Uniqueness and stability analysis of final data inverse source problems for evolution equations

Vladimir Romanov; Alemdar Hasanov

doi:10.1515/jiip-2021-0072

Article

Uniqueness and stability analysis of final data inverse source problems for evolution equations

Vladimir Romanov and Alemdar Hasanov

Published/Copyright: April 28, 2022

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From the journal Journal of Inverse and Ill-posed Problems Volume 30 Issue 3

Abstract

This article proposes a unified approach to the issues of uniqueness and Lipschitz stability for the final data inverse source problems of determining the unknown spatial load F ⁢ ( x ) in the evolution equations. The approach is based on integral identities outlined here for the one-dimensional and multidimensional heat equations

ρ ⁢ ( x ) ⁢ u t - ( k ⁢ ( x ) ⁢ u x ) x = F ⁢ ( x ) ⁢ G ⁢ ( t ) , ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] ,

and

ρ ⁢ ( x ) ⁢ u t - div ⁡ ( k ⁢ ( x ) ⁢ ∇ ⁡ u ) = F ⁢ ( x ) ⁢ G ⁢ ( x , t ) , ( x , t ) ∈ Ω × ( 0 , T ] , Ω ∈ ℝ n ,

for the damped wave, and the Euler–Bernoulli beam and Kirchhoff plate equations

ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t - ( r ⁢ ( x ) ⁢ u x ) x = F ⁢ ( x ) ⁢ G ⁢ ( x , t ) , ρ ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) ,

for ( x , t ) ∈ ( 0 , ℓ ) × ( 0 , T ] , and

ρ ⁢ ( x ) ⁢ h ⁢ ( x ) ⁢ u t ⁢ t + μ ⁢ ( x ) ⁢ u t + ( D ⁢ ( x ) ⁢ ( u x 1 , x 1 + ν ⁢ u x 2 , x 2 ) ) x 1 , x 1 + ( D ⁢ ( x ) ⁢ ( u x 2 , x 2 + ν ⁢ u x 1 , x 1 ) ) x 2 , x 2 + 2 ⁢ ( 1 - ν ) ⁢ ( D ⁢ ( x ) ⁢ u x 1 , x 2 ) x 1 , x 2

= F ⁢ ( x ) ⁢ G ⁢ ( t ) , ( x , t ) ∈ Ω T := Ω × ( 0 , T ) , Ω := ( 0 , ℓ 1 ) × ( 0 , ℓ 2 ) ,

and allows us to prove the uniqueness and stability of the solutions for all considered inverse problems under the same conditions imposed on the load G ⁢ ( t ) or G ⁢ ( x , t ) .

Keywords: Heat; damped wave; Euler–Bernoulli and Kirchhoff equations; inverse source problem; final time output; uniqueness; stability; stability radius

MSC 2010: 74G75; 65M32

Funding statement: The first author was supported by Mathematical Center in Akademgorodok at the Novosibirsk State University (the agreement with Ministry of Science and High Education of the Russian Federation number 075-15-2019-1613). The research of the second author has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK) through the Incentive Program for International Scientific Publications (UBYT).

References

[1] O. Baysal and A. Hasanov, Solvability of the clamped Euler–Bernoulli beam equation, Appl. Math. Lett. 93 (2019), 85–90. 10.1016/j.aml.2019.02.006Search in Google Scholar

[2] I. Bushuyev, Global uniqueness for inverse parabolic problems with final observation, Inverse Problems 11 (1995), no. 4, 11–16. 10.1088/0266-5611/11/4/001Search in Google Scholar

[3] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19, American Mathematical Society, Providence, 2002. Search in Google Scholar

[4] N. L. Gol’dman, On a counterexample of inverse parabolic problems with final overdetermination, Dokl. Math. 88 (2013), no. 5, 714–716. 10.1134/S1064562413060240Search in Google Scholar

[5] A. Hasanov, V. Romanov and O. Baysal, Unique recovery of unknown spatial load in damped Euler–Bernoulli beam equation from final time measured output, Inverse Problems 37 (2021), no. 7, Paper No. 075005. 10.1088/1361-6420/ac01fbSearch in Google Scholar

[6] A. Hasanov Hasanoğlu and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017. 10.1007/978-3-319-62797-7Search in Google Scholar

[7] V. Isakov, Inverse Source Problems, Math. Surveys Monogr. 34, American Mathematical Society, Providence, 1990. 10.1090/surv/034Search in Google Scholar

[8] V. Isakov, Inverse parabolic problems with the final overdetermination, Commun. Pure Appl. Math. 44 (1991), no. 2, 185–209. 10.1002/cpa.3160440203Search in Google Scholar

[9] V. L. Kamynin, On the unique solvability of an inverse problem for parabolic equations with a final overdetermination condition, Mat. Zametki 73 (2003), no. 2, 217–227; translation in Math. Notes, 73 (2003), 202-211. Search in Google Scholar

[10] A. Kawano, Uniqueness in the identification of asynchronous sources and damage in vibrating beams, Inverse Problems 30 (2014), no. 6, Article ID 065008. 10.1088/0266-5611/30/6/065008Search in Google Scholar

[11] A. B. Kostin, Counterexamples in inverse problems for parabolic, elliptic, and hyperbolic equations, Comput. Math. Math. Phys. 54 (2014), no. 5, 797–810. 10.1134/S0965542514020092Search in Google Scholar

[12] S. Nicaise and O. Zaïr, Determination of point sources in vibrating beams by boundary measurements: Identifiability, stability, and reconstruction results, Electron. J. Differential Equations 2004 (2004), Paper No. 20. Search in Google Scholar

[13] P. W. Poon, The stability radius of a quasi-Fredholm operator, Proc. Amer. Math. Soc. 126 (1998), no. 4, 1071–1080. 10.1090/S0002-9939-98-04253-1Search in Google Scholar

[14] A. I. Prilepko and A. B. Kostin, On certain inverse problems for parabolic equations with final and integral observation, Sb. Math. 75 (1993), no. 2, Paper No. 473. 10.1070/SM1993v075n02ABEH003394Search in Google Scholar

[15] A. I. Prilepko and V. V. Solovev, Solvability theorems and the Rothe method in inverse problems for an equation of parabolic type, Differ. Equ. 23 (1987), 1341–1349. Search in Google Scholar

[16] M. Slodička, Uniqueness for an inverse source problem of determining a space dependent source in a non-autonomous parabolic equation, Appl. Math. Lett. 107 (2020), Article ID 106395. 10.1016/j.aml.2020.106395Search in Google Scholar

[17] K. Van Bockstal, Identification of an unknown spatial load distribution in a vibrating beam or plate from the final state, J. Inverse Ill-Posed Probl. 27 (2019), no. 5, 623–642. 10.1515/jiip-2018-0068Search in Google Scholar

Received: 2021-11-21

Accepted: 2022-03-07

Published Online: 2022-04-28

Published in Print: 2022-06-01

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Articles in the same Issue

https://doi.org/10.1515/jiip-2021-0072

Keywords for this article

Heat; damped wave; Euler–Bernoulli and Kirchhoff equations; inverse source problem; final time output; uniqueness; stability; stability radius