Startseite On unique determination of an unknown spatial load in damped Euler–Bernoulli beam equation from final time output
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On unique determination of an unknown spatial load in damped Euler–Bernoulli beam equation from final time output

  • Anjuna Dileep , Alemdar Hasanov EMAIL logo , Kumarasamy Sakthivel und Cristiana Sebu ORCID logo
Veröffentlicht/Copyright: 28. Juli 2021
Journal of Inverse and Ill-posed Problems
Aus der Zeitschrift Journal of Inverse and Ill-posed Problems Band 30 Heft 4

Abstract

In this paper, we discuss the role of the damping term μ ⁢ u t in unique determination of unknown spatial load F ⁢ ( x ) in a damped Euler–Bernoulli beam equation u t ⁢ t + μ ⁢ u t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) from the measured final time displacement u T ⁢ ( x ) := u ⁢ ( x , T ) or velocity ν T ⁢ ( x ) := u t ⁢ ( x , T ) . In [A. Hasanov Hasanoğlu and V. G. Romanov, Introduction to Inverse Problems for Differential Equations, Springer, Cham, 2017] it was shown that the unique determination of the spatial load F ⁢ ( x ) in the undamped wave equation u t ⁢ t - ( k ⁢ ( x ) ⁢ u x ) x = F ⁢ ( x ) ⁢ G ⁢ ( t ) from final data (displacement or velocity) is not possible. This result is also valid for the undamped beam equation u t ⁢ t + ( r ⁢ ( x ) ⁢ u x ⁢ x ) x ⁢ x = F ⁢ ( x ) ⁢ G ⁢ ( t ) . We show that in the presence of the damping term, the spatial load can be uniquely determined from the final time output under some acceptable conditions with respect to the final time T > 0 and the damping coefficient μ > 0 . The cases of pure spatial load G ⁢ ( t ) ≡ 1 , and the exponentially decaying load G ⁢ ( t ) = e - η ⁢ t are analyzed separately, leading also to sufficient conditions for the uniqueness of the inverse problem solution. The results presented in this paper clearly demonstrate the key role of the damping term in the unique determination of the unknown spatial load F ⁢ ( x ) in the damped Euler–Bernoulli beam equation.

Keywords: Damped Euler–Bernoulli equations; damping coefficient; final time output; inverse source problem; uniqueness; singular values
MSC 2010: 74G75; 65M32

Funding statement: The research of the first author has been supported by the Scientific and Technological Research Council of Turkey (TUBITAK).

Acknowledgements

The authors would like to thank the reviewer’s comments for improving this article further.

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Received: 2021-01-20
Revised: 2021-06-15
Accepted: 2021-06-24
Published Online: 2021-07-28
Published in Print: 2022-08-01

© 2021 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

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  2. Inverse problem of shape identification from boundary measurement for Stokes equations: Shape differentiability of Lagrangian
  3. The modulus of the Fourier transform on a sphere determines 3-dimensional convex polytopes
  4. Convexity of a discrete Carleman weighted objective functional in an inverse medium scattering problem
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  6. Recovery of a Lamé parameter from displacement fields in nonlinear elasticity models
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  8. A reduced basis Landweber method for the identification of piecewise constant Robin coefficient in an elliptic equation
  9. On unique determination of an unknown spatial load in damped Euler–Bernoulli beam equation from final time output
  10. Can a single scalar second order PDE govern well the propagation of the electric wave field in a heterogeneous 3D medium?
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https://doi.org/10.1515/jiip-2021-0009
Schlagwörter für diesen Artikel
Damped Euler–Bernoulli equations; damping coefficient; final time output; inverse source problem; uniqueness; singular values

Artikel in diesem Heft

  1. Frontmatter
  2. Inverse problem of shape identification from boundary measurement for Stokes equations: Shape differentiability of Lagrangian
  3. The modulus of the Fourier transform on a sphere determines 3-dimensional convex polytopes
  4. Convexity of a discrete Carleman weighted objective functional in an inverse medium scattering problem
  5. Fixed angle inverse scattering in the presence of a Riemannian metric
  6. Recovery of a Lamé parameter from displacement fields in nonlinear elasticity models
  7. Three-dimensional transient electromagnetic inversion with optimal transport
  8. A reduced basis Landweber method for the identification of piecewise constant Robin coefficient in an elliptic equation
  9. On unique determination of an unknown spatial load in damped Euler–Bernoulli beam equation from final time output
  10. Can a single scalar second order PDE govern well the propagation of the electric wave field in a heterogeneous 3D medium?
  11. Research biography of a distinguished expert in the field of inverse problems: Professor Eric Todd Quinto
  12. Optimal convergence rates for inexact Newton regularization with CG as inner iteration
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