Various stability estimates for the problem of determining an initial heat distribution from a single measurement

2015/12/23 by Mourad Choulli, Choulli, Mourad
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1512.07421

openalex publication_date 2015/12/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem of determining the initial heat distribution in the heat equation from a point measurement. We show that this inverse problem is naturally related to the one of recovering the coefficients of Dirichlet series from its sum. Taking the advantage of existing literature on Dirichlet series, in connection with Müntz 's theorem, we establish various stability estimates of Hölder and logarithmic type. These stability estimates are then used to derive the corresponding ones for the original inverse problem, mainly in the case of one space dimension. In higher space dimensions, we are interested to an internal or a boundary measurement. This issue is closely related to the problem of observability arising in Control Theory. We complete and improve the existing results.

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