Observability of Baouendi-Grushin-Type Equations Through Resolvent Estimates

TL;DR

This paper investigates the observability and controllability of Baouendi-Grushin-type equations, revealing how the degeneracy affects propagation speed and establishing resolvent estimates that determine conditions for observability in different regimes.

Contribution

It provides the first resolvent estimates for Baouendi-Grushin operators and characterizes observability conditions for related Schrödinger, heat, and wave equations based on the operator's degeneracy.

Findings

Resolvent estimate for $ ext{Baouendi-Grushin}$ operator $ riangle_ ext{γ}$

Conditions for observability depending on the ratio $( ext{γ}+1)/s$

Observability can hold in small or large time, or fail entirely, based on parameters

Abstract

In this article, we study the observability (or, equivalently, the controllability) of some subelliptic evolution equations depending on their step. This sheds light on the speed of propagation of these equations, notably in the "degenerated directions" of the subelliptic structure. First, for any $\gamma\geq 1$, we establish a resolvent estimate for the Baouendi-Grushin-type operator $\Delta_\gamma=\partial_x^2+|x|^{2\gamma}\partial_y^2$, which has step $\gamma+1$. We then derive consequences for the observability of the Schr\''odinger type equation $i\partial_tu-(-\Delta_\gamma)^{s}u=0$ where $s\in N$. We identify three different cases: depending on the value of the ratio $(\gamma+1)/s$, observability may hold in arbitrarily small time, or only for sufficiently large times, or even fail for any time. As a corollary of our resolvent estimate, we also obtain observability for heat-type equations $\partial_tu+(-\Delta_\gamma)^su=0$ and establish a decay rate for the damped wave equation associated with $\Delta_{\gamma}$.