June 2018, 7(2): 197-216. doi: 10.3934/eect.2018010

The recovery of a parabolic equation from measurements at a single point

Department of Mathematics, University of West Georgia, Carrollton, GA 30118, USA

* Corresponding author: boumenir@westga.edu

Received  December 2016 Revised  November 2017 Published  May 2018

By measuring the temperature at an arbitrary single point located inside an unknown object or on its boundary, we show how we can uniquely reconstruct all the coefficients appearing in a general parabolic equation which models its cooling. We also reconstruct the shape of the object. The proof hinges on the fact that we can detect infinitely many eigenfunctions whose Wronskian does not vanish. This allows us to evaluate these coefficients by solving a simple linear algebraic system. The geometry of the domain and its boundary are found by reconstructing the first eigenfunction.

Citation: Amin Boumenir, Vu Kim Tuan, Nguyen Hoang. The recovery of a parabolic equation from measurements at a single point. Evolution Equations & Control Theory, 2018, 7 (2) : 197-216. doi: 10.3934/eect.2018010
References:
[1]

Sh. A. AlimovV. A. Il'in and E. M. Nikishin, Questions on the convergence of multiple trigonometric series and spectral expansions Ⅰ, Russian Mathematical Surveys, 31 (1976), 28-83.

[2]

H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging, Mathématiques & Applications 62, Springer, Berlin, 2008.

[3]

S. A. Avdonin and A. Bulanova, Boundary control approach to the spectral estimation problem: The case of multiple poles, Math. Control Signals Systems, 22 (2011), 245-265. doi: 10.1007/s00498-010-0052-5.

[4]

S. A. AvdoninF. Gesztesy and A. Makarov, Spectral estimation and inverse initial boundary value problems, Inverse Probl. and Imaging, 4 (2010), 1-9. doi: 10.3934/ipi.2010.4.1.

[5]

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

[6]

L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. doi: 10.1007/978-1-4419-7805-9.

[7]

A. Bostan and P. Dumas, Wronskians and linear independence, Amer. Math. Monthly, 117 (2010), 722-727, arXiv: 1301.6598v1. doi: 10.4169/000298910x515785.

[8]

A. Boumenir and V. K. Tuan, Inverse problems for multidimensional heat equations by measurements at a single point on the boundary, Numer. Funct. Anal. Optim., 30 (2009), 1215-1230. doi: 10.1080/01630560903498979.

[9]

A. Boumenir and V. K. Tuan, An inverse problem for the wave equation, Journal of Inverse and Ill-posed Problems, 19 (2011), 273-592. doi: 10.1515/JIIP.2011.056.

[10]

A. S. Demidov and M. Moussaoui, An inverse problem originating from magnetohydrodynamics, Inverse Problems, 20 (2004), 137-154. doi: 10.1088/0266-5611/20/1/008.

[11]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Archive for Rational Mechanics and Analysis, 43 (1971), 272-292. doi: 10.1007/BF00250466.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[13]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser, Basel, 2006.

[14]

V. A. Il'in and Sh. A. Alimov, Conditions for the convergence of spectral decompositions that correspond to self-adjoint extentions of elliptic operators, Ⅴ, Differential Equations, 10 (1974), 360-377.

[15]

V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ Ed, Applied Mathematical Sciences 127, Springer, New York, 2006.

[16]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, The Netherlands, 2004. doi: 10.1515/9783110915549.

[17]

V. Mikhailov, Equations aux Derivees Partielles, French transl., Mir, Moscow, 1980.

[18]

K. Wolsson, Linear dependence of a function set of $m$ variables with vanishing generalized Wronskians, Linear Algebra and its Applications, 117 (1989), 73-80. doi: 10.1016/0024-3795(89)90548-X.

[19]

Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, Singapore, 2006. doi: 10.1142/6238.

[20]

L. V. Zhizhiashshvilli, Some problems in the theory of simple and multiple trigonometric series, Russian Mathematical Surveys, 28 (1973), 62-119.

