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Our main results guarantees that $P$ is null controllable if and only if it is weakly degenerate, that is, $\\alpha \\in (0,1)$. So, in order to steer the system to zero, one needs controls to act on both sides of the point of degeneracy in the strongly degenerate case $\\alpha\\in [1,2)$.\n  Our"},"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":"1807.00026","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2018-06-29T18:20:18Z","cross_cats_sorted":["math.AP"],"title_canon_sha256":"30fd6b3fb9c214f207f2bc7009b0e0a1f08e42c68328f4b573a2d230e748fb35","abstract_canon_sha256":"57b3af0156f07cb564722c79b5c3aad78016cb3528ac6f68437cc21e351324d1"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:11:56.354612Z","signature_b64":"uTvZPbM12TetDlV6hCGZsSTfKF2/7gw+qzhmEES0dODVln3QB8CVdt8GO04AvK2ehAbY/e7Y2b+G7C5v+TFNBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"196acb365b82b3a7523ec78ea2fe55cb603a40a404e76515978f25f8b2c18df0","last_reissued_at":"2026-05-18T00:11:56.353920Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:11:56.353920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Null controllability of parabolic equations with interior degeneracy and one-sided control","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.OC","authors_text":"Patrick Martinez, Piermarco Cannarsa, Roberto Ferretti","submitted_at":"2018-06-29T18:20:18Z","abstract_excerpt":"For $\\alpha\\in (0,2)$ we study the null controllability of the parabolic operator $$Pu= u_t - (\\vert x \\vert ^\\alpha u_x)_x\\qquad (1<x<1),$$ which degenerates at the interior point $x=0$, for locally distributed controls acting only one side of the origin (that is, on some interval $(a,b)$ with $0<a<b<1$). 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