Boundary null-controllability for the beam equation with classical structural damping
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916012000018432 |
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| author | Avdonin, Sergei Edward, Julian |
| author_facet | Avdonin, Sergei Edward, Julian |
| contents | Let $Δ$ be the Dirichlet Laplacian on the interval $(0,π)$, and let $T>0$. We prove a well-posedness results for the structurally damped beam equation $$u_{tt}+Δ^2 u-ρΔu_t=0, x\in (0,π),t>0$$ with various boundary conditions including $$ u(0,t)=u_{xx}(0,t)=0; u(π,t)=f(t),u_{xx}(π,t)=0, $$ and $f\in H_0^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null controllability for all $ρ\leq 2$. For $ρ>2$, we show null controllability for arbitrary $T>0$ holds for almost all $ρ$, but fails for a dense subset of $(2,\infty)$.
An analagous result is proven for Neumann control. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_14371 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Boundary null-controllability for the beam equation with classical structural damping Avdonin, Sergei Edward, Julian Optimization and Control Analysis of PDEs 93C20 (primary), 93B05, 74K10 (secondary) Let $Δ$ be the Dirichlet Laplacian on the interval $(0,π)$, and let $T>0$. We prove a well-posedness results for the structurally damped beam equation $$u_{tt}+Δ^2 u-ρΔu_t=0, x\in (0,π),t>0$$ with various boundary conditions including $$ u(0,t)=u_{xx}(0,t)=0; u(π,t)=f(t),u_{xx}(π,t)=0, $$ and $f\in H_0^2(0,T)$ and appropriate initial conditions. Viewing $f$ as a control, we prove null controllability for all $ρ\leq 2$. For $ρ>2$, we show null controllability for arbitrary $T>0$ holds for almost all $ρ$, but fails for a dense subset of $(2,\infty)$. An analagous result is proven for Neumann control. |
| title | Boundary null-controllability for the beam equation with classical structural damping |
| topic | Optimization and Control Analysis of PDEs 93C20 (primary), 93B05, 74K10 (secondary) |
| url | https://arxiv.org/abs/2605.14371 |