{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:S354SU3R3E7DGIQAAZZ75RNCXE","short_pith_number":"pith:S354SU3R","canonical_record":{"source":{"id":"0901.3577","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-01-22T23:15:40Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"24f04ef304b0c1cf4f4df7846c57eff1849c068889d5eb137f34ae3b889e646a","abstract_canon_sha256":"77c63ef56c3ac6e936e81d9bacb3e53df086e3d146f741e8121ff0c68d930483"},"schema_version":"1.0"},"canonical_sha256":"96fbc95371d93e3322000673fec5a2b924dad8642fd7651218e05e40ba0ac273","source":{"kind":"arxiv","id":"0901.3577","version":6},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.3577","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"arxiv_version","alias_value":"0901.3577v6","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.3577","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"pith_short_12","alias_value":"S354SU3R3E7D","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"S354SU3R3E7DGIQA","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"S354SU3R","created_at":"2026-05-18T12:26:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:S354SU3R3E7DGIQAAZZ75RNCXE","target":"record","payload":{"canonical_record":{"source":{"id":"0901.3577","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-01-22T23:15:40Z","cross_cats_sorted":["math.OC"],"title_canon_sha256":"24f04ef304b0c1cf4f4df7846c57eff1849c068889d5eb137f34ae3b889e646a","abstract_canon_sha256":"77c63ef56c3ac6e936e81d9bacb3e53df086e3d146f741e8121ff0c68d930483"},"schema_version":"1.0"},"canonical_sha256":"96fbc95371d93e3322000673fec5a2b924dad8642fd7651218e05e40ba0ac273","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:21:20.940115Z","signature_b64":"CnoQOiZewOf1/Q2rylTdddVle1xbDnmIT9os43NsSOGDSG63dfndRr5WwzdQd56Q0n3V1OYwfaJmKQKwAt90Dg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"96fbc95371d93e3322000673fec5a2b924dad8642fd7651218e05e40ba0ac273","last_reissued_at":"2026-05-18T03:21:20.939717Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:21:20.939717Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0901.3577","source_version":6,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:21:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zTG8MnpZsK1Pf99TOuh5JaC45VIgxKYtYnkiDLDAjfph0/7vIgpCkdvNUE3J5JD7hN/8cWH1w9lK6ZYJFkkoBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T03:23:15.023608Z"},"content_sha256":"7cf23794027472f137a80dafb6926366240846cefc786b11ec6d10eda4b5c273","schema_version":"1.0","event_id":"sha256:7cf23794027472f137a80dafb6926366240846cefc786b11ec6d10eda4b5c273"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:S354SU3R3E7DGIQAAZZ75RNCXE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lyapunov-like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"math.DS","authors_text":"A. Gorban, E. Steur, H. Nijmeijer, I. Tyukin","submitted_at":"2009-01-22T23:15:40Z","abstract_excerpt":"We provide Lyapunov-like characterizations of boundedness and convergence of non-trivial solutions for a class of systems with unstable invariant sets. Examples of systems to which the results may apply include interconnections of stable subsystems with one-dimensional unstable dynamics or critically stable dynamics. Systems of this type arise in problems of nonlinear output regulation, parameter estimation and adaptive control.\n  In addition to providing boundedness and convergence criteria the results allow to derive domains of initial conditions corresponding to solutions leaving a given ne"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.3577","kind":"arxiv","version":6},"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":"2026-05-18T03:21:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VeOBkJCDTCMsKyFuC+OTqRbHG84SFqZcBZ5BRYzeGMaey5sT7CYCoxM/Srj6LAoujw0Wy6luQaSQyeCzoR7ZBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-09T03:23:15.024334Z"},"content_sha256":"1f357155b33aad141f487954ac18b5c347bedc684c0c763b5c1b5865be4a4f63","schema_version":"1.0","event_id":"sha256:1f357155b33aad141f487954ac18b5c347bedc684c0c763b5c1b5865be4a4f63"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S354SU3R3E7DGIQAAZZ75RNCXE/bundle.json","state_url":"https://pith.science/pith/S354SU3R3E7DGIQAAZZ75RNCXE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