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Logarithmic stable recovery of the source and the initial state of time fractional diffusion equations

Abstract : In this paper we study the inverse problem of identifying a source or an initial state in a timefractional diffusion equation from the knowledge of a single boundary measurement. We derive logarithmic stability estimates for both inversions. These results show that the ill-posedness increases exponentially when the fractional derivative order tends to zero, while it exponentially decreases when the regularity of the source or the initial state becomes larger. The stability estimate concerning the problem of recovering the initial state can be considered as a weak observability inequality in control theory. The analysis is mainly based on Laplace inversion techniques and a precise quantification of the unique continuation property for the resolvent of the time-fractional diffusion operator as a function of the frequency in the complex plane. We also determine a global time regularity for the time-fractional diffusion equation which is of interest itself.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03498181
Contributor : Yavar Kian Connect in order to contact the contributor
Submitted on : Monday, December 20, 2021 - 9:08:14 PM
Last modification on : Monday, April 25, 2022 - 3:44:48 AM

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  • HAL Id : hal-03498181, version 1
  • ARXIV : 2112.10835

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Yavar Kian, Éric Soccorsi, Faouzi Triki. Logarithmic stable recovery of the source and the initial state of time fractional diffusion equations. 2021. ⟨hal-03498181⟩

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