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It is also shown that if $f$ is assigned as an internal linear feedback, the solution of the system is uniformly exponential decay to a constant state in $H^{s+3}(S)\\times H^s("},"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":"1811.05943","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-11-14T18:23:34Z","cross_cats_sorted":[],"title_canon_sha256":"8b1f0189bbe3444c5f934837181bb564e2446445eb9bb47f437ec8a343b30e2c","abstract_canon_sha256":"ff448657e8fe74b15b12b24df7e5da95bdf7be2738325fe68320c32feb834344"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:00:41.797902Z","signature_b64":"+PdQpjzSMdK8uNJ3E8YDG5kxbS8XVDCwQP0cd+JLXqYxWJDqSl14VYAJFUsop9vNyLD/8wwiRp8R0cUacbefBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"10956909f4b116755a4145df109d8cadcaa0a3512cdf23c4a0617128817b19f2","last_reissued_at":"2026-05-18T00:00:41.797489Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:00:41.797489Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact controllability and stability of the Sixth Order Boussinesq equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bing-Yu Zhang, Min Chen, Shenghao Li","submitted_at":"2018-11-14T18:23:34Z","abstract_excerpt":"The article studies the exact controllability and the stability of the sixth order Boussinesq equation\n  \\[ u_{tt}-u_{xx}+\\beta u_{xxxx}-u_{xxxxxx}+(u^2)_{xx}=f, \\quad \\beta=\\pm1, \\]\n  on the interval $S:=[0,2\\pi]$ with periodic boundary conditions.\n  It is shown that the system is locally exactly controllable in the classic Sobolev space, $H^{s+3}(S)\\times H^s(S)$ for $s\\geq 0$, for \"small\" initial and terminal states. 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