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If \\(C_{\\mathrm H}(T,L)\\) denotes the optimal \\(L^2\\) null-control cost for initial data in \\(H^{-1}(0,L)\\), we prove that \\[\n  C_{\\mathrm H}(T,L)\n  =\n  \\exp\\left(\\frac{\\kappa_*L^2+o(1)}{T}\\right),\n  \\qquad\n  \\kappa_*\n  =\n  \\frac{\\Gamma(\\frac14)^4}{8\\pi^3}\n  \\simeq\n  0.696601964842838,\n  \\qquad T\\to0^+. \\] The constant \\(\\kappa_*\\) coincides with the upper-bound constant obtained by Dard\\'e and Ervedoza (2019, ANPDE), which was expressed there through a c"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2608.08041","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2026-08-08T09:57:47Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"f3306c43bd8b24a6e1e5f29943e70e460e18bf16a497bbb7ecf09ec44757888d","abstract_canon_sha256":"90566b40dc16c9281c56dddd2010b306d0b20f44f2598560d5ebd56033d77cad"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-11T01:21:29.366724Z","signature_b64":"b0dZochWlg3A7Mx3m+C9PTTRiy7RZ+5G777Eu0J+lMJrQ2T5lzwUxdfDXTPhi3Z9Mlv4rm3wDene9Jf4Hd1tBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ab1ce07755844591feb2f5456f0ccdeb7a78f67983bd043e5eeec28f9ab218c2","last_reissued_at":"2026-08-11T01:21:29.364314Z","signature_status":"signed_v1","first_computed_at":"2026-08-11T01:21:29.364314Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Optimal cost of fast boundary controls for the one-dimensional heat equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.OC","authors_text":"Pierre Lissy","submitted_at":"2026-08-08T09:57:47Z","abstract_excerpt":"We consider the heat equation on $(0,L)$ with homogeneous Dirichlet condition at one endpoint and a Dirichlet boundary control at the other. If \\(C_{\\mathrm H}(T,L)\\) denotes the optimal \\(L^2\\) null-control cost for initial data in \\(H^{-1}(0,L)\\), we prove that \\[\n  C_{\\mathrm H}(T,L)\n  =\n  \\exp\\left(\\frac{\\kappa_*L^2+o(1)}{T}\\right),\n  \\qquad\n  \\kappa_*\n  =\n  \\frac{\\Gamma(\\frac14)^4}{8\\pi^3}\n  \\simeq\n  0.696601964842838,\n  \\qquad T\\to0^+. \\] The constant \\(\\kappa_*\\) coincides with the upper-bound constant obtained by Dard\\'e and Ervedoza (2019, ANPDE), which was expressed there through a c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.08041","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.08041/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2608.08041","created_at":"2026-08-11T01:21:29.365108+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.08041v1","created_at":"2026-08-11T01:21:29.365108+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.08041","created_at":"2026-08-11T01:21:29.365108+00:00"},{"alias_kind":"pith_short_12","alias_value":"VMOOA52VQRCZ","created_at":"2026-08-11T01:21:29.365108+00:00"},{"alias_kind":"pith_short_16","alias_value":"VMOOA52VQRCZD7VS","created_at":"2026-08-11T01:21:29.365108+00:00"},{"alias_kind":"pith_short_8","alias_value":"VMOOA52V","created_at":"2026-08-11T01:21:29.365108+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/VMOOA52VQRCZD7VS6VCW6DGN5N","json":"https://pith.science/pith/VMOOA52VQRCZD7VS6VCW6DGN5N.json","graph_json":"https://pith.science/api/pith-number/VMOOA52VQRCZD7VS6VCW6DGN5N/graph.json","events_json":"https://pith.science/api/pith-number/VMOOA52VQRCZD7VS6VCW6DGN5N/events.json","paper":"https://pith.science/paper/VMOOA52V"},"agent_actions":{"view_html":"https://pith.science/pith/VMOOA52VQRCZD7VS6VCW6DGN5N","download_json":"https://pith.science/pith/VMOOA52VQRCZD7VS6VCW6DGN5N.json","view_paper":"https://pith.science/paper/VMOOA52V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.08041&json=true","fetch_graph":"https://pith.science/api/pith-number/VMOOA52VQRCZD7VS6VCW6DGN5N/graph.json","fetch_events":"https://pith.science/api/pith-number/VMOOA52VQRCZD7VS6VCW6DGN5N/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/VMOOA52VQRCZD7VS6VCW6DGN5N/action/timestamp_anchor","attest_storage":"https://pith.science/pith/VMOOA52VQRCZD7VS6VCW6DGN5N/action/storage_attestation","attest_author":"https://pith.science/pith/VMOOA52VQRCZD7VS6VCW6DGN5N/action/author_attestation","sign_citation":"https://pith.science/pith/VMOOA52VQRCZD7VS6VCW6DGN5N/action/citation_signature","submit_replication":"https://pith.science/pith/VMOOA52VQRCZD7VS6VCW6DGN5N/action/replication_record"}},"created_at":"2026-08-11T01:21:29.365108+00:00","updated_at":"2026-08-11T01:21:29.365108+00:00"}