July  2020, 25(7): 2533-2554. doi: 10.3934/dcdsb.2020021

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

Received  May 2018 Revised  January 2019 Published  July 2020 Early access  April 2020

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.

Citation: Zhi-Xue Zhao, Mapundi K. Banda, Bao-Zhu Guo. Boundary switch on/off control approach to simultaneous identification of diffusion coefficient and initial state for one-dimensional heat equation. Discrete and Continuous Dynamical Systems - B, 2020, 25 (7) : 2533-2554. doi: 10.3934/dcdsb.2020021
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1964.

[2] S. A. Avdonin and S. A. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, 1995. 
[3]

F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), 137-141.  doi: 10.1007/BF01386217.

[4]

A. BenabdallahP. Gaitan and J. L. Rousseau, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim., 46 (2007), 1849-1881.  doi: 10.1137/050640047.

[5]

M. Choulli and M. Yamamoto, Uniqueness and stability in determining the heat radiative coefficient, the initial temperature and a boundary coefficient in a parabolic equation, Nonlinear Anal., 69 (2008), 3983-3998.  doi: 10.1016/j.na.2007.10.031.

[6]

J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proc. Cambridge Philos. Soc., 43 (1947), 50-67.  doi: 10.1017/S0305004100023197.

[7]

F. R. Gantmacher, The Theory of Matrices: Volume One, Chelsea Publishing Company, New York, 1959.

[8]

G. H. GolubM. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.  doi: 10.1080/00401706.1979.10489751.

[9]

B. Z. Guo and J. D. Chang, Simultaneous identifiability of coefficients, initial state and source for string and beam equations via boundary control and observation, Proc. 8th Asian Control Conference, Kaohsiung, 2011,365–370.

[10]

S. Gutman and J. H. Ha, Identifiability of piecewise constant conductivity in a heat conduction process, SIAM J. Control Optim., 46 (2007), 694-713.  doi: 10.1137/060657364.

[11]

P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, Philadelphia, 2010. doi: 10.1137/1.9780898718836.

[12]

Y. B. Hua and T. K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise, IEEE Trans. Acoust. Speech Signal Process., 38 (1990), 814-824.  doi: 10.1109/29.56027.

[13]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.

[14]

S. Kitamura and S. Nakagiri, Identifiability of spatially-varying and constant parameters in distributed systems of parabolic type, SIAM J. Control Optim., 15 (1977), 785-802.  doi: 10.1137/0315050.

[15]

R. Murayama, The Gel'fand-Levitan theory and certain inverse problems for the parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28 (1981), 317-330. 

[16]

S. Nakagiri, Identifiability of linear systems in Hilbert spaces, SIAM J. Control Optim., 21 (1983), 501-530.  doi: 10.1137/0321031.

[17]

Y. Orlov and J. Bentsman, Adaptive distributed parameter systems identification with enforceable identifiability conditions and reduced-order spatial differentiation, IEEE Trans. Automat. Control, 45 (2000), 203-216.  doi: 10.1109/9.839944.

[18]

A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic equation, SIAM J. Control Optim., 17 (1979), 494-499.  doi: 10.1137/0317035.

[19]

K. RamdaniM. Tucsnak and G. Weiss, Recovering the initial state of an infinite-dimensional system using observers, Automatica, 46 (2010), 1616-1625.  doi: 10.1016/j.automatica.2010.06.032.

[20]

S. Saks and A. Zygmund, Analytic Functions, 2$^nd$ edition, Wydawnietwo Naukowe, Warsaw, 1965.

[21]

A. SmyshlyaevY. Orlov and M. Krstic, Adaptive identification of two unstable PDEs with boundary sensing and actuation, Internat. J. Adapt. Control Signal Process, 23 (2009), 131-149.  doi: 10.1002/acs.1056.

[22]

G. W. Stewart, On the perturbation of pseudo-inverses, projections and linear least squares problems, SIAM Rev., 19 (1977), 634-662.  doi: 10.1137/1019104.

[23]

T. Suzuki and R. Murayama, A uniqueness theorem in an identification problem for coefficients of parabolic equations, Proc. Japan Acad. Ser. A Math. Sci., 56 (1980), 259-263.  doi: 10.3792/pjaa.56.259.

[24]

T. Suzuki, Uniqueness and nonuniqueness in an inverse problem for the parabolic equation, J. Differential Equations, 47 (1983), 296-316.  doi: 10.1016/0022-0396(83)90038-4.

[25]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986.

[26]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[27]

Y. B. WangJ. ChengJ. Nakagawa and M. Yamamoto, A numerical method for solving the inverse heat conduction problem without initial value, Inverse Probl. Sci. Eng., 18 (2010), 655-671.  doi: 10.1080/17415971003698615.

[28]

P. A. Wedin, Perturbation theory for pseudo-inverses, BIT, 13 (1973), 217-232.  doi: 10.1007/BF01933494.

[29]

G. Q. Xu, State reconstruction of a distributed parameter system with exact observability, J. Math. Anal. Appl., 409 (2014), 168-179.  doi: 10.1016/j.jmaa.2013.06.014.

[30]

M. Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, 17 (2001), 1181-1202.  doi: 10.1088/0266-5611/17/4/340.

[31]

G. H. Zheng and T. Wei, Recovering the source and initial value simultaneously in a parabolic equation, Inverse Problems, 30 (2014), 065013, 35pp. doi: 10.1088/0266-5611/30/6/065013.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover Publications, New York, 1964.

