{"paper":{"title":"Boundary null-controllability for the beam equation with classical structural damping","license":"http://creativecommons.org/licenses/by-sa/4.0/","headline":"The damped beam equation reaches null state from the boundary for every damping strength up to 2 and for almost every strength above 2.","cross_cats":["math.AP"],"primary_cat":"math.OC","authors_text":"Julian Edward, Sergei Avdonin","submitted_at":"2026-05-14T04:49:29Z","abstract_excerpt":"Let $\\Delta$ be the Dirichlet Laplacian on the interval $(0,\\pi)$, and let $T>0$. We prove a well-posedness results for the structurally damped beam equation $$u_{tt}+\\Delta^2 u-\\rho \\Delta u_t=0, x\\in (0,\\pi),t>0$$ with various boundary conditions including $$ u(0,t)=u_{xx}(0,t)=0; u(\\pi,t)=f(t),u_{xx}(\\pi,t)=0, $$ and $f\\in H_0^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null controllability for all $\\rho \\leq 2$. For $\\rho >2$, we show null controllability for arbitrary $T>0$ holds for almost all $\\rho$, but fails for a dense subset of $(2,\\infty)$.\n  An a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove null controllability for all ρ ≤ 2. For ρ >2, we show null controllability for arbitrary T>0 holds for almost all ρ, but fails for a dense subset of (2,∞).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The well-posedness of the damped beam equation under the specified boundary conditions and the validity of the controllability criteria (such as observability inequalities) for the given ranges of ρ.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Null controllability holds for the beam equation with structural damping ρ for all ρ ≤ 2 and almost all ρ > 2.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The damped beam equation reaches null state from the boundary for every damping strength up to 2 and for almost every strength above 2.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"9e165c2820666d8e46d8902638316b78797a6e09b8c613773e87f5ab80864a87"},"source":{"id":"2605.14371","kind":"arxiv","version":1},"verdict":{"id":"599d82fb-b44b-46b6-9d15-6e0ae4a2577d","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T02:04:47.169926Z","strongest_claim":"We prove null controllability for all ρ ≤ 2. For ρ >2, we show null controllability for arbitrary T>0 holds for almost all ρ, but fails for a dense subset of (2,∞).","one_line_summary":"Null controllability holds for the beam equation with structural damping ρ for all ρ ≤ 2 and almost all ρ > 2.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The well-posedness of the damped beam equation under the specified boundary conditions and the validity of the controllability criteria (such as observability inequalities) for the given ranges of ρ.","pith_extraction_headline":"The damped beam equation reaches null state from the boundary for every damping strength up to 2 and for almost every strength above 2."},"references":{"count":25,"sample":[{"doi":"","year":2003,"title":"G. Avalos and I. Lasiecka, ”Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation”. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 3, 601–616","work_id":"bc8135c6-bf25-4b9b-b74e-595884945a99","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2014,"title":"F. Ammar-Khodja, A. Benabdallah, M. Gonz´ alez-Burgos, L. de Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index ofcomplex sequences”, J. Funct","work_id":"6cba35b7-b20e-4d1c-81ef-bc65433d8872","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2025,"title":"Null-controllability for the beam equation with struc- tural damping","work_id":"a1bfe92f-02d7-4a28-93a4-efe1e2cf4323","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Null-controllability for the beam equation with frac- tional structural damping","work_id":"05fb1044-dfa3-462a-9b51-d852fd929697","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"In preparation","work_id":"daaf8c35-5cd4-44fc-9f2e-fd2d0b2c3212","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":25,"snapshot_sha256":"e0c90a90e1958db78f198785cc626d30c4e42fd7cf8a27e230e0143f173afbb8","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"}