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Boundary switch on/off control approach to simultaneous identification of diffusion coefficient and initial state for one-dimensional heat equation

  • * Corresponding author: Zhi-Xue Zhao

    * Corresponding author: Zhi-Xue Zhao 
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  • In this paper, we consider simultaneous reconstruction of diffusion coefficient and initial state for a one-dimensional heat equation through boundary control and measurement. The boundary measurement is proposed to make the system approximately observable, and both the coefficient and initial state are shown to be identifiable by this measurement under a boundary switch on/off control. By a Dirichlet series representation for the observation, we can transform the problem into an inverse process of reconstruction of the spectrum and coefficients for the Dirichlet series in terms of observation. This happens to be the reconstruction of spectral data for an exponential sequence with measurement error, and it enables us to develop an algorithm based on the matrix pencil method in signal analysis. A theoretical error analysis for the algorithm concerning the coefficient reconstruction is carried out for the proposed method. The numerical simulations are presented to verify the proposed algorithm.

    Mathematics Subject Classification: Primary: 35K05, 35R30; Secondary: 65M32, 65N21.


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  • Figure 1.  Block diagram of identifiability analysis

    Figure 2.  The initial values for Example 5.1 for various levels of noise

    Figure 3.  The initial values for Example 3 for various levels of noise

    Table 1.  The estimated $ \left\{\widetilde{C}_{n_k},\widetilde{\lambda}_{n_k},\lambda_n^\prime,\alpha,n_k,\alpha_k\right\} $ following Steps 1-4

    $ n(\mbox{or}\;k) $ $ \widetilde{C}_{n_k} $ $ \widetilde{\lambda}_{n_k} $ $ 100*\lambda_n^\prime $ $ \alpha $ $ n_k $ $ \alpha_k $
    0 0.5000 0.0000 -0.0105 0
    1 -9.4053 0.9869 1.0630 0.1077 1 0.1000
    2 4.9549 8.8761 4.3056 0.1091 3 0.0999
    3 -0.0162 24.6246 9.9040 0.1115 5 0.0998
    4 -0.0083 48.1762 18.2243 0.1154 7 0.0996
     | Show Table
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