{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:ES3XQJIA2RZ2IXRSGWZYFESDCX","short_pith_number":"pith:ES3XQJIA","canonical_record":{"source":{"id":"1111.1549","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-11-07T11:42:51Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"212993574b0caffb07b1fe0ecff9a345a92b4a961007db85e424147c3ca285b7","abstract_canon_sha256":"0664a977146f69958ce32173b81d8bd6d25d5f7a091c1db2ec9e70ee83e4dca2"},"schema_version":"1.0"},"canonical_sha256":"24b7782500d473a45e3235b382924315e6a9add4c8f5e463054f7f1bf346aea3","source":{"kind":"arxiv","id":"1111.1549","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.1549","created_at":"2026-05-18T04:09:25Z"},{"alias_kind":"arxiv_version","alias_value":"1111.1549v1","created_at":"2026-05-18T04:09:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.1549","created_at":"2026-05-18T04:09:25Z"},{"alias_kind":"pith_short_12","alias_value":"ES3XQJIA2RZ2","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"ES3XQJIA2RZ2IXRS","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"ES3XQJIA","created_at":"2026-05-18T12:26:28Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:ES3XQJIA2RZ2IXRSGWZYFESDCX","target":"record","payload":{"canonical_record":{"source":{"id":"1111.1549","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-11-07T11:42:51Z","cross_cats_sorted":["math.DG"],"title_canon_sha256":"212993574b0caffb07b1fe0ecff9a345a92b4a961007db85e424147c3ca285b7","abstract_canon_sha256":"0664a977146f69958ce32173b81d8bd6d25d5f7a091c1db2ec9e70ee83e4dca2"},"schema_version":"1.0"},"canonical_sha256":"24b7782500d473a45e3235b382924315e6a9add4c8f5e463054f7f1bf346aea3","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:09:25.835952Z","signature_b64":"0vuXw3/JE1U4Bj+sMGMeOevUA32SVwC5KInmsIc8tjGuOt3FQs3B1X8XNxlxA6BJSgeCuQuOtkxt+M0kQnNwAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24b7782500d473a45e3235b382924315e6a9add4c8f5e463054f7f1bf346aea3","last_reissued_at":"2026-05-18T04:09:25.835270Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:09:25.835270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.1549","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:09:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ObPSuAx5GBrFILZ4OqWGxOZsH6rhSvLhtrJyM2k2szGHdr4xrbJrnp0NyKnBuxZEybiGPhmwgshj/Z5reaf+Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T14:49:09.304616Z"},"content_sha256":"f5d272567db30abdc5f31b74ff8550b9fdf0eeaa1dcff28f1cef7aea4bbfe6af","schema_version":"1.0","event_id":"sha256:f5d272567db30abdc5f31b74ff8550b9fdf0eeaa1dcff28f1cef7aea4bbfe6af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:ES3XQJIA2RZ2IXRSGWZYFESDCX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Optimal Control Theory on almost-Lie Algebroids","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DG"],"primary_cat":"math.OC","authors_text":"Michal Jozwikowski","submitted_at":"2011-11-07T11:42:51Z","abstract_excerpt":"We extend the Pontryagin Maximum Principle (PMP) to the geometric setting of almost-Lie (AL) algebroids -- objects which generalize Lie algebroids. The result may be understood as a very general reduction scheme for optimal control problems (OCPs). It covers the standard PMP, as well as gives necessary optimality conditions for symmetric OCPs on Lie groups, principal bundles, and Lie groupoids. We do not assume the symmetry of boundary conditions. The ideas are based on a very general concept of homotopy of admissible paths on AL algebroids. Our framework works for OCPs with fixed-end-points a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1549","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:09:25Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U9QEcMz1ZJ1e0VLZ5RtrgHW3VFOWgBhbYkX87YE8gPC9WVioivAPmT+PqrqmS2gSner8cxSM2M4JuybACvkPCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T14:49:09.304956Z"},"content_sha256":"68ff7b8872ad9d21b9c6cfb79ce497c04212ae987065d8a30cfdb6db3cff81ee","schema_version":"1.0","event_id":"sha256:68ff7b8872ad9d21b9c6cfb79ce497c04212ae987065d8a30cfdb6db3cff81ee"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ES3XQJIA2RZ2IXRSGWZYFESDCX/bundle.json","state_url":"https://pith.science/pith/ES3XQJIA2RZ2IXRSGWZYFESDCX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ES3XQJIA2RZ2IXRSGWZYFESDCX/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-28T14:49:09Z","links":{"resolver":"https://pith.science/pith/ES3XQJIA2RZ2IXRSGWZYFESDCX","bundle":"https://pith.science/pith/ES3XQJIA2RZ2IXRSGWZYFESDCX/bundle.json","state":"https://pith.science/pith/ES3XQJIA2RZ2IXRSGWZYFESDCX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ES3XQJIA2RZ2IXRSGWZYFESDCX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:ES3XQJIA2RZ2IXRSGWZYFESDCX","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"0664a977146f69958ce32173b81d8bd6d25d5f7a091c1db2ec9e70ee83e4dca2","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-11-07T11:42:51Z","title_canon_sha256":"212993574b0caffb07b1fe0ecff9a345a92b4a961007db85e424147c3ca285b7"},"schema_version":"1.0","source":{"id":"1111.1549","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.1549","created_at":"2026-05-18T04:09:25Z"},{"alias_kind":"arxiv_version","alias_value":"1111.1549v1","created_at":"2026-05-18T04:09:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.1549","created_at":"2026-05-18T04:09:25Z"},{"alias_kind":"pith_short_12","alias_value":"ES3XQJIA2RZ2","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_16","alias_value":"ES3XQJIA2RZ2IXRS","created_at":"2026-05-18T12:26:28Z"},{"alias_kind":"pith_short_8","alias_value":"ES3XQJIA","created_at":"2026-05-18T12:26:28Z"}],"graph_snapshots":[{"event_id":"sha256:68ff7b8872ad9d21b9c6cfb79ce497c04212ae987065d8a30cfdb6db3cff81ee","target":"graph","created_at":"2026-05-18T04:09:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We extend the Pontryagin Maximum Principle (PMP) to the geometric setting of almost-Lie (AL) algebroids -- objects which generalize Lie algebroids. The result may be understood as a very general reduction scheme for optimal control problems (OCPs). It covers the standard PMP, as well as gives necessary optimality conditions for symmetric OCPs on Lie groups, principal bundles, and Lie groupoids. We do not assume the symmetry of boundary conditions. The ideas are based on a very general concept of homotopy of admissible paths on AL algebroids. Our framework works for OCPs with fixed-end-points a","authors_text":"Michal Jozwikowski","cross_cats":["math.DG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-11-07T11:42:51Z","title":"Optimal Control Theory on almost-Lie Algebroids"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.1549","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f5d272567db30abdc5f31b74ff8550b9fdf0eeaa1dcff28f1cef7aea4bbfe6af","target":"record","created_at":"2026-05-18T04:09:25Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"0664a977146f69958ce32173b81d8bd6d25d5f7a091c1db2ec9e70ee83e4dca2","cross_cats_sorted":["math.DG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2011-11-07T11:42:51Z","title_canon_sha256":"212993574b0caffb07b1fe0ecff9a345a92b4a961007db85e424147c3ca285b7"},"schema_version":"1.0","source":{"id":"1111.1549","kind":"arxiv","version":1}},"canonical_sha256":"24b7782500d473a45e3235b382924315e6a9add4c8f5e463054f7f1bf346aea3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"24b7782500d473a45e3235b382924315e6a9add4c8f5e463054f7f1bf346aea3","first_computed_at":"2026-05-18T04:09:25.835270Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:09:25.835270Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0vuXw3/JE1U4Bj+sMGMeOevUA32SVwC5KInmsIc8tjGuOt3FQs3B1X8XNxlxA6BJSgeCuQuOtkxt+M0kQnNwAA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:09:25.835952Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.1549","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f5d272567db30abdc5f31b74ff8550b9fdf0eeaa1dcff28f1cef7aea4bbfe6af","sha256:68ff7b8872ad9d21b9c6cfb79ce497c04212ae987065d8a30cfdb6db3cff81ee"],"state_sha256":"d897c7dbbe3dcd7ed0bf970269f260e5c0d63452af19f7e244367cfaed152db5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gY/u8qqssFIowt2+Qj8802qmYdIHAbTz+lx33/dfgrz5/T5vLQP/m6HxRgpx77A9to4Tp/nKimzDVD2SYsqXBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T14:49:09.306808Z","bundle_sha256":"68889ec698bf9a9cdb0e364885501834c38fb64c695468bb2d63549443cde7c3"}}