{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:IXBW6FD7TW3E32ALXO6CWRLI47","short_pith_number":"pith:IXBW6FD7","canonical_record":{"source":{"id":"1809.05308","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-14T08:57:43Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"d3f8f62a6dc1d46649c83ca999906e9c51c35ecab3f8115665980916418bcdc7","abstract_canon_sha256":"4cf59c9486e35bccd7ad7b975f05ffcd1b9a92e4347042f19c36f456c5e7a86f"},"schema_version":"1.0"},"canonical_sha256":"45c36f147f9db64de80bbbbc2b4568e7f1558ce4868a181c9210f6b4eda7282a","source":{"kind":"arxiv","id":"1809.05308","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.05308","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"arxiv_version","alias_value":"1809.05308v1","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.05308","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"pith_short_12","alias_value":"IXBW6FD7TW3E","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"IXBW6FD7TW3E32AL","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"IXBW6FD7","created_at":"2026-05-18T12:32:31Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:IXBW6FD7TW3E32ALXO6CWRLI47","target":"record","payload":{"canonical_record":{"source":{"id":"1809.05308","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-14T08:57:43Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"d3f8f62a6dc1d46649c83ca999906e9c51c35ecab3f8115665980916418bcdc7","abstract_canon_sha256":"4cf59c9486e35bccd7ad7b975f05ffcd1b9a92e4347042f19c36f456c5e7a86f"},"schema_version":"1.0"},"canonical_sha256":"45c36f147f9db64de80bbbbc2b4568e7f1558ce4868a181c9210f6b4eda7282a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:44.032072Z","signature_b64":"xDaZacybsmhMesObS79hUrh2lQQSISzg5Mtu8x0MI/eEk4Ql6zUtF2RCyJuQfV+0DKVPyLUd/SftkINFouOCCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"45c36f147f9db64de80bbbbc2b4568e7f1558ce4868a181c9210f6b4eda7282a","last_reissued_at":"2026-05-18T00:05:44.031540Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:44.031540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1809.05308","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":"2026-05-18T00:05:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Np5iqjOoh+ml6fROrUm5By4NoT32tYzBz8Hoxrdulk5LFvLEjyDEviYq8rIXLWKLv6jgE3UBI60LQ6sEL4VDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T12:53:29.852341Z"},"content_sha256":"cc96e711d2a20136cd4255574c7cd020470d6b60d89e83b1facbed22132ce40e","schema_version":"1.0","event_id":"sha256:cc96e711d2a20136cd4255574c7cd020470d6b60d89e83b1facbed22132ce40e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:IXBW6FD7TW3E32ALXO6CWRLI47","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Stochastic LQ and Associated Riccati equation of PDEs Driven by State-and Control-Dependent White Noise","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.OC","authors_text":"Shanjian Tang (School of Mathematical Sciences), Ying Hu (IRMAR)","submitted_at":"2018-09-14T08:57:43Z","abstract_excerpt":"The optimal stochastic control problem with a quadratic cost functional for linear partial differential equations (PDEs) driven by a state-and control-dependent white noise is formulated and studied. Both finite-and infinite-time horizons are considered. The multi-plicative white noise dynamics of the system give rise to a new phenomenon of singularity to the associated Riccati equation and even to the Lyapunov equation. Well-posedness of both Riccati equation and Lyapunov equation are obtained for the first time. The linear feedback coefficient of the optimal control turns out to be singular "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.05308","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":"2026-05-18T00:05:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"FfdzEW+cLOv4Tts5OiR0gepltvsIbNPbfT/PXrU3NYJg876EtHkBVHkN5dVECpMFWe19yb4p6Y4zOrZUlaeVBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-04T12:53:29.852699Z"},"content_sha256":"2466867ca36564861f1bb58af9b541bda56c155ab7355a34406994845cbad686","schema_version":"1.0","event_id":"sha256:2466867ca36564861f1bb58af9b541bda56c155ab7355a34406994845cbad686"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IXBW6FD7TW3E32ALXO6CWRLI47/bundle.json","state_url":"https://pith.science/pith