Work

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

Year: 2014

Type: article

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 an arbitrary space dimension. For this purpose, we follow a mixed strategy which combines the approach due to Lebeau and Robbiano, relying on Fourier decomposition and Carleman inequalities for heat equations with non-smooth 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.

Source: Inverse Problems

DOI: 10.1088/0266-5611/30/2/025006

Authors: K Beauchard, P Cannarsa, M Yamamoto

Institutions: École Polytechnique, Centre de Mathématiques Laurent Schwartz, University of Rome Tor Vergata, The University of Tokyo

Language: English

Topic: Stability and Controllability of Differential Equations

Subfield: Control and Systems Engineering

Field: Engineering

Domain: Physical Sciences

Open Access status: bronze

Funder: Agence Nationale de la Recherche

Awards: G4596822079, G7905394774