Geodesic
Uniform controllability for the beam equation with vanishing structural damping
Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 869-881 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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This paper is devoted to studying the effects of a vanishing structural damping on the controllability properties of the one dimensional linear beam equation. The vanishing term depends on a small parameter $\varepsilon \in (0,1)$. We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls $v_{\varepsilon }$ as $\varepsilon $ goes to zero. It is shown that for any time $T$ sufficiently large but independent of $\varepsilon $ and for each initial data in a suitable space there exists a uniformly bounded family of controls $(v_\varepsilon )_\varepsilon $ in $L^2(0, T)$ acting on the extremity $x = \pi $. Any weak limit of this family is a control for the beam equation. This analysis is based on Fourier expansion and explicit construction and evaluation of biorthogonal sequences. This method allows us to measure the magnitude of the control needed for each eigenfrequency and to show their uniform boundedness when the structural damping tends to zero.
This paper is devoted to studying the effects of a vanishing structural damping on the controllability properties of the one dimensional linear beam equation. The vanishing term depends on a small parameter $\varepsilon \in (0,1)$. We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls $v_{\varepsilon }$ as $\varepsilon $ goes to zero. It is shown that for any time $T$ sufficiently large but independent of $\varepsilon $ and for each initial data in a suitable space there exists a uniformly bounded family of controls $(v_\varepsilon )_\varepsilon $ in $L^2(0, T)$ acting on the extremity $x = \pi $. Any weak limit of this family is a control for the beam equation. This analysis is based on Fourier expansion and explicit construction and evaluation of biorthogonal sequences. This method allows us to measure the magnitude of the control needed for each eigenfrequency and to show their uniform boundedness when the structural damping tends to zero.
DOI : 10.1007/s10587-014-0140-7
Classification : 30E05, 35L35, 35Q74, 58J45, 74K20, 93B05
Keywords: beam equation; null-controllability; structural damping; moment problem; biorthogonals 
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Bugariu, Ioan Florin. Uniform controllability for the beam equation with vanishing structural damping. Czechoslovak Mathematical Journal, Tome 64 (2014) no. 4, pp. 869-881. doi: 10.1007/s10587-014-0140-7

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