[2] S. A. Avdonin and S. A. Ivanov, Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge University Press, 1995. 
[3]

F. L. Bauer and C. T. Fike, Norms and exclusion theorems, Numer. Math., 2 (1960), 137-141.  doi: 10.1007/BF01386217.

[4]

A. BenabdallahP. Gaitan and J. L. Rousseau, Stability of discontinuous diffusion coefficients and initial conditions in an inverse problem for the heat equation, SIAM J. Control Optim., 46 (2007), 1849-1881.  doi: 10.1137/050640047.

[5]

M. Choulli and M. Yamamoto, Uniqueness and stability in determining the heat radiative coefficient, the initial temperature and a boundary coefficient in a parabolic equation, Nonlinear Anal., 69 (2008), 3983-3998.  doi: 10.1016/j.na.2007.10.031.

[6]

J. Crank and P. Nicolson, A practical method for numerical evaluation of solutions of partial differential equations of the heat conduction type, Proc. Cambridge Philos. Soc., 43 (1947), 50-67.  doi: 10.1017/S0305004100023197.

[7]

F. R. Gantmacher, The Theory of Matrices: Volume One, Chelsea Publishing Company, New York, 1959.

[8]

G. H. GolubM. Heath and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics, 21 (1979), 215-223.  doi: 10.1080/00401706.1979.10489751.

[9]

B. Z. Guo and J. D. Chang, Simultaneous identifiability of coefficients, initial state and source for string and beam equations via boundary control and observation, Proc. 8th Asian Control Conference, Kaohsiung, 2011,365–370.

[10]

S. Gutman and J. H. Ha, Identifiability of piecewise constant conductivity in a heat conduction process, SIAM J. Control Optim., 46 (2007), 694-713.  doi: 10.1137/060657364.

[11]

P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms, SIAM, Philadelphia, 2010. doi: 10.1137/1.9780898718836.

[12]

Y. B. Hua and T. K. Sarkar, Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise, IEEE Trans. Acoust. Speech Signal Process., 38 (1990), 814-824.  doi: 10.1109/29.56027.

[13]

V. Isakov, Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.

[14]

S. Kitamura and S. Nakagiri, Identifiability of spatially-varying and constant parameters in distributed systems of parabolic type, SIAM J. Control Optim., 15 (1977), 785-802.  doi: 10.1137/0315050.

[15]

R. Murayama, The Gel'fand-Levitan theory and certain inverse problems for the parabolic equation, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 28 (1981), 317-330. 

[16]

S. Nakagiri, Identifiability of linear systems in Hilbert spaces, SIAM J. Control Optim., 21 (1983), 501-530.  doi: 10.1137/0321031.

[17]

Y. Orlov and J. Bentsman, Adaptive distributed parameter systems identification with enforceable identifiability conditions and reduced-order spatial differentiation, IEEE Trans. Automat. Control, 45 (2000), 203-216.  doi: 10.1109/9.839944.

[18]

A. Pierce, Unique identification of eigenvalues and coefficients in a parabolic equation, SIAM J. Control Optim., 17 (1979), 494-499.  doi: 10.1137/0317035.

[19]

K. RamdaniM. Tucsnak and G. Weiss, Recovering the initial state of an infinite-dimensional system using observers, Automatica, 46 (2010), 1616-1625.  doi: 10.1016/j.automatica.2010.06.032.

[20]

S. Saks and A. Zygmund, Analytic Functions, 2$^nd$ edition, Wydawnietwo Naukowe, Warsaw, 1965.

[21]

A. SmyshlyaevY. Orlov and M. Krstic, Adaptive identification of two unstable PDEs with boundary sensing and actuation, Internat. J. Adapt. Control Signal Process, 23 (2009), 131-149.  doi: 10.1002/acs.1056.

[22]

G. W. Stewart, On the perturbation of pseudo-inverses, projections and linear least squares problems, SIAM Rev., 19 (1977), 634-662.  doi: 10.1137/1019104.

[23]

T. Suzuki and R. Murayama, A uniqueness theorem in an identification problem for coefficients of parabolic equations, Proc. Japan Acad. Ser. A Math. Sci., 56 (1980), 259-263.  doi: 10.3792/pjaa.56.259.

[24]

T. Suzuki, Uniqueness and nonuniqueness in an inverse problem for the parabolic equation, J. Differential Equations, 47 (1983), 296-316.  doi: 10.1016/0022-0396(83)90038-4.

[25]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Third edition. Chelsea Publishing Co., New York, 1986.

[26]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[27]

Y. B. WangJ. ChengJ. Nakagawa and M. Yamamoto, A numerical method for solving the inverse heat conduction problem without initial value, Inverse Probl. Sci. Eng., 18 (2010), 655-671.  doi: 10.1080/17415971003698615.

[28]

P. A. Wedin, Perturbation theory for pseudo-inverses, BIT, 13 (1973), 217-232.  doi: 10.1007/BF01933494.

[29]

G. Q. Xu, State reconstruction of a distributed parameter system with exact observability, J. Math. Anal. Appl., 409 (2014), 168-179.  doi: 10.1016/j.jmaa.2013.06.014.

[30]

M. Yamamoto and J. Zou, Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, 17 (2001), 1181-1202.  doi: 10.1088/0266-5611/17/4/340.

[31]

G. H. Zheng and T. Wei, Recovering the source and initial value simultaneously in a parabolic equation, Inverse Problems, 30 (2014), 065013, 35pp. doi: 10.1088/0266-5611/30/6/065013.

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
$ 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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