{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:CINLG6WUG2XREG4MSI2JYESSAD","short_pith_number":"pith:CINLG6WU","canonical_record":{"source":{"id":"1501.01503","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-01-07T14:32:29Z","cross_cats_sorted":[],"title_canon_sha256":"7bbf0826b6b5c6cbc978565455f4837bf1f134bc4c394aa1328946026067e473","abstract_canon_sha256":"c8e11e9de20ebd95790a028785d4a34f997a6af9d76ecde7228202445c18ecbf"},"schema_version":"1.0"},"canonical_sha256":"121ab37ad436af121b8c92349c125200e77ebf8feaed7a745a5a8f7828432d38","source":{"kind":"arxiv","id":"1501.01503","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.01503","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"arxiv_version","alias_value":"1501.01503v2","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.01503","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"pith_short_12","alias_value":"CINLG6WUG2XR","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"CINLG6WUG2XREG4M","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"CINLG6WU","created_at":"2026-05-18T12:29:14Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:CINLG6WUG2XREG4MSI2JYESSAD","target":"record","payload":{"canonical_record":{"source":{"id":"1501.01503","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-01-07T14:32:29Z","cross_cats_sorted":[],"title_canon_sha256":"7bbf0826b6b5c6cbc978565455f4837bf1f134bc4c394aa1328946026067e473","abstract_canon_sha256":"c8e11e9de20ebd95790a028785d4a34f997a6af9d76ecde7228202445c18ecbf"},"schema_version":"1.0"},"canonical_sha256":"121ab37ad436af121b8c92349c125200e77ebf8feaed7a745a5a8f7828432d38","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:31:53.230647Z","signature_b64":"XtL6+P7SBJl86appksu5FFJLq6xZiCNGbJrBWJbQCeswg4cjuoln/+Qol1yCN9k4ZCPSY6b+qGg5Df69f/auCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"121ab37ad436af121b8c92349c125200e77ebf8feaed7a745a5a8f7828432d38","last_reissued_at":"2026-05-18T01:31:53.230199Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:31:53.230199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1501.01503","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:31:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cnL5v6xBFMBPbh8ASAedoczcF9U5bmlE+1J9JxnIasWbAO1IFE5GVUaaUb3v20NiHB6NsBbphyXzYSwqFr8QCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T05:56:36.615040Z"},"content_sha256":"211ea43720cc317b0bf75ee9dc339bfef7300226a5c3ff064fa57372c450bf16","schema_version":"1.0","event_id":"sha256:211ea43720cc317b0bf75ee9dc339bfef7300226a5c3ff064fa57372c450bf16"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:CINLG6WUG2XREG4MSI2JYESSAD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Conjugate times and regularity of the minimum time function with differential inclusions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Piermarco Cannarsa, Teresa Scarinci","submitted_at":"2015-01-07T14:32:29Z","abstract_excerpt":"This paper studies the regularity of the minimum time function, $T(\\cdot)$, for a control system with a general closed target, taking the state equation in the form of a differential inclusion. Our first result is a sensitivity relation which guarantees the propagation of the proximal subdifferential of $T$ along any optimal trajectory. Then, we obtain the local $C^2$ regularity of the minimum time function along optimal trajectories by using such a relation to exclude the presence of conjugate times."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01503","kind":"arxiv","version":2},"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":"2026-05-18T01:31:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YRw+Cb12I1ELUDROGIbN+GN9GaIzVz6fgrgjSwO/SYGeY+eGi/rL4USLyiL10mv/3guJvAZLe+ZI15zHuYY7Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T05:56:36.615393Z"},"content_sha256":"981f20d8f991950293f170b8c4144df1488e996e4cb3a4524964f8c41fcd57cd","schema_version":"1.0","event_id":"sha256:981f20d8f991950293f170b8c4144df1488e996e4cb3a4524964f8c41fcd57cd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CINLG6WUG2XREG4MSI2JYESSAD/bundle.json","state_url":"https://pith.science/pith/CINLG6WUG2XREG4MSI2JYESSAD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CINLG6WUG2XREG4MSI2JYESSAD/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":"2026-06-24T05:56:36Z","links":{"resolver":"https://pith.science/pith/CINLG6WUG2XREG4MSI2JYESSAD","bundle":"https://pith.science/pith/CINLG6WUG2XREG4MSI2JYESSAD/bundle.json","state":"https://pith.science/pith/CINLG6WUG2XREG4MSI2JYESSAD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CINLG6WUG2XREG4MSI2JYESSAD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:CINLG6WUG2XREG4MSI2JYESSAD","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":"c8e11e9de20ebd95790a028785d4a34f997a6af9d76ecde7228202445c18ecbf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-01-07T14:32:29Z","title_canon_sha256":"7bbf0826b6b5c6cbc978565455f4837bf1f134bc4c394aa1328946026067e473"},"schema_version":"1.0","source":{"id":"1501.01503","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.01503","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"arxiv_version","alias_value":"1501.01503v2","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.01503","created_at":"2026-05-18T01:31:53Z"},{"alias_kind":"pith_short_12","alias_value":"CINLG6WUG2XR","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_16","alias_value":"CINLG6WUG2XREG4M","created_at":"2026-05-18T12:29:14Z"},{"alias_kind":"pith_short_8","alias_value":"CINLG6WU","created_at":"2026-05-18T12:29:14Z"}],"graph_snapshots":[{"event_id":"sha256:981f20d8f991950293f170b8c4144df1488e996e4cb3a4524964f8c41fcd57cd","target":"graph","created_at":"2026-05-18T01:31:53Z","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":"This paper studies the regularity of the minimum time function, $T(\\cdot)$, for a control system with a general closed target, taking the state equation in the form of a differential inclusion. Our first result is a sensitivity relation which guarantees the propagation of the proximal subdifferential of $T$ along any optimal trajectory. Then, we obtain the local $C^2$ regularity of the minimum time function along optimal trajectories by using such a relation to exclude the presence of conjugate times.","authors_text":"Piermarco Cannarsa, Teresa Scarinci","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-01-07T14:32:29Z","title":"Conjugate times and regularity of the minimum time function with differential inclusions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01503","kind":"arxiv","version":2},"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":"sha256:211ea43720cc317b0bf75ee9dc339bfef7300226a5c3ff064fa57372c450bf16","target":"record","created_at":"2026-05-18T01:31:53Z","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":"c8e11e9de20ebd95790a028785d4a34f997a6af9d76ecde7228202445c18ecbf","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2015-01-07T14:32:29Z","title_canon_sha256":"7bbf0826b6b5c6cbc978565455f4837bf1f134bc4c394aa1328946026067e473"},"schema_version":"1.0","source":{"id":"1501.01503","kind":"arxiv","version":2}},"canonical_sha256":"121ab37ad436af121b8c92349c125200e77ebf8feaed7a745a5a8f7828432d38","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"121ab37ad436af121b8c92349c125200e77ebf8feaed7a745a5a8f7828432d38","first_computed_at":"2026-05-18T01:31:53.230199Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:31:53.230199Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XtL6+P7SBJl86appksu5FFJLq6xZiCNGbJrBWJbQCeswg4cjuoln/+Qol1yCN9k4ZCPSY6b+qGg5Df69f/auCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:31:53.230647Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.01503","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:211ea43720cc317b0bf75ee9dc339bfef7300226a5c3ff064fa57372c450bf16","sha256:981f20d8f991950293f170b8c4144df1488e996e4cb3a4524964f8c41fcd57cd"],"state_sha256":"edd37008abe36d04c2c9ddd5636d590d0eada1df4d19d28909ca801a2a93272e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"3pXmVtbj3lUZfi8KTAiIhqVcwkGhGJJcl5BsDPnprjzEwtGuVibrX5DaRVh/e6gWzYq71aj4gMiUYm3PJIY+AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T05:56:36.617363Z","bundle_sha256":"5a5c4b0e5e8c152999bccfa5b4df8c4678a809f18a21f1023eee08c823efe469"}}