*Proceedings of the IEEE Conference on Decision and Control*, vol. 2015-February, no. February, 7040306, pp. 5855-5861. https://doi.org/10.1109/CDC.2014.7040306

}

TY - JOUR

T1 - Geometric optimal control for symmetry breaking cost functions

AU - Borum, Andy D.

AU - Bretl, Timothy

PY - 2014/1/1

Y1 - 2014/1/1

N2 - We consider an optimal control problem defined on a Lie group whose associated Hamiltonian function is left-invariant under the action of a subgroup of the Lie group. Necessary conditions for optimality are derived using Lie-Poisson reduction for semidirect products, which allows us to study the Hamiltonian system in a space of lower dimension. Our main contribution is a reduced sufficient condition for optimality that relies on the nonexistence of conjugate points. We derive coordinate formulae for computing conjugate points in the reduced Hamiltonian system, and we relate these conjugate points to local optimality in the original optimal control problem. These conditions are applied to an optimal control problem that can be used to model either a kinematic airplane or a Kirchhoff elastic rod in a gravitational field.

AB - We consider an optimal control problem defined on a Lie group whose associated Hamiltonian function is left-invariant under the action of a subgroup of the Lie group. Necessary conditions for optimality are derived using Lie-Poisson reduction for semidirect products, which allows us to study the Hamiltonian system in a space of lower dimension. Our main contribution is a reduced sufficient condition for optimality that relies on the nonexistence of conjugate points. We derive coordinate formulae for computing conjugate points in the reduced Hamiltonian system, and we relate these conjugate points to local optimality in the original optimal control problem. These conditions are applied to an optimal control problem that can be used to model either a kinematic airplane or a Kirchhoff elastic rod in a gravitational field.

UR - http://www.scopus.com/inward/record.url?scp=84988290510&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84988290510&partnerID=8YFLogxK

U2 - 10.1109/CDC.2014.7040306

DO - 10.1109/CDC.2014.7040306

M3 - Conference article

AN - SCOPUS:84988290510

VL - 2015-February

SP - 5855

EP - 5861

JO - Proceedings of the IEEE Conference on Decision and Control

JF - Proceedings of the IEEE Conference on Decision and Control

SN - 0191-2216

IS - February

M1 - 7040306

ER -