Authors

Fu Zheng, Hao Zhou

Abstract

In this paper, the state reconstruction of the wave equation with general viscosity and non-collocated observation and control are given from both the theoretical aspect and the numerical aspect. The forward-backward observers-based algorithm is utilized to solve this problem. The formula for calculating the initial value is derived. Moreover, the iterative sequence is also built for any given guess value and it is showed that it strongly converges to initial value. However, because the exponential stabilities of the error systems between the original systems and the forward∖backward observers play important roles in the involved algorithm, the exponential stability of some related system is firstly discussed. Furthermore, combining the theory of port-Hamiltonian system and the finite difference, the semi-discretization scheme of the finite difference with order reduction is given and the uniform exponential stability of the semi-discretization system is verified by the method paralleling to the continuous system. Finally, the convergence analysis of the finite difference scheme is given and the convergence of the solution of the mixed finite element scheme induced by the finite difference scheme with order reduction to the solution of the continuous counterpart is also presented.

Keywords

Wave equation; Non-collocated observation and control; Exponential stability; State reconstruction; Semi-discretization

Citation

  • Journal: Journal of Mathematical Analysis and Applications
  • Year: 2021
  • Volume: 502
  • Issue: 1
  • Pages: 125257
  • Publisher: Elsevier BV
  • DOI: 10.1016/j.jmaa.2021.125257

BibTeX

@article{Zheng_2021,
  title={{State reconstruction of the wave equation with general viscosity and non-collocated observation and control}},
  volume={502},
  ISSN={0022-247X},
  DOI={10.1016/j.jmaa.2021.125257},
  number={1},
  journal={Journal of Mathematical Analysis and Applications},
  publisher={Elsevier BV},
  author={Zheng, Fu and Zhou, Hao},
  year={2021},
  pages={125257}
}

Download the bib file

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