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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 $\\rho \\leq 2$. 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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. 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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. 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