Exact observability properties of subelliptic wave and Schr{\"o}dinger equations

TL;DR

This survey explores recent advances in the exact observability and controllability of subelliptic wave and Schrödinger equations, highlighting how propagation slows down in certain directions and employing advanced mathematical tools.

Contribution

It compiles and discusses recent results on the observability of subelliptic equations, integrating geometric, spectral, and harmonic analysis methods.

Findings

Propagation slowdown in transverse directions

Connections between geometry and controllability

Use of semi-classical and spectral analysis techniques

Abstract

In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{\"o}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. The proofs combine sub-Riemannian geometry, semi-classical analysis, spectral theory and non-commutative harmonic analysis.