S354SU3R3E7DGIQAAZZ75RNCXE/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":"2026-06-09T03:23:15Z","links":{"resolver":"https://pith.science/pith/S354SU3R3E7DGIQAAZZ75RNCXE","bundle":"https://pith.science/pith/S354SU3R3E7DGIQAAZZ75RNCXE/bundle.json","state":"https://pith.science/pith/S354SU3R3E7DGIQAAZZ75RNCXE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S354SU3R3E7DGIQAAZZ75RNCXE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:S354SU3R3E7DGIQAAZZ75RNCXE","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":"77c63ef56c3ac6e936e81d9bacb3e53df086e3d146f741e8121ff0c68d930483","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-01-22T23:15:40Z","title_canon_sha256":"24f04ef304b0c1cf4f4df7846c57eff1849c068889d5eb137f34ae3b889e646a"},"schema_version":"1.0","source":{"id":"0901.3577","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0901.3577","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"arxiv_version","alias_value":"0901.3577v6","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0901.3577","created_at":"2026-05-18T03:21:20Z"},{"alias_kind":"pith_short_12","alias_value":"S354SU3R3E7D","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_16","alias_value":"S354SU3R3E7DGIQA","created_at":"2026-05-18T12:26:01Z"},{"alias_kind":"pith_short_8","alias_value":"S354SU3R","created_at":"2026-05-18T12:26:01Z"}],"graph_snapshots":[{"event_id":"sha256:1f357155b33aad141f487954ac18b5c347bedc684c0c763b5c1b5865be4a4f63","target":"graph","created_at":"2026-05-18T03:21:20Z","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 provide Lyapunov-like characterizations of boundedness and convergence of non-trivial solutions for a class of systems with unstable invariant sets. Examples of systems to which the results may apply include interconnections of stable subsystems with one-dimensional unstable dynamics or critically stable dynamics. Systems of this type arise in problems of nonlinear output regulation, parameter estimation and adaptive control.\n  In addition to providing boundedness and convergence criteria the results allow to derive domains of initial conditions corresponding to solutions leaving a given ne","authors_text":"A. Gorban, E. Steur, H. Nijmeijer, I. Tyukin","cross_cats":["math.OC"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-01-22T23:15:40Z","title":"Lyapunov-like Conditions of Forward Invariance and Boundedness for a Class of Unstable Systems"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0901.3577","kind":"arxiv","version":6},"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":"sha256:7cf23794027472f137a80dafb6926366240846cefc786b11ec6d10eda4b5c273","target":"record","created_at":"2026-05-18T03:21:20Z","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":"77c63ef56c3ac6e936e81d9bacb3e53df086e3d146f741e8121ff0c68d930483","cross_cats_sorted":["math.OC"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2009-01-22T23:15:40Z","title_canon_sha256":"24f04ef304b0c1cf4f4df7846c57eff1849c068889d5eb137f34ae3b889e646a"},"schema_version":"1.0","source":{"id":"0901.3577","kind":"arxiv","version":6}},"canonical_sha256":"96fbc95371d93e3322000673fec5a2b924dad8642fd7651218e05e40ba0ac273","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"96fbc95371d93e3322000673fec5a2b924dad8642fd7651218e05e40ba0ac273","first_computed_at":"2026-05-18T03:21:20.939717Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:21:20.939717Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CnoQOiZewOf1/Q2rylTdddVle1xbDnmIT9os43NsSOGDSG63dfndRr5WwzdQd56Q0n3V1OYwfaJmKQKwAt90Dg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:21:20.940115Z","signed_message":"canonical_sha256_bytes"},"source_id":"0901.3577","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7cf23794027472f137a80dafb6926366240846cefc786b11ec6d10eda4b5c273","sha256:1f357155b33aad141f487954ac18b5c347bedc684c0c763b5c1b5865be4a4f63"],"state_sha256":"d17dd90c6dc25898ef23dcc57dab5bd11cf1fbf1d3ecac72eee24b0a0d0ba58a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"2d3Z1cYIAJ85CVLRGtMczLclD64Ehoppu8qsh1kXGpt8jUE1isdGy0GJ7MamtuNiocchgh5cUjQsy5KDwhvrCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-09T03:23:15.028784Z","bundle_sha256":"cf94e78ff43d8b99284e9c6f938bcc1c963bf7abfd15fcebeb4dc3659c46fce6"}}