/IXBW6FD7TW3E32ALXO6CWRLI47/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IXBW6FD7TW3E32ALXO6CWRLI47/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":"2026-07-04T12:53:29Z","links":{"resolver":"https://pith.science/pith/IXBW6FD7TW3E32ALXO6CWRLI47","bundle":"https://pith.science/pith/IXBW6FD7TW3E32ALXO6CWRLI47/bundle.json","state":"https://pith.science/pith/IXBW6FD7TW3E32ALXO6CWRLI47/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IXBW6FD7TW3E32ALXO6CWRLI47/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:IXBW6FD7TW3E32ALXO6CWRLI47","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":"4cf59c9486e35bccd7ad7b975f05ffcd1b9a92e4347042f19c36f456c5e7a86f","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-14T08:57:43Z","title_canon_sha256":"d3f8f62a6dc1d46649c83ca999906e9c51c35ecab3f8115665980916418bcdc7"},"schema_version":"1.0","source":{"id":"1809.05308","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1809.05308","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"arxiv_version","alias_value":"1809.05308v1","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.05308","created_at":"2026-05-18T00:05:44Z"},{"alias_kind":"pith_short_12","alias_value":"IXBW6FD7TW3E","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_16","alias_value":"IXBW6FD7TW3E32AL","created_at":"2026-05-18T12:32:31Z"},{"alias_kind":"pith_short_8","alias_value":"IXBW6FD7","created_at":"2026-05-18T12:32:31Z"}],"graph_snapshots":[{"event_id":"sha256:2466867ca36564861f1bb58af9b541bda56c155ab7355a34406994845cbad686","target":"graph","created_at":"2026-05-18T00:05:44Z","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":"The optimal stochastic control problem with a quadratic cost functional for linear partial differential equations (PDEs) driven by a state-and control-dependent white noise is formulated and studied. Both finite-and infinite-time horizons are considered. The multi-plicative white noise dynamics of the system give rise to a new phenomenon of singularity to the associated Riccati equation and even to the Lyapunov equation. Well-posedness of both Riccati equation and Lyapunov equation are obtained for the first time. The linear feedback coefficient of the optimal control turns out to be singular ","authors_text":"Shanjian Tang (School of Mathematical Sciences), Ying Hu (IRMAR)","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-14T08:57:43Z","title":"Stochastic LQ and Associated Riccati equation of PDEs Driven by State-and Control-Dependent White Noise"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.05308","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:cc96e711d2a20136cd4255574c7cd020470d6b60d89e83b1facbed22132ce40e","target":"record","created_at":"2026-05-18T00:05:44Z","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":"4cf59c9486e35bccd7ad7b975f05ffcd1b9a92e4347042f19c36f456c5e7a86f","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-09-14T08:57:43Z","title_canon_sha256":"d3f8f62a6dc1d46649c83ca999906e9c51c35ecab3f8115665980916418bcdc7"},"schema_version":"1.0","source":{"id":"1809.05308","kind":"arxiv","version":1}},"canonical_sha256":"45c36f147f9db64de80bbbbc2b4568e7f1558ce4868a181c9210f6b4eda7282a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45c36f147f9db64de80bbbbc2b4568e7f1558ce4868a181c9210f6b4eda7282a","first_computed_at":"2026-05-18T00:05:44.031540Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:05:44.031540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xDaZacybsmhMesObS79hUrh2lQQSISzg5Mtu8x0MI/eEk4Ql6zUtF2RCyJuQfV+0DKVPyLUd/SftkINFouOCCg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:05:44.032072Z","signed_message":"canonical_sha256_bytes"},"source_id":"1809.05308","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cc96e711d2a20136cd4255574c7cd020470d6b60d89e83b1facbed22132ce40e","sha256:2466867ca36564861f1bb58af9b541bda56c155ab7355a34406994845cbad686"],"state_sha256":"18be4ce6b2372bfa005203aa39d208b6e4cb2c2dd9a5ba1bce08e4c45c7a34cf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X1bCIltz4iBz5cuu0Fo8Otl1l6w944u9TjeaFZizpfloNmkdpw3fRBJNMoCnHN+EenxlTcyZ79u/nZdtTyrcAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-04T12:53:29.854629Z","bundle_sha256":"ba1cc1ca3dca942cf43062c6d53fde20a9ef0107ad074bf630480c017cb9207b"}}