Inverse source problem and null controllability for multidimensional parabolic operators of Grushin type

TL;DR

This paper establishes Lipschitz stability and null controllability results for inverse source problems involving multidimensional Grushin-type parabolic operators, overcoming challenges posed by interior degeneracy using a combined Fourier and Carleman approach.

Contribution

It introduces a novel method combining Fourier decomposition and Carleman inequalities to analyze Grushin-type operators, providing new stability and controllability results.

Findings

Proved Lipschitz stability for inverse source problems with interior degeneracy.

Established observability and null controllability for multidimensional Grushin-type equations.

Developed a mixed strategy combining Fourier analysis and Carleman estimates.

Abstract

The approach to Lipschitz stability for uniformly parabolic equations introduced by Imanuvilov and Yamamoto in 1998, based on Carleman estimates, seems hard to apply to the case of Grushin-type operators of interest to this paper. Indeed, such estimates are still missing for parabolic operators degenerating in the interior of the space domain. Nevertheless, we are able to prove Lipschitz stability results for inverse source problems for such operators, with locally distributed measurements in arbitrary space dimension. For this purpose, we follow a mixed strategy which combines the appraoch due to Lebeau and Robbiano, relying on Fourier decomposition, with Carleman inequalities for heat equations with nonsmooth coefficients (solved by the Fourier modes). As a corollary, we obtain a direct proof of the observability of multidimensional Grushin-type parabolic equations, with locally distributed observations, which is equivalent to null controllability with locally distributed controls.