On the -semigroup generation and exponential stability resulting from a shear force feedback on a rotating beam |
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| Authors: | Bao-Zhu Guo Jun-min Wang Siu-Pang Yung |
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| Affiliation: | aInstitute of Systems Science, Academy of Mathematics and System Sciences, Academia Sinica, Beijing 100080, PR China;bSchool of Computational and Applied Mathematics, University of the Witwatersrand, Private 3, Wits 2050, Johannesburg, South Africa;cDepartment of Mathematics, The University of Hong Kong, Hong Kong, PR China |
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| Abstract: | In this paper, we show that a linear unbounded operator associated with an Euler–Bernoulli beam equation under shear boundary feedback generates a C0-semigroup in the underlying state Hilbert space. This provides an answer to a long time unsolved problem due to the lack of dissipativity for the operator. The main steps are a careful estimation of the Green's function and the verification of the Riesz basis property for the generalized eigenfunctions. As a consequence, we show that this semigroup is differentiable and exponentially stable, which is in sharp contrast to the properties possessed by most feedback controlled beams based on a passive design principle. |
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| Keywords: | Riesz basis C0-semigroup Differentiable semigroup Stability |
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