Authors

Xiaofeng Wang, Wenlong Xue, Yong He, Fu Zheng

Abstract

In this note, Timoshenko beams with interior damping and boundary damping are studied from the viewpoints of control theory and numerical approximation. Especially, the uniform exponential stabilities of the beams are studied. The meaning of uniform exponential stability in this paper is two-fold: The first one is in the classical sense and also is concisely called exponential stability by many authors; The second one is that the semi-discretization systems, which are derived from an exponentially stable continuous beam by some semi-discretization schemes, are uniformly exponentially stable with respect to the discretized parameter. To investigate uniform exponential stability of continuous and discrete systems, five completely different methods, which are stability theory of port-Hamiltonian system, direct method of Lyapunov functional, perturbation theory of C 0 -semigroup, spectral analysis of unbounded operator and frequency standard of exponential stability for contractive semigroup, are involved. Especially, a new method, which is based on the frequency domain characteristics of uniform exponential stability of C 0 -semigroup of contractions, is established to verify the uniform exponential stability of semi-discretization systems derived from coupled system. The effectiveness of the numerical approximating algorithms is verified by numerical simulations.

Keywords

Timoshenko beam; Exponential stability; Semi-discretization; Finite difference; \( C_0 \)-Semigroup

Citation

  • Journal: Applied Mathematics and Computation
  • Year: 2023
  • Volume: 451
  • Issue:
  • Pages: 128028
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.amc.2023.128028

BibTeX

@article{Wang_2023,
  title={{Uniformly exponentially stable approximations for Timoshenko beams}},
  volume={451},
  ISSN={0096-3003},
  DOI={10.1016/j.amc.2023.128028},
  journal={Applied Mathematics and Computation},
  publisher={Elsevier BV},
  author={Wang, Xiaofeng and Xue, Wenlong and He, Yong and Zheng, Fu},
  year={2023},
  pages={128028}
}

Download the bib file

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