Geometric optimal control for symmetry breaking cost functions

Andy D. Borum, Timothy Bretl

Research output: Contribution to journalConference article

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 languageEnglish (US)
Article number7040306
Pages (from-to)5855-5861
Number of pages7
JournalProceedings of the IEEE Conference on Decision and Control
Volume2015-February
Issue numberFebruary
DOIs
StatePublished - Jan 1 2014
Event2014 53rd IEEE Annual Conference on Decision and Control, CDC 2014 - Los Angeles, United States
Duration: Dec 15 2014Dec 17 2014

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Conjugate points
Hamiltonians
Symmetry Breaking
Cost functions
Cost Function
Optimal Control Problem
Lie groups
Optimal Control
Hamiltonian Systems
Optimality
Local Optimality
Elastic Rods
Gravitational Field
Nonexistence
Kinematics
Siméon Denis Poisson
Aircraft
Subgroup
Necessary Conditions
Invariant

ASJC Scopus subject areas

  • Control and Systems Engineering
  • Modeling and Simulation
  • Control and Optimization

Cite this

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

In: Proceedings of the IEEE Conference on Decision and Control, Vol. 2015-February, No. February, 7040306, 01.01.2014, p. 5855-5861.

Research output: Contribution to journalConference article

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