Minimal time for heat observability on almost-Riemannian surfaces

Abstract

We determine the minimal observation time for the heat equation on a closed almost-Riemannian surface, for a class of observation sets leaving sufficiently thin collars around simple Baouendi-Grushin circles. The structure and density are analytic near these circles, and the coefficients may depend on both variables. The formula is expressed through a complex phase describing concentration near the singular set and extends the thresholds known for separated models. Regular unobserved regions away from the circles do not change this time, provided all other singularities are observed. For singular circles of constant bracket step greater than two, local analyticity implies failure of observability at every time for observation sets at positive distance. We also construct quasimodes concentrated near simple singular circles, with exponentially small errors. Their superpositions give a lower bound for the Schrödinger observation time.