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

Zhi-Xue Zhao; Mapundi K. Banda; Bao-Zhu Guo

doi:10.3934/dcdsb.2020021

Article Contents

2020, Volume 25, Issue 7: 2539-2554. Doi: 10.3934/dcdsb.2020021

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

a.
College of Mathematical Science, Tianjin Normal University, Tianjin 300387, China
b.
Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa
c.
School of Mathematics and Physics, North China Electric Power University, Beijing 102206, China
d.
School of Computer Science and Applied Mathematics, University of the Witwatersrand, Wits 2050, Johannesburg, South Africa

^* Corresponding author: Zhi-Xue Zhao
^* Corresponding author: Zhi-Xue Zhao

Received: April 30, 2018

Revised: December 31, 2018

Early access: April 2020

Published: July 2020

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Abstract

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.

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

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Figure 1. Block diagram of identifiability analysis

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

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

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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

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