Logarithmic Stable Recovery of the Source and the Initial State of Time Fractional Diffusion Equations
Yavar Kian, Éric Soccorsi, Faouzi Triki
Abstract
In this paper we study the inverse problem of identifying a source or an\ninitial state in a time-fractional diffusion equation from the knowledge of a\nsingle boundary measurement. We derive logarithmic stability estimates for both\ninversions. These results show that the ill-posedness increases exponentially\nwhen the fractional derivative order tends to zero, while it exponentially\ndecreases when the regularity of the source or the initial state becomes\nlarger. The stability estimate concerning the problem of recovering the initial\nstate can be considered as a weak observability inequality in control theory.\nThe analysis is mainly based on Laplace inversion techniques and a precise\nquantification of the unique continuation property for the resolvent of the\ntime-fractional diffusion operator as a function of the frequency in the\ncomplex plane. We also determine a global time regularity for the\ntime-fractional diffusion equation which is of interest itself.\n