Classical and semi-classical observability for the Bouendi-Grushin operator

Abstract: The observability for the classical Schrödinger equation usually holds for very short time, under suitable geometric conditions. However, it is not the case when the underlying geometry is sub-elliptic. In this talk we consider the Schrodinger equation associated with the Bouendi- Grushin operator. The Bouendi-Grushin operator is a subelliptic operator which is degenerate along a line. In the Bouendi case, the associated Schrödinger equation exhibits a transport effect which leads to a "sub-elliptic" geometric control condition and a minimal time to ensure the observability. For general Bouendi-Grushin with stronger sub-elliptic effect, the observability for the Schrödinger equation is never true. These observability results can be seen from a semi- classical point of view, through a optimal resolvent esti- mate. Consequently, our resolvent estimate leads to an energy decay rate for the associated damped wave equation. This talk is based on a joint work with Nicolas Burq and another with Cyril Letrouit.

analysis of PDEsoptimization and control

Audience: learners

Control in Times of Crisis. Online seminar.

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