[21]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Equations, Vol. Ⅲ, Elsevier/North-Holland, Amsterdam, 3 (2007), 527-621. doi: 10.1016/S1874-5717(07)80010-7.

show all references

References:
[1]

Sh. A. AlimovV. A. Il'in and E. M. Nikishin, Questions on the convergence of multiple trigonometric series and spectral expansions Ⅰ, Russian Mathematical Surveys, 31 (1976), 28-83.

[2]

H. Ammari, An Introduction to Mathematics of Emerging Biomedical Imaging, Mathématiques & Applications 62, Springer, Berlin, 2008.

[3]

S. A. Avdonin and A. Bulanova, Boundary control approach to the spectral estimation problem: The case of multiple poles, Math. Control Signals Systems, 22 (2011), 245-265. doi: 10.1007/s00498-010-0052-5.

[4]

S. A. AvdoninF. Gesztesy and A. Makarov, Spectral estimation and inverse initial boundary value problems, Inverse Probl. and Imaging, 4 (2010), 1-9. doi: 10.3934/ipi.2010.4.1.

[5]

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

[6]

L. Beilina and M. V. Klibanov, Approximate Global Convergence and Adaptivity for Coefficient Inverse Problems, Springer, New York, 2012. doi: 10.1007/978-1-4419-7805-9.

[7]

A. Bostan and P. Dumas, Wronskians and linear independence, Amer. Math. Monthly, 117 (2010), 722-727, arXiv: 1301.6598v1. doi: 10.4169/000298910x515785.

[8]

A. Boumenir and V. K. Tuan, Inverse problems for multidimensional heat equations by measurements at a single point on the boundary, Numer. Funct. Anal. Optim., 30 (2009), 1215-1230. doi: 10.1080/01630560903498979.

[9]

A. Boumenir and V. K. Tuan, An inverse problem for the wave equation, Journal of Inverse and Ill-posed Problems, 19 (2011), 273-592. doi: 10.1515/JIIP.2011.056.

[10]

A. S. Demidov and M. Moussaoui, An inverse problem originating from magnetohydrodynamics, Inverse Problems, 20 (2004), 137-154. doi: 10.1088/0266-5611/20/1/008.

[11]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Archive for Rational Mechanics and Analysis, 43 (1971), 272-292. doi: 10.1007/BF00250466.

[12]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001.

[13]

A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Frontiers in Mathematics, Birkhäuser, Basel, 2006.

[14]

V. A. Il'in and Sh. A. Alimov, Conditions for the convergence of spectral decompositions that correspond to self-adjoint extentions of elliptic operators, Ⅴ, Differential Equations, 10 (1974), 360-377.

[15]

V. Isakov, Inverse Problems for Partial Differential Equations, 2$^{nd}$ Ed, Applied Mathematical Sciences 127, Springer, New York, 2006.

[16]

M. V. Klibanov and A. Timonov, Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, The Netherlands, 2004. doi: 10.1515/9783110915549.

[17]

V. Mikhailov, Equations aux Derivees Partielles, French transl., Mir, Moscow, 1980.

[18]

K. Wolsson, Linear dependence of a function set of $m$ variables with vanishing generalized Wronskians, Linear Algebra and its Applications, 117 (1989), 73-80. doi: 10.1016/0024-3795(89)90548-X.

[19]

Z. Wu, J. Yin and C. Wang, Elliptic and Parabolic Equations, World Scientific, Singapore, 2006. doi: 10.1142/6238.

[20]

L. V. Zhizhiashshvilli, Some problems in the theory of simple and multiple trigonometric series, Russian Mathematical Surveys, 28 (1973), 62-119.

[21]

E. Zuazua, Controllability and observability of partial differential equations: Some results and open problems, in Handbook of Differential Equations: Evolutionary Equations, Vol. Ⅲ, Elsevier/North-Holland, Amsterdam, 3 (2007), 527-621. doi: 10.1016/S1874-5717(07)80010-7.

Figure 1.  $ {{\varphi }_{1}} $
Figure 2.  $\Omega$
Figure 3.  $ {{\varphi }_{2}} $
Figure 4.  $ {{\varphi }_{12}} $
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