{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/72845"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/72845","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Optimal control problems on lie groups with symmetry breaking cost functions","abstract":"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.","abstract_html":"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.","abstract_has_math":false,"creators":["Borum, Andy"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"M.S.","degree_level":"Thesis","degree_discipline":"Aerospace Engineering","degree_department":null,"school":null,"contributors":["Bretl, Timothy W."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2015,"date_issued":"2015-01-21T19:48:47Z","date_published":"2015-01-21T19:48:47Z","updated_at":"2026-07-22T22:26:07Z","subjects":["geometric optimal control","symmetry reduction","conjugate points"],"languages":["en"],"rights":["Copyright 2014 Andy Borum"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/72845","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Bretl, Timothy W."]},{"key":"dc:creator","label":"Author","values":["Borum, Andy"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2015-01-21T19:48:47Z","2014-12","2015-01-21"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Aerospace Engineering"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["M.S."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["geometric optimal control","symmetry reduction","conjugate points"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2014 Andy Borum"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/72845"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2014-12-11T15:16:28Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Borum_Andy.pdf: 278226 bytes, checksum: 906cef61cbc484bd7b35e4eda220f6cf (MD5)","Made available in DSpace on 2015-01-21T19:48:47Z (GMT). No. of bitstreams: 1 Andy_Borum.pdf: 278226 bytes, checksum: 906cef61cbc484bd7b35e4eda220f6cf (MD5)"]},{"key":"dc:title","label":"Title","values":["Optimal control problems on lie groups with symmetry breaking cost functions"]}]}],"canonical_facts":{"dc:contributor":["Bretl, Timothy W."],"dc:creator":["Borum, Andy"],"dc:date":["2015-01-21T19:48:47Z","2014-12","2015-01-21"],"dc:description":["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.","Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2014-12-11T15:16:28Z Item was in collections: University of Illinois Theses & Dissertations (ID: 1) No. of bitstreams: 1 Borum_Andy.pdf: 278226 bytes, checksum: 906cef61cbc484bd7b35e4eda220f6cf (MD5)","Made available in DSpace on 2015-01-21T19:48:47Z (GMT). No. of bitstreams: 1 Andy_Borum.pdf: 278226 bytes, checksum: 906cef61cbc484bd7b35e4eda220f6cf (MD5)"],"dc:identifier":["http://hdl.handle.net/2142/72845"],"dc:language":["en"],"dc:rights":["Copyright 2014 Andy Borum"],"dc:subject":["geometric optimal control","symmetry reduction","conjugate points"],"dc:title":["Optimal control problems on lie groups with symmetry breaking cost functions"],"dc:type":["text"],"thesis:degree_discipline":["Aerospace Engineering"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["M.S."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:26:07Z"}