{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:WIOGUG3ND2S4TCDJDL45OZO2OL","short_pith_number":"pith:WIOGUG3N","canonical_record":{"source":{"id":"1608.02712","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-08-09T07:57:24Z","cross_cats_sorted":[],"title_canon_sha256":"98676cbf0e3d93090737fdd7eb44d3fb257bee284f2e81bbab5083c534fa6ca6","abstract_canon_sha256":"baacd6a559d8eca687659577c3e4b9f9702ba63fd308cdb6dfd5ebbdb9928518"},"schema_version":"1.0"},"canonical_sha256":"b21c6a1b6d1ea5c988691af9d765da72c9750e644112b6f66759de5fc2a0d2df","source":{"kind":"arxiv","id":"1608.02712","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.02712","created_at":"2026-05-18T00:16:24Z"},{"alias_kind":"arxiv_version","alias_value":"1608.02712v1","created_at":"2026-05-18T00:16:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.02712","created_at":"2026-05-18T00:16:24Z"},{"alias_kind":"pith_short_12","alias_value":"WIOGUG3ND2S4","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"WIOGUG3ND2S4TCDJ","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"WIOGUG3N","created_at":"2026-05-18T12:30:48Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:WIOGUG3ND2S4TCDJDL45OZO2OL","target":"record","payload":{"canonical_record":{"source":{"id":"1608.02712","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-08-09T07:57:24Z","cross_cats_sorted":[],"title_canon_sha256":"98676cbf0e3d93090737fdd7eb44d3fb257bee284f2e81bbab5083c534fa6ca6","abstract_canon_sha256":"baacd6a559d8eca687659577c3e4b9f9702ba63fd308cdb6dfd5ebbdb9928518"},"schema_version":"1.0"},"canonical_sha256":"b21c6a1b6d1ea5c988691af9d765da72c9750e644112b6f66759de5fc2a0d2df","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:24.036214Z","signature_b64":"yUeZeVC/SDFu1wcIFHXGILwOOzKMoGnHZkzM3HgfICm321qRWdq55X/LZ9mgxAVO0qEX5TGa8CjThC7RPAc3DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b21c6a1b6d1ea5c988691af9d765da72c9750e644112b6f66759de5fc2a0d2df","last_reissued_at":"2026-05-18T00:16:24.035638Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:24.035638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1608.02712","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":"2026-05-18T00:16:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SA1HebDTwo4w+EOIxA0KuB6Scnx3x1JwgD2CCA+wpYov6fz7PuSy/OQzyj+7sucYzSsF5DmJcEulf84msxdHBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T21:55:18.103748Z"},"content_sha256":"1bc02c6605650504c483bbab6ab861b58ee1a5d019fe253568eac266295b6976","schema_version":"1.0","event_id":"sha256:1bc02c6605650504c483bbab6ab861b58ee1a5d019fe253568eac266295b6976"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:WIOGUG3ND2S4TCDJDL45OZO2OL","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lyapunov-like functions involving Lie brackets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Franco Rampazzo, Monica Motta","submitted_at":"2016-08-09T07:57:24Z","abstract_excerpt":"For a given closed target we embed the dissipative relation that defines a control Lyapunov function in a more general differential inequality involving Hamiltonians built from iterated Lie brackets. The solutions of the resulting extended relation, here called degree-k control Lyapunov functions (k>=1), turn out to be still sufficient for the system to be globally asymptotically controllable to the target. Furthermore, we work out some examples where no standard (i.e., degree-1) smooth control Lyapunov functions exist while a smooth degree-k control Lyapunov function does exist, for some k>1."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.02712","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":"2026-05-18T00:16:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"7nfdJS68n4MI77zw8C/cUsQHTpW/XptzyLDoSp6XrqsjJYUHJfagPX0pZU7TVX5Zv2Mb2eEv6lllF6xcmeCwBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-25T21:55:18.104104Z"},"content_sha256":"dae31e6610ecc46e7e887e7ee9fea1213fc9ef0e1337532847db4278342449f3","schema_version":"1.0","event_id":"sha256:dae31e6610ecc46e7e887e7ee9fea1213fc9ef0e1337532847db4278342449f3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WIOGUG3ND2S4TCDJDL45OZO2OL/bundle.json","state_url":"https://pith.science/pith/WIOGUG3ND2S4TCDJDL45OZO2OL/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WIOGUG3ND2S4TCDJDL45OZO2OL/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":"2026-06-25T21:55:18Z","links":{"resolver":"https://pith.science/pith/WIOGUG3ND2S4TCDJDL45OZO2OL","bundle":"https://pith.science/pith/WIOGUG3ND2S4TCDJDL45OZO2OL/bundle.json","state":"https://pith.science/pith/WIOGUG3ND2S4TCDJDL45OZO2OL/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WIOGUG3ND2S4TCDJDL45OZO2OL/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:WIOGUG3ND2S4TCDJDL45OZO2OL","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":"baacd6a559d8eca687659577c3e4b9f9702ba63fd308cdb6dfd5ebbdb9928518","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-08-09T07:57:24Z","title_canon_sha256":"98676cbf0e3d93090737fdd7eb44d3fb257bee284f2e81bbab5083c534fa6ca6"},"schema_version":"1.0","source":{"id":"1608.02712","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.02712","created_at":"2026-05-18T00:16:24Z"},{"alias_kind":"arxiv_version","alias_value":"1608.02712v1","created_at":"2026-05-18T00:16:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.02712","created_at":"2026-05-18T00:16:24Z"},{"alias_kind":"pith_short_12","alias_value":"WIOGUG3ND2S4","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"WIOGUG3ND2S4TCDJ","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"WIOGUG3N","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:dae31e6610ecc46e7e887e7ee9fea1213fc9ef0e1337532847db4278342449f3","target":"graph","created_at":"2026-05-18T00:16:24Z","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":"For a given closed target we embed the dissipative relation that defines a control Lyapunov function in a more general differential inequality involving Hamiltonians built from iterated Lie brackets. The solutions of the resulting extended relation, here called degree-k control Lyapunov functions (k>=1), turn out to be still sufficient for the system to be globally asymptotically controllable to the target. Furthermore, we work out some examples where no standard (i.e., degree-1) smooth control Lyapunov functions exist while a smooth degree-k control Lyapunov function does exist, for some k>1.","authors_text":"Franco Rampazzo, Monica Motta","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-08-09T07:57:24Z","title":"Lyapunov-like functions involving Lie brackets"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.02712","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":"sha256:1bc02c6605650504c483bbab6ab861b58ee1a5d019fe253568eac266295b6976","target":"record","created_at":"2026-05-18T00:16:24Z","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":"baacd6a559d8eca687659577c3e4b9f9702ba63fd308cdb6dfd5ebbdb9928518","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2016-08-09T07:57:24Z","title_canon_sha256":"98676cbf0e3d93090737fdd7eb44d3fb257bee284f2e81bbab5083c534fa6ca6"},"schema_version":"1.0","source":{"id":"1608.02712","kind":"arxiv","version":1}},"canonical_sha256":"b21c6a1b6d1ea5c988691af9d765da72c9750e644112b6f66759de5fc2a0d2df","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b21c6a1b6d1ea5c988691af9d765da72c9750e644112b6f66759de5fc2a0d2df","first_computed_at":"2026-05-18T00:16:24.035638Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:16:24.035638Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yUeZeVC/SDFu1wcIFHXGILwOOzKMoGnHZkzM3HgfICm321qRWdq55X/LZ9mgxAVO0qEX5TGa8CjThC7RPAc3DA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:16:24.036214Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.02712","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1bc02c6605650504c483bbab6ab861b58ee1a5d019fe253568eac266295b6976","sha256:dae31e6610ecc46e7e887e7ee9fea1213fc9ef0e1337532847db4278342449f3"],"state_sha256":"b6e5debbdf0ac6f3fc7c6acc67ed13daa75015db647dfe7fefdc0b79d2a13ec5"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xyCCcByCH3wqYKSr8mm6DisVZ1S5QlGtcYtiCkBDXgbweD8UNMtjNZ0GtWQ/W5jsFRaSl1hUSPHXSFtyoeOaAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-25T21:55:18.106049Z","bundle_sha256":"601aef878495bf558e88ddbcd2fb5382fb08769f2aa862fc8074245e18f6d45c"}}