Optimal control problems on lie groups with symmetry breaking cost functions

In this thesis, we consider smooth optimal control systems that evolve on Lie groups. Pontryagin's maximum principle allows us to search for local solutions of the optimal control problem by studying an associated Hamiltonian dynamical system. When the associated Hamiltonian function possess symmetries, we can often study the Hamiltonian system in a vector space whose dimension is lower than the original system. We apply these symmetry reduction techniques to optimal control problems on Lie groups for which the associated Hamiltonian function is left-invariant under the action of a subgroup of the Lie group. Necessary conditions for optimality are derived by applying Lie-Poisson reduction for semidirect products, a previously developed method of symmetry group reduction in the field of geometric mechanics. Our main contribution is a reduced sufficient condition for optimality that relies on the nonexistence of conjugate points. Coordinate formulae are derived 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 optimality conditions are then applied to an example optimal control problem on the Lie group SE(3) that exhibits symmetries with respect to SE(2), a subgroup of SE(3). This optimal control problem can be used to model either a kinematic airplane, i.e. a rigid body moving at a constant speed whose angular velocities can be controlled, or a Kirchhoff elastic rod in a gravitational field.

Name thesis:degree_name

M.S.

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Aerospace Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• Borum, Andy

Contributors dc:contributor

• Bretl, Timothy W.

• geometric optimal control
• symmetry reduction
• conjugate points

Statement dc:rights

• Copyright 2014 Andy Borum

Language dc:language

en

Handle dc:identifier

http://hdl.handle.net/2142/72845

